Aero planetary gear mechanism dynamic fatigue reliability collaborative design method

Abstract

The present application relates to the technical field of mechanism model, in particular to a kind of aviation planetary gear mechanism dynamic fatigue reliability collaborative design method, comprising the following steps: obtaining the full working condition dynamic load history and tooth root bending stress data of target aviation planetary gear mechanism;Equivalent tooth root alternating bending stress data set is established;Gear overall fatigue life data is obtained;Conversion obtains gear unit probability life distribution;Determine the thin-walled inner gear ring rim flexibility coefficient and planetary carrier side plate flexibility coefficient.The present application can quantify the topology regulation law of the flexibility of thin-walled component to the dispersion of dynamic tooth root stress amplitude by integrating multi-body contact dynamics deformation mechanism and extreme value statistical theory into probability reliability evaluation, solves the defects that rigidity assumption error is large in traditional gear design, local fatigue life is insufficient and traditional empirical formula cannot consider light weight and high survival rate, and improves the forward timing research and development precision of flexible planetary transmission system.

Description

Technical Field

[0001] This invention relates to the field of mechanism modeling technology, and in particular to a collaborative design method for dynamic fatigue reliability of aerospace planetary gear mechanisms. Background Technology

[0002] Heavy-duty high-speed planetary gear systems in aviation (such as the planetary gear units of helicopter main gearboxes) are key components for high-speed, heavy-load power transmission, and their operational reliability directly determines the aircraft's availability and flight safety. To achieve the ultimate power-to-weight ratio and lightweight design, the internal gear rings and planetary carriers in aviation planetary gear systems are typically designed as flexible structures with significant thin-walled characteristics. During actual service, these large-sized, thin-walled load-bearing components undergo non-negligible macroscopic elastic deformation.

[0003] Existing technologies for gear transmission structure design and life prediction often rely on rigidity assumptions or traditional international standards (such as ISO 6336) empirical formulas. These traditional methods typically simplify gears as rigid disks and reduce multi-tooth meshing processes to single-point contact by integrating meshing stiffness. This makes it difficult to accurately simulate the coupling interaction mechanism between the macroscopic flexibility of thin-walled components and the microscopic contact behavior of gears, and even more difficult to quantify the nonlinear control law of structural flexible deformation on the mean and dispersion of dynamic tooth root bending stress. Meanwhile, existing reliability optimization design (RBDO) methods face problems such as excessive computational costs and insufficient system-level correlation failure modeling when dealing with complex large gear transmission devices. Due to the lack of accurate feedback on the flexible characteristic parameters of thin-walled components, existing designs often result in excessive structural redundancy (overweight) or insufficient local fatigue life, making it difficult to achieve a balance between lightweight design and high survivability. Summary of the Invention

[0004] The purpose of this invention is to overcome the shortcomings of existing technologies and propose a collaborative design method for dynamic fatigue reliability of aerospace planetary gear mechanisms.

[0005] To achieve the above objectives, the present invention adopts the following technical solution: a collaborative design method for dynamic fatigue reliability of aerospace planetary gear mechanisms, comprising the following steps: Acquire the dynamic load history and tooth root bending stress data of the target aerospace planetary gear mechanism under all working conditions; Establish an equivalent tooth root alternating bending stress dataset; Obtain overall fatigue life data for gears; The conversion yields the probability lifetime distribution of the gear tooth element; Determine the flexibility coefficient of the thin-walled internal gear ring rim and the flexibility coefficient of the planetary carrier side plate; Construct a system-level dynamic fatigue reliability assessment model; Construct a Kriging reliability proxy model; Solve and select the optimal combination of structural parameters.

[0006] Preferably, the steps for obtaining the full-condition dynamic load history and tooth root bending stress data of the target aerospace planetary gear mechanism are as follows: Select the target aerospace planetary gear mechanism and obtain its geometric parameters, transmission component parameters, meshing pair parameters, support boundary parameters, input torque data, and input speed data. The thin-walled internal gear ring and the planet carrier side plate in the target aerospace planetary gear mechanism are discretized in three dimensions using finite element method to obtain the internal gear ring flexible body and the planet carrier flexible body. The internal gear ring flexible body and the planet carrier flexible body are condensed using the free interface modal synthesis method to obtain the internal gear ring condensed flexible body and the planet carrier condensed flexible body. The internal gear ring condensed flexible body, the planet carrier condensed flexible body, the sun gear rigid body, the planet gear rigid body and the bearing six-degree-of-freedom stiffness matrix are coupled to establish a global three-dimensional dynamic rigid-flexible coupling model. Based on the global three-dimensional dynamic rigid-flexible coupling model and nonlinear multibody contact dynamics, the edge contact, stress concentration and dynamic load of the gear pair of the target aerospace planetary gear mechanism in the dynamic alternating meshing process are solved to obtain the dynamic load history under all working conditions. The dynamic load history under all working conditions is discretized into multiple equivalent impact loads along the tooth surface contact line. The multiple equivalent impact loads are applied sequentially to the corresponding contact lines, and the tooth root bending stress time history of the critical section is extracted to obtain the tooth root bending stress data.

[0007] Preferably, the steps for establishing an equivalent tooth root alternating bending stress dataset are as follows: The input torque data, which follows a normal distribution, is obtained and then discretized into multiple typical load levels. The multiple typical load levels are respectively input into the global three-dimensional dynamic rigid-flexible coupling model to obtain the transient tooth root bending stress time history corresponding to each typical load level. Rainflow counting is performed on the transient tooth root bending stress time history to obtain several sets of stress cycle parameters, including stress amplitude, average stress, and number of cycles; Goodman average stress correction is applied to the stress amplitude and average stress in the stress cycle parameters to obtain the corrected stress cycle. Based on Miner's linear cumulative damage theory and the median SN curve under the stress ratio of -1, the modified stress cycle is equivalently converted into a constant amplitude equivalent bending stress, which satisfies the following: ; In the formula, This represents the constant amplitude equivalent bending stress. This indicates the stress level of the modified stress cycle. Indicates the stress level number. Indicates the first The number of cycles corresponding to each stress level Indicates the first The Goodman corrected stress amplitude corresponding to each stress level The slope exponent of the median SN curve; The mean and standard deviation of the constant amplitude equivalent bending stress are statistically analyzed to establish an equivalent tooth root alternating bending stress dataset that conforms to a normal distribution.

[0008] Preferably, the steps for obtaining overall fatigue life data of gears are as follows: Based on the gear tooth structure parameters of the target aerospace planetary gear mechanism, a corresponding main test gear is designed. The gear to be evaluated includes at least one of a sun gear, a planet gear, and an internal gear ring, so that the material, heat treatment process, tooth profile parameters, and tooth root stress state of each main test gear are matched with the corresponding gear tooth structure parameters of the gear to be evaluated. Each of the main test gears was installed on a power flow closed gear rotation tester, and bending fatigue accelerated life test was carried out under preset oil bath lubrication temperature and multiple preset stress levels. The failure criterion was to use the appearance of a visible crack at the root of the first tooth in each of the main test gears and its development into fatigue fracture. Vibration signals were collected by an accelerometer placed near the corresponding bearing housing. The power flow closed gear rotation testing machine is stopped based on the vibration signal. The cumulative number of rotations of each main test gear is read and used as the overall fatigue life data of the corresponding gear to be evaluated.

[0009] Preferably, the specific steps for converting the gear tooth element probability lifetime distribution are as follows: The overall fatigue life data of the gear was fitted using a two-parameter Weibull distribution to obtain the overall gear life distribution; Using the minimum order statistic theory, the overall gear life distribution is transformed into a single gear tooth unit life distribution. The probability density function of the extreme value distribution transformation model is: ; In the formula, The probability density function representing the overall lifespan distribution of the gear. Represents a random variable representing lifespan. This indicates the number of teeth on the main test gear. The cumulative distribution function represents the population probability lifetime distribution of a single gear tooth element. The probability density function represents the population distribution of the probability lifetime of a single gear tooth element; When the probability lifetime distribution of a single gear tooth unit follows a two-parameter Weibull distribution, the shape and scale parameters of the overall gear lifetime distribution and the probability lifetime distribution of a single gear tooth unit satisfy the following: ; The parameters of the probability life distribution of the single gear tooth unit, calculated from the overall gear life distribution, are as follows: ; In the formula, Shape parameters representing the overall life distribution of the gear. A scale parameter representing the overall life distribution of the gear. The shape parameter represents the probability lifetime distribution of a single gear tooth element. The scale parameter representing the probability lifetime distribution of a single gear tooth element; The maximum likelihood estimation method using random truncated data is employed to analyze the truncated life information of the unbroken teeth in the main test gear, and the shape and scale parameters of the probability life distribution of the single gear tooth unit are corrected and estimated. The joint maximum likelihood estimation function is: ; In the formula, Denotes the joint maximum likelihood estimation function. This represents the total observed sample size. Indicates the gear tooth unit number. Indicates the first The number of test cycles for each gear tooth unit Represents a state indicator variable. This represents the probability density function value of the probability lifetime distribution of a single gear tooth unit at the number of test cycles. This represents the cumulative distribution function value of the probability lifetime distribution of a single gear tooth unit at the number of test cycles. When the gear tooth unit fails due to bending fatigue fracture, the state indicator variable is set to 1. When the gear tooth unit does not fracture and the test stops with the whole machine, the state indicator variable is set to 0. The gear tooth PSN curve is fitted according to the corrected estimated single gear tooth unit life distribution to obtain the gear tooth unit probability life distribution.

[0010] Preferably, the steps for determining the flexibility coefficient of the thin-walled internal gear ring rim and the flexibility coefficient of the planetary carrier side plate are as follows: Extract the outer radius of the inner gear ring rim, the root radius of the inner gear ring, the thickness of the planetary carrier side plate, and the radius of the planetary carrier side plate from the geometric parameters of the target aerospace planetary gear mechanism; The flexibility coefficient of the thin-walled internal gear ring, characterized by the relative thickness of the rim, is calculated based on the outer radius of the rim and the root radius of the internal gear ring. The formula is as follows: ; In the formula, This represents the rim flexibility coefficient of a thin-walled internal gear ring. This indicates the outer radius of the rim of the internal gear ring. Indicates the radius of the root circle of the internal gear ring; The planetary carrier side plate flexibility coefficient, characterized by the relative thickness of the side plates, is calculated based on the thickness and radius of the planetary carrier side plate. The formula is as follows: ; In the formula, This indicates the flexibility coefficient of the planetary carrier side plate. Indicates the thickness of the planetary carrier side plates. Indicates the radius of the planetary carrier side plate circle; The range of the flexibility coefficient of the thin-walled internal gear ring rim is set to 0.01 to 0.05, and the range of the flexibility coefficient of the planetary carrier side plate is set to 0.02 to 0.06, thus obtaining the flexible structure design parameter space.

[0011] Preferably, the specific steps for constructing a system-level dynamic fatigue reliability assessment model are as follows: Each gear tooth unit in the target aerospace planetary gear mechanism is taken as a potential failure functional unit. Based on the probability lifetime distribution of the gear tooth unit and the equivalent tooth root alternating bending stress dataset, the lifetime probability density function of each gear tooth unit under random load conditions is obtained. The stress intensity interference model is extended based on the conditional expectation algorithm, and the inter-tooth load correlation and alternating meshing timing are incorporated into the stress intensity interference model to obtain the system survival probability model, as shown in the formula: ; In the formula, This represents the system survival probability under the target runtime. Indicates the target runtime. Represents a standard normal random variable. This represents the probability density function of the standard normal distribution. This indicates the total number of gear tooth functional units within the system. Indicates the gear tooth functional unit number. Represents the lifespan integral variable. Indicates the first The cumulative number of constant amplitude alternating loads applied to each gear tooth functional unit within the target operating time. Indicates the first The lifetime probability density function of each gear tooth functional unit under random load conditions Indicates the first The average value of the equivalent stress of discrete loads corresponding to each gear tooth functional unit. Indicates the first The standard deviation of the discrete load equivalent stress corresponding to each gear tooth functional unit; The cumulative number of constant-amplitude alternating load applications on representative teeth of each meshing pair within a specified operating time is calculated based on the absolute angular velocities of the sun gear, planet gears, internal gear ring, and planet carrier. This cumulative number of constant-amplitude alternating load applications is then used as a factor in the system survival probability model. A system-level dynamic fatigue reliability assessment model was obtained.

[0012] Preferably, the steps for constructing the Kriging reliability proxy model are as follows: Multiple sample points are selected in the flexible structure design parameter space, and the flexible coefficient of the thin-walled internal gear ring rim and the flexible coefficient of the planetary carrier side plate corresponding to each sample point are combined to form a multidimensional flexible structure design variable space vector. The system-level dynamic fatigue reliability assessment model is called for each sample point to obtain the corresponding system fatigue reliability response value, and the training sample set is composed of the multiple sample points and the system fatigue reliability response value. When constructing the training sample set, the mean of the equivalent tooth root alternating bending stress dataset is kept constant, so that the stress amplitude characteristics of the equivalent tooth root alternating bending stress dataset exhibit a gradient distribution characteristic with increasing dispersion as the flexibility coefficient of the thin-walled inner gear ring rim and the flexibility coefficient of the planetary carrier side plate decreases. The Kriging model is trained using regression basis functions and correlation function hyperparameters. The fatigue reliability of the system with unknown design is represented as a linear weighted combination of known sample responses, and the prediction expression is: ; In the formula, This represents the reliability prediction value at the space vector of design variables for a multidimensional flexible structure. Represents the space vector of design variables for multidimensional flexible structures. Represents the total number of observed samples. Indicates the observation sample number, Indicates the first The spatial correlation weighting coefficients of each observed sample relative to the spatial vector of the multidimensional flexible structure design variables. Indicates the first The system fatigue reliability response value corresponding to each observation sample.

[0013] Preferably, the steps for solving and selecting the optimal combination of structural parameters are as follows: Using the Kriging reliability surrogate model as the fitness calculation model, a non-dominated sorting genetic algorithm is used to perform a co-evolutionary search on the rim flexibility coefficient of the thin-walled internal gear ring and the side plate flexibility coefficient of the planetary carrier, so as to obtain the Pareto non-dominated solution front of system quality and system reliability. The quality objective function, reliability objective function, weighted composite objective function, and constraints of the co-evolutionary search are as follows: ; ; ; In the formula, This represents the rim flexibility coefficient of a thin-walled internal gear ring. This indicates the flexibility coefficient of the planetary carrier side plate. This represents the system quality normalization objective function. This represents the minimized objective function formed by transforming the objective function for maximizing system reliability with a negative sign. This represents the weighted composite objective function. This represents the total structural mass of the planetary transmission system. This represents the maximum system quality among the candidate solutions. This represents the system reliability at the target runtime predicted by the Kriging reliability surrogate model. Indicates the target runtime. This represents the weighting coefficient of the quality target. This represents the target weighting coefficient for system reliability; The candidate structural parameter combinations in the Pareto non-dominated solution front are sorted by TOPSIS approximation of the ideal solution, and the candidate structural parameter combination with the highest relative closeness is taken as the optimal structural parameter combination and output.

[0014] Compared with the prior art, the advantages and positive effects of the present invention are as follows: This invention integrates the multibody contact dynamic deformation mechanism and extreme value statistical theory into probabilistic reliability assessment, which can quantify the topological control law of the dynamic tooth root stress amplitude dispersion of thin-walled component flexibility. It effectively solves the defects of large rigidity assumption error, insufficient local fatigue life and the inability of traditional empirical formulas to take into account both lightweight and high survival rate in traditional gear design, and improves the forward timing development accuracy of flexible planetary transmission system. Attached Figure Description

[0015] Figure 1 This is a general configuration diagram of the target aerospace planetary gear mechanism in an embodiment of the present invention; Figure 2 The finite element model diagram of the planetary carrier and gear ring is shown. Figure 3A schematic diagram showing the flexibility characteristics of the gear ring rim and the planetary carrier side plate; Figure 4 The diagram shows the quasi-static elastic deformation and radial, axial, and tangential displacement distribution of the gear ring. Figure 5 Diagram showing the quasi-static elastic deformation of the planet carrier and the tilting state of the planetary pins; Figure 6 A flowchart for rainflow counting, Goodman correction, and equivalent constant amplitude stress conversion of dynamic tooth root stress; Figure 7 A comparison chart of equivalent stress distribution and stiffness levels at the tooth root under random input torque. Figure 8 A schematic diagram of a gear prototype design for reliability assessment of a target aerospace planetary gear mechanism; Figure 9 This is a response surface plot for dynamic fatigue reliability assessment fitted based on the Kriging reliability surrogate model. Detailed Implementation

[0016] To make the objectives, technical solutions, and advantages of this invention clearer, the invention will be further described in detail below with reference to the accompanying drawings and embodiments. It should be understood that the specific embodiments described herein are merely illustrative and not intended to limit the invention.

[0017] Please see Figure 1-9 This invention provides a technical solution: a collaborative design method for dynamic fatigue reliability of aerospace planetary gear mechanisms, comprising the following steps: Acquire the dynamic load history and tooth root bending stress data of the target aerospace planetary gear mechanism under all working conditions; Establish an equivalent tooth root alternating bending stress dataset; Obtain overall fatigue life data for gears; The conversion yields the probability lifetime distribution of the gear tooth element; Determine the flexibility coefficient of the thin-walled internal gear ring rim and the flexibility coefficient of the planetary carrier side plate; Construct a system-level dynamic fatigue reliability assessment model; Construct a Kriging reliability proxy model; Solve and select the optimal combination of structural parameters.

[0018] In this embodiment, the target aerospace planetary gear mechanism can be an NGW-type aerospace planetary gear mechanism, which includes a sun gear, multiple planet gears, a thin-walled internal gear ring, a planet carrier, planet pins, bearings, and a support housing. The sun gear serves as the input component, the planet carrier as the output component, and the thin-walled internal gear ring can be fixed to the housing by bolts or other fixing structures. Both the thin-walled internal gear ring and the planet carrier side plates serve as flexible load-bearing components in the dynamic modeling. In implementation, a dynamic model is first established based on the geometric parameters, material parameters, transmission ratio, input torque, input speed, and support boundary of the target aerospace planetary gear mechanism. Then, the dynamic stress input at the tooth root and the tooth life strength input are obtained through simulation and experiment, respectively. Subsequently, the fatigue life distribution of the tooth unit is established using probabilistic statistical methods. Next, the fatigue life distribution of the tooth unit, load randomness, and meshing sequence are introduced into the system-level dynamic fatigue reliability assessment model. Finally, the Kriging reliability proxy model is used to reduce the optimization calculation cost, and the optimal combination of structural parameters is output through multi-objective optimization and multi-criteria decision-making.

[0019] In one specific embodiment, the target aerospace planetary gear mechanism can be selected from the high-speed, heavy-load flexible planetary gear system in the main reducer of a certain type of civil transport helicopter. This planetary gear system includes one sun gear, eight planet gears, and one internal ring gear. The sun gear is the power input, and the planet gears are evenly distributed on the planet carrier via bearings and planetary pins. The planet carrier serves as the power output unit. Since cruise operation accounts for a high proportion of flight time, it can be used as the primary operating condition for reliability assessment and structural optimization. The input torque can be described using a normal distribution, and the input speed can be set to a constant speed. The aforementioned operating condition data is used for acquiring the dynamic load history under all operating conditions, statistical analysis of equivalent tooth root alternating bending stress, and reliability optimization calculations.

[0020] The specific steps for obtaining the full-condition dynamic load history and tooth root bending stress data of the target aerospace planetary gear mechanism are as follows: Select the target aerospace planetary gear mechanism and obtain its geometric parameters, transmission component parameters, meshing pair parameters, support boundary parameters, input torque data, and input speed data. Three-dimensional finite element discretization was performed on the thin-walled internal gear ring and planet carrier side plate in the target aerospace planetary gear mechanism to obtain the internal gear ring flexible body and the planet carrier flexible body. The free interface modal synthesis method was used to condense the internal gear ring flexible body and the planet carrier flexible body to obtain the internal gear ring condensed flexible body and the planet carrier condensed flexible body. A global three-dimensional dynamic rigid-flexible coupling model is established by coupling the internal gear ring condensed flexible body, the planet carrier condensed flexible body, the sun gear rigid body, the planet gear rigid body and the bearing six-degree-of-freedom stiffness matrix. Based on the global three-dimensional dynamic rigid-flexible coupling model and nonlinear multibody contact dynamics, the edge contact, stress concentration and dynamic load of the gear pair of the target aerospace planetary gear mechanism in the dynamic alternating meshing process are solved to obtain the dynamic load history under all working conditions. The dynamic load history under all working conditions is discretized into multiple equivalent impact loads along the tooth surface contact line. These multiple equivalent impact loads are then applied sequentially to the corresponding contact lines. The time history of tooth root bending stress at the critical section is extracted to obtain tooth root bending stress data.

[0021] In this embodiment, the geometric parameters include the number of teeth, module, pressure angle, helix angle, tooth width, base circle diameter, root circle radius, tip circle radius, and root fillet radius of the sun gear, planet gears, and internal gear ring; the transmission component parameters include the material's elastic modulus, Poisson's ratio, density, mass, moment of inertia, and component connection relationships; the meshing pair parameters include meshing stiffness, contact line position, tooth surface error, installation error, profile modification parameters, and meshing phase difference; and the support boundary parameters include bearing stiffness, housing constraints, bolt hole constraints, and planetary pin support stiffness.

[0022] When performing three-dimensional finite element discretization on the thin-walled internal gear ring and planetary carrier side plate, ten-node tetrahedral or hexahedral elements can be used to mesh the thin-walled internal gear ring, planetary carrier side plate, and planetary pin regions, extracting the mass matrix, stiffness matrix, and modal information of the finite element model. Since system-level dynamic analysis focuses on the overall structural flexibility, inertia, and modal response, local stress concentration areas can be refined in the subsequent tooth root stress extraction stage. When using the free interface modal synthesis method for condensation, connection interface nodes, load application nodes, bearing connection nodes, and key modes are retained, while internal degrees of freedom with little impact on the system's dynamic response are deleted, thereby reducing solution costs while ensuring computational accuracy.

[0023] When establishing a global three-dimensional dynamic rigid-flexible coupling model, the sun gear and planet gears can be modeled as rigid bodies, while the thin-walled internal gear ring and planet carrier side plates are modeled as condensed flexible bodies. The bearings are represented by a six-degree-of-freedom stiffness matrix to represent radial, axial, tilting, and torsional support characteristics. In this model, the planet carrier side plates are not repeatedly modeled as independent planet carrier rigid bodies, but rather their macroscopic elastic deformation and modal response are represented by the condensed flexible bodies of the planet carrier. Tooth surface contact can be considered as elastic contact distributed along the contact line, and the contact line is discretized into multiple micro-elements, each of which is assigned contact stiffness and normal clearance. The dynamic load, edge contact, and stress concentration during meshing are calculated using nonlinear multibody contact dynamics. For multiple meshing pairs in a planetary gear mechanism, the meshing phase, meshing frequency, and load transmission path of the sun gear-planet gear meshing pair and the planet gear-internal gear ring meshing pair need to be considered separately.

[0024] When the dynamic load history under all operating conditions is discretized along the tooth contact line into multiple equivalent impact loads, the dynamic meshing force time history within a complete meshing cycle can be discretized into multiple load steps. Each load step is applied sequentially to the corresponding contact line in the order of engagement to disengagement. Subsequently, the tooth root bending stress time history is extracted at the critical section of the tooth. The critical section can be determined based on the tooth root transition curve, the location of the maximum principal stress, or the location of the maximum tooth root bending stress. The resulting tooth root bending stress data serves as the basis for subsequent rainflow counting, Goodman correction, and equivalent constant amplitude stress conversion.

[0025] The specific steps for establishing an equivalent tooth root alternating bending stress dataset are as follows: Obtain input torque data that follows a normal distribution, and discretize the input torque data into multiple typical load levels; By inputting multiple typical load levels into the global three-dimensional dynamic rigid-flexible coupling model, the transient tooth root bending stress time history corresponding to each typical load level is obtained. Rainflow counting was performed on the time history of transient tooth root bending stress to obtain several sets of stress cycle parameters, including stress amplitude, average stress, and number of cycles. Goodman average stress correction was applied to the stress amplitude and average stress in the stress cycle parameters to obtain the corrected stress cycle. Based on Miner's linear cumulative damage theory and the median SN curve under the stress ratio of -1, the modified stress cycle is equivalently transformed into constant amplitude equivalent bending stress, which satisfies the following: ; In the formula, This represents the constant amplitude equivalent bending stress. This indicates the stress level of the modified stress cycle. Indicates the stress level number. Indicates the first The number of cycles corresponding to each stress level Indicates the first The Goodman corrected stress amplitude corresponding to each stress level The slope exponent of the median SN curve; The mean and standard deviation of the constant amplitude equivalent bending stress are statistically analyzed to establish an equivalent tooth root alternating bending stress dataset that conforms to a normal distribution.

[0026] In this embodiment, the input torque data can be derived from flight load spectra, bench loading spectra, engine output torque monitoring data, or design mission profiles. If the input torque follows a normal distribution, it can be discretized into multiple typical load levels using equal probability discretization, three-sigma interval discretization, or Gaussian integral sampling methods. For example, several typical load points can be selected near the mean of the input torque, and each load point can cover low-load, medium-load, and high-load regions.

[0027] For each typical load level, the load is input into a global three-dimensional dynamic rigid-flexible coupling model to calculate the transient tooth root bending stress time history of the target aerospace planetary gear mechanism under the corresponding torque condition. Since tooth root bending stress is typically a non-constant amplitude alternating load, it cannot be directly used for fatigue life distribution calculations; therefore, stress cycles need to be extracted using the rainflow counting method. After rainflow counting, the stress amplitude, mean stress, and number of cycles for each stress cycle level are obtained. The Goodman mean stress correction targets the stress amplitude and mean stress in the aforementioned stress cycle parameters, not the rainflow counting itself or the alternating cycle counting itself. Subsequently, the Goodman mean stress correction method is used to uniformly convert the stress cycles under different mean stress conditions into corrected stress cycles under equivalent stress ratio conditions.

[0028] After applying the Goodman mean stress correction, the variable amplitude load is converted into a constant amplitude equivalent bending stress equivalent to its fatigue damage, based on Miner's linear cumulative damage theory. In the above formula, the numerator... The denominator represents the weighted cumulative contribution of all modified stress cycles to fatigue damage. Indicates the total number of loops, overall open The power of the formula yields the constant-amplitude equivalent bending stress with the same fatigue damage effect. The dimension of this formula is stress, and both sides of the formula have consistent dimensions, making it suitable for subsequent life distribution and reliability assessments.

[0029] After calculating the equivalent bending stress under multiple typical load levels, statistical analysis was performed on the obtained samples to obtain the mean and standard deviation of the equivalent tooth root alternating bending stress. If the normality test or engineering experience determines that it meets the normal distribution, a dataset of equivalent tooth root alternating bending stress conforming to the normal distribution is established. This dataset can correspond to the sun gear teeth, planet gear teeth, and internal gear ring teeth respectively, or multiple sub-datasets can be established according to different meshing pairs and different tooth positions.

[0030] The specific steps for obtaining overall fatigue life data of gears are as follows: Based on the gear tooth structure parameters of the target aerospace planetary gear mechanism, the corresponding main test gear is designed. The gear to be evaluated includes at least one of the sun gear, planet gear and internal gear ring, so that the material, heat treatment process, tooth profile parameters and tooth root stress state of each main test gear are matched with the corresponding gear tooth structure parameters of the gear to be evaluated. Each main test gear was installed on a power flow closed gear rotary testing machine, and bending fatigue accelerated life test was carried out under preset oil bath lubrication temperature and multiple preset stress levels. The failure criterion was to use the appearance of a visible crack at the root of the first tooth in each main test gear and its development into fatigue fracture. Vibration signals were collected by an accelerometer placed near the corresponding bearing housing. The power flow closed gear rotation testing machine is stopped based on the vibration signal. The cumulative number of cycles of each main test gear is read and used as the overall fatigue life data of the corresponding gear to be evaluated.

[0031] In this embodiment, the gear to be evaluated can be one or more of the following: sun gear, planet gear, and internal gear ring. When the sun gear, planet gear, and internal gear ring in the target aerospace planetary gear mechanism are made of the same material, have the same or similar heat treatment process, and the bending fatigue resistance of the gear teeth has a high degree of consistency, the gear with stronger load representativeness can be selected as the main test gear design object; when the materials, processes, or tooth root stress states of the three types of gears differ greatly, corresponding main test gears can be designed for the sun gear, planet gear, and internal gear ring respectively.

[0032] The power-flow enclosed gear rotation testing machine can simulate the meshing, lubrication, and time-varying stress environment of gears during actual service, making it more suitable for obtaining overall gear fatigue life data than simple pulsating bending fatigue testing. During testing, oil bath lubrication can be used, with the oil temperature controlled and stabilized within a preset range, such as 70±2℃. Auxiliary components such as shafts, bearings, and mating gears are designed with a lifespan exceeding that of the main test gear to ensure that the main test gear preferentially experiences bending fatigue failure during the test.

[0033] The design of the main test gear should ensure that its material, heat treatment process, tooth profile parameters, and tooth root stress state match the corresponding gear to be evaluated in the target aerospace planetary gear mechanism. Specifically, parameters such as module, pressure angle, tooth root fillet radius, material grade, surface hardness, core hardness, carburized layer depth, and grinding process can be kept consistent or equivalent. Only within the feasible range of the experiment can the tooth width, tooth root height, or center distance be appropriately adjusted to induce controllable tooth root bending fatigue failure in the main test gear.

[0034] During the test, vibration signals near the bearing housing can be collected using an accelerometer. These signals, combined with time-domain thresholds, frequency-domain characteristics, and impact response characteristics, are used to determine whether crack propagation or tooth breakage occurs in the gear teeth. When the first tooth develops a visible crack at its root and progresses to fatigue fracture, the vibration signal triggers the testing machine to stop. The cumulative number of revolutions of the main test gear at this point is recorded and used as the overall fatigue life data for that main test gear. Since the overall gear failure is determined by the first tooth to fail, this data essentially corresponds to the smallest order statistic in the tooth life sample.

[0035] The specific steps for converting the gear tooth element probability lifetime distribution are as follows: The overall fatigue life data of the gears were fitted using a two-parameter Weibull distribution to obtain the overall gear life distribution; Using the minimum order statistic theory, the overall gear life distribution is transformed into the life distribution of a single gear tooth unit. The probability density function of the extreme value distribution transformation model is: ; In the formula, The probability density function representing the overall lifespan distribution of the gear. Represents a random variable representing lifespan. This indicates the number of teeth on the main test gear. The cumulative distribution function represents the population probability lifetime distribution of a single gear tooth element. The probability density function represents the population distribution of the probability lifetime of a single gear tooth element; When the probability lifetime distribution of a single gear tooth element follows a two-parameter Weibull distribution, the shape and scale parameters of the overall gear lifetime distribution and the probability lifetime distribution of a single gear tooth element satisfy the following: ; The parameters for the probabilistic life distribution of a single gear tooth element, obtained by back-calculating from the overall gear life distribution, are as follows: ; In the formula, Shape parameters representing the overall life distribution of the gear. A scale parameter representing the overall life distribution of the gear. The shape parameter represents the probability lifetime distribution of a single gear tooth element. The scale parameter representing the probability lifetime distribution of a single gear tooth element; The maximum likelihood estimation method using random truncated data is employed to analyze the truncated life information of the unbroken teeth in the main test gear. The shape and scale parameters of the probability life distribution of a single gear tooth element are then corrected and estimated. The joint maximum likelihood estimation function is: ; In the formula, Denotes the joint maximum likelihood estimation function. This represents the total observed sample size. Indicates the gear tooth unit number. Indicates the first The number of test cycles for each gear tooth unit Represents a state indicator variable. This represents the probability density function value of the probability lifetime distribution of a single gear tooth element at the number of test cycles. This represents the cumulative distribution function value of the probability lifetime distribution of a single gear tooth element at the number of test cycles. When the gear tooth unit fails due to bending fatigue fracture, the state indicator variable is set to 1. When the gear tooth unit does not fracture and the test stops with the whole machine, the state indicator variable is set to 0. The gear tooth PSN curve is fitted according to the corrected estimated single gear tooth unit life distribution to obtain the gear tooth unit probability life distribution.

[0036] In this embodiment, the overall fatigue life data of the gear can be grouped according to different stress levels. Under each stress level, a two-parameter Weibull distribution is used to fit the overall life sample of the main test gear to obtain the overall gear life distribution. Since the failure of the entire gear is determined when any tooth of the main test gear breaks first, the overall gear life is equivalent to the minimum value among the lifespans of all teeth of the gear. Let the main test gear have a total of If the lifespan of each gear tooth unit follows the same parent distribution, then the overall lifespan distribution of the gear is... The minimum order statistic distribution of the lifetime samples of each gear tooth unit.

[0037] In the above extreme value distribution transformation model, This indicates that the lifespan of a single gear tooth unit is less than or equal to The probability, Indicates the rest The lifespan of each gear unit is greater than The probability, Indicates any gear tooth unit in The density contribution of nearby failures is considered. Therefore, this model can reasonably describe the statistical law that "failure of the first tooth leads to overall gear failure." The integral result of this formula satisfies the probability density function normalization requirement and can be used to infer the life distribution of a single tooth unit from the overall gear life distribution.

[0038] When the population distribution of the probability life of a single gear tooth unit follows a two-parameter Weibull distribution, the distribution of the minimum order statistic also follows a two-parameter Weibull distribution. Its shape parameter remains unchanged, while its scale parameter varies with the number of teeth. Changes. Therefore, if the overall gear life distribution parameters are obtained from experimental fitting... and You can pass and The shape and scale parameters of the probabilistic life distribution of a single gear tooth element are calculated in reverse. This mapping relationship can eliminate the statistical influence of the number of teeth on the overall life dispersion of the gear, making the tooth life data of gears with different numbers of teeth comparable.

[0039] Simultaneously, at each gear fatigue test shutdown, in addition to the failure life data provided by the first fractured tooth, the other unfractured teeth also provide randomly truncated life information. For example, when the first tooth of a main test gear with 25 teeth fractures, one failure data point and 24 truncated data points can be obtained. Using the maximum likelihood estimation method with randomly truncated data, the life information of the unfractured teeth can be fully utilized to correct and estimate the probability life distribution parameters of a single tooth element, improving the reliability of the tooth probability life distribution. Subsequently, the tooth element life distributions are obtained under multiple stress levels, and the life points under the same reliability or the same failure probability are connected to fit the tooth PSN curve, ultimately obtaining the tooth element probability life distribution.

[0040] In one specific embodiment, the main test gear has 25 teeth, and bending fatigue life tests are conducted at three stress levels: 818 MPa, 766 MPa, and 692 MPa. Since the first tooth of each main test gear fails due to fatigue, each main test gear generates one tooth life failure data point and 24 tooth life truncation data points. The number of main test gears, the number of tooth life failure data points, and the number of tooth life truncation data points at the three stress levels are shown in Table 1.

[0041] Table 1. Statistics on gear tooth life failure and truncation data

[0042] As shown in Table 1, under all three stress levels, the number of truncated tooth life data points is significantly greater than the number of failure data points, providing a larger sample base for maximum likelihood estimation of randomly truncated data. Comparing the tooth life distribution probability density curves obtained by this statistical method with those obtained by the minimum order statistic transformation method, the two sets of curves show good overlap, with mean differences of 0.8%–1.2% and standard deviation differences of 3%–5%, indicating that the minimum order statistic transformation method is effective and has low computational cost.

[0043] The specific steps for determining the flexibility coefficient of the thin-walled internal gear ring rim and the flexibility coefficient of the planetary carrier side plate are as follows: Extract the outer radius of the inner gear ring rim, the root radius of the inner gear ring, the thickness of the planetary carrier side plate, and the radius of the planetary carrier side plate from the geometric parameters of the target aerospace planetary gear mechanism. The flexibility coefficient of the thin-walled internal gear ring, characterized by its relative rim thickness, is calculated based on the outer radius and root radius of the internal gear ring rim. The formula is as follows: ; In the formula, This represents the rim flexibility coefficient of a thin-walled internal gear ring. This indicates the outer radius of the rim of the internal gear ring. Indicates the radius of the root circle of the internal gear ring; The flexibility coefficient of the planetary carrier side plates, characterized by the relative thickness of the side plates, is calculated based on the thickness and radius of the planetary carrier side plates. The formula is as follows: ; In the formula, This indicates the flexibility coefficient of the planetary carrier side plate. Indicates the thickness of the planetary carrier side plates. Indicates the radius of the planetary carrier side plate circle; The range of flexible coefficient values ​​for the thin-walled internal gear ring rim is set to 0.01 to 0.05, and the range of flexible coefficient values ​​for the planetary carrier side plate is set to 0.02 to 0.06, thus obtaining the design parameter space for the flexible structure.

[0044] In this embodiment, the rim flexibility coefficient of the thin-walled internal gear ring is... Used to characterize the relative thickness of the rim of an internal gear ring. The radial thickness of the internal gear ring rim is represented by, divided by This leads to the formation of a dimensionless flexibility characterization parameter. A smaller parameter indicates a relatively thinner internal gear ring rim and stronger structural flexibility; a larger parameter indicates a relatively thicker internal gear ring rim and higher structural stiffness. Since both the numerator and denominator of this formula are in the dimension of length, therefore... It is a dimensionless parameter, suitable for comparing the flexibility between internal gear rings of different sizes.

[0045] Planetary carrier side plate flexibility coefficient Used to characterize the relative thickness of the planetary carrier side plates. Indicates the thickness of the planetary carrier side plates. This represents the radius of the planetary carrier side plate circle, and the ratio between the two is a dimensionless parameter. The smaller this parameter, the thinner the planetary carrier side plate, making it more prone to tilting and deflection at the planetary pin supports; the larger this parameter, the thicker the planetary carrier side plate, and the higher the overall support stiffness of the planetary carrier. This can be achieved by simultaneously setting... and This allows the thickness of the internal gear ring rim and the planetary carrier side plate to be unified into a dimensionless design variable space, which facilitates subsequent Kriging surrogate model training and multi-objective optimization.

[0046] In practical implementation, the range of flexibility coefficient for the thin-walled internal gear ring rim is set to 0.01 to 0.05, and the range of flexibility coefficient for the planetary carrier side plate is set to 0.02 to 0.06. These ranges can be comprehensively determined based on aerospace planetary transmission design specifications, machining allowances, structural strength boundaries, assembly space, and resonance avoidance requirements. Within this parameter space, sample points can be selected using Latin hypercube sampling, uniform grid sampling, or optimal Latin sampling. Each sample point corresponds to a combination of a thin-walled internal gear ring rim flexibility coefficient and a planetary carrier side plate flexibility coefficient, thus corresponding to a flexible structure design scheme.

[0047] The specific steps for constructing a system-level dynamic fatigue reliability assessment model are as follows: Each gear tooth unit in the target aerospace planetary gear mechanism is taken as a potential failure functional unit. Based on the probability lifetime distribution of the gear tooth unit and the equivalent tooth root alternating bending stress dataset, the lifetime probability density function of each gear tooth unit under random load conditions is obtained. The stress intensity interference model is extended based on the conditional expectation algorithm, and the inter-tooth load correlation and alternating meshing timing are incorporated into the stress intensity interference model to obtain the system survival probability model, the formula of which is: ; In the formula, This represents the system survival probability under the target runtime. Indicates the target runtime. Represents a standard normal random variable. This represents the probability density function of the standard normal distribution. This indicates the total number of gear tooth functional units within the system. Indicates the gear tooth functional unit number. Represents the lifespan integral variable. Indicates the first The cumulative number of constant amplitude alternating loads applied to each gear tooth functional unit within the target operating time. Indicates the first The lifetime probability density function of each gear tooth functional unit under random load conditions Indicates the first The average value of the equivalent stress of discrete loads corresponding to each gear tooth functional unit. Indicates the first The standard deviation of the discrete load equivalent stress corresponding to each gear tooth functional unit; Based on the absolute angular velocities of the sun gear, planet gears, internal gear ring, and planet carrier, the cumulative number of constant-amplitude alternating load applications on representative teeth of each meshing pair within a specified operating time is calculated. This cumulative number of constant-amplitude alternating load applications is then used as a criterion in the system survival probability model. A system-level dynamic fatigue reliability assessment model was obtained.

[0048] In this embodiment, the root bending fatigue fracture of any tooth in the target aerospace planetary gear mechanism may lead to a decrease in gear transmission capacity, load redistribution, increased vibration and shock, and even system failure. Therefore, each tooth unit in the system can be regarded as a potentially failing functional unit, and the system approximately behaves as a series system in terms of fatigue reliability. If the load correlation is ignored, the system reliability may be simply represented as the product of the reliability of each tooth. However, in the planetary gear mechanism, the load borne by each tooth is affected by the input torque, manufacturing error, flexible deformation, and meshing phase. There is a statistical correlation between tooth failure events, so it is necessary to introduce a conditional expectation algorithm for extended modeling.

[0049] In the above system survival probability model, These are standardized normal random variables used to characterize the combined effect of the randomness of the system's input load on the equivalent stress of each gear tooth element. For a given... Once the life distribution conditions of each gear tooth unit are determined, the system conditional survival probability can be expressed as the product of the conditional survival probabilities of each gear tooth; then... Integrating over the full probability space yields the system survival probability considering load correlation. This model avoids simply treating tooth failure events as independent, thus more accurately reflecting the failure correlation of multi-tooth systems.

[0050] Furthermore, planetary gear mechanisms exhibit significant alternating meshing timing. The sun gear, planet gears, internal ring gear, and planet carrier have different absolute angular velocities, and the representative teeth on different gears experience different numbers of meshing cycles within the same target operating time. Therefore, it is necessary to calculate the cumulative number of constant-amplitude alternating loads applied to representative teeth in each meshing pair based on the absolute angular velocities of the sun gear, planet gears, internal ring gear, and planet carrier. For example, the sun gear teeth periodically mesh with multiple planet gears, the internal gear ring teeth mesh sequentially with the planet gears, and the planet gear teeth are simultaneously affected by both external and internal meshing. The number of cycles for different types of gear teeth needs to be calculated separately. As the lower limit of the lifetime integral, it means that the first... The probability that a gear tooth unit has not experienced fatigue failure within the target operating time.

[0051] The specific steps for constructing the Kriging reliability proxy model are as follows: Multiple sample points are selected in the flexible structure design parameter space, and the flexible coefficients of the thin-walled internal gear ring rim and the planetary carrier side plate corresponding to each sample point are combined to form a multidimensional flexible structure design variable space vector. For each sample point, the system-level dynamic fatigue reliability assessment model is invoked to obtain the corresponding system fatigue reliability response value, and the training sample set is composed of multiple sample points and system fatigue reliability response values. When constructing the training sample set, the mean of the equivalent tooth root alternating bending stress dataset is kept constant, so that the stress amplitude characteristics of the equivalent tooth root alternating bending stress dataset exhibit a gradient distribution characteristic with increasing dispersion as the flexibility coefficient of the thin-walled inner gear ring rim and the flexibility coefficient of the planetary carrier side plate change. The Kriging model is trained using regression basis functions and correlation function hyperparameters. The fatigue reliability of the system with unknown design is represented as a linear weighted combination of known sample responses, and the prediction expression is: ; In the formula, This represents the reliability prediction value at the space vector of design variables for a multidimensional flexible structure. Represents the space vector of design variables for multidimensional flexible structures. Represents the total number of observed samples. Indicates the observation sample number, Indicates the first The spatial correlation weighting coefficient of each observed sample relative to the space vector of multidimensional flexible structure design variables Indicates the first The system fatigue reliability response value corresponding to each observation sample.

[0052] In this embodiment, the space vector of design variables for the multidimensional flexible structure can be represented as: ,in The rim flexibility coefficient of the thin-walled internal gear ring. This represents the flexibility coefficient of the planetary carrier side plate. Since each sample point requires rigid-flexible coupling dynamic simulation, equivalent stress statistics, system reliability integration, and fatigue life probability calculation, repeatedly calling the high-fidelity evaluation model during the optimization process would lead to excessively high computational costs. Therefore, this embodiment constructs a Kriging reliability surrogate model to approximately express the mathematical mapping relationship between the flexible structure design variables and the system fatigue reliability response.

[0053] The training sample set can be obtained by selecting multiple sample points within the flexible structure design parameter space. For each sample point, based on the corresponding... and The rim thickness of the internal gear ring and the side plate thickness of the planetary carrier are updated, and the structural flexibility characteristics, equivalent stress statistics at the tooth root, and system-level dynamic fatigue reliability response values ​​are recalculated. Multiple sample points and their response values ​​together constitute the training sample set. To highlight the regulatory role of flexible elastic deformation on stress amplitude dispersion, the mean of the equivalent tooth root alternating bending stress dataset can be kept constant when constructing the training sample set. The variation in stress amplitude standard deviation caused by different flexible structural parameters is used as the main input feature, thereby isolating the pure dispersion topological contribution caused by flexible deformation.

[0054] The Kriging model includes regression and stochastic correlation terms. The regression basis function describes the global trend, while the correlation function hyperparameters describe the spatial correlation between sample points. For any unknown design point... Its reliability prediction value Based on known sample responses The weights are obtained by linear weighted combination. From unknown design points and the first The spatial distances and correlation functions between the observed sample points are determined. Therefore, when an unknown design point is close to a known sample point, that sample point contributes more to the predicted value; conversely, when an unknown design point is far from a known sample point, that sample point contributes less to the predicted value. In this way, a continuous reliability response surface can be obtained with a smaller number of high-fidelity samples.

[0055] The specific steps for solving and selecting the optimal combination of structural parameters are as follows: Using the Kriging reliability surrogate model as the fitness calculation model, a non-dominated sorting genetic algorithm was employed to perform a co-evolutionary search on the rim flexibility coefficient of the thin-walled internal gear ring and the side plate flexibility coefficient of the planetary carrier, thereby obtaining the Pareto non-dominated solution front for system quality and system reliability. The quality objective function, reliability objective function, weighted composite objective function, and constraints of the co-evolutionary search are as follows: ; ; In the formula, This represents the rim flexibility coefficient of a thin-walled internal gear ring. This indicates the flexibility coefficient of the planetary carrier side plate. This represents the system quality normalization objective function. This represents the minimized objective function formed by transforming the objective function for maximizing system reliability with a negative sign. This represents the weighted composite objective function. This represents the total structural mass of the planetary transmission system. This represents the maximum system mass among candidate combinations of structural parameters. This represents the system reliability at the target runtime predicted by the Kriging reliability surrogate model. Indicates the target runtime. This represents the weighting coefficient of the quality target. This represents the target weighting coefficient for system reliability; The candidate structural parameter combinations in the Pareto undominated solution front are ranked using TOPSIS to approximate the ideal solution, and the quality objective weighting coefficient is used. And system reliability target weighting coefficient A weighted normalized evaluation is performed on the candidate structural parameter combinations, where Based on the actual working condition index weights, the distance between each candidate structural parameter combination and the positive and negative ideal solutions is calculated, and the candidate structural parameter combination with the highest relative closeness is taken as the optimal structural parameter combination and output.

[0056] In this embodiment, there is often a trade-off between system mass and system fatigue reliability. Increasing the rim thickness of the thin-walled internal gear ring or the side plate thickness of the planetary carrier can improve structural stiffness and fatigue reliability to some extent, but it will increase the system mass. Decreasing the rim thickness of the thin-walled internal gear ring or the side plate thickness of the planetary carrier can reduce the system mass, but it may make the structure too flexible, leading to meshing misalignment, off-center loading, increased stress dispersion, and reduced fatigue reliability. Therefore, it is necessary to consider both mass and reliability objectives simultaneously through multi-objective optimization.

[0057] During the optimization process, a non-dominated sorting genetic algorithm is used. and A co-evolutionary search is performed. Each individual represents a combination of structural parameters. The fitness calculation model is the Kriging reliability surrogate model, and the system quality calculation model is also used. The system quality normalization objective function is... The reliability objective function is used to measure the lightweightness of candidate solutions. By negating the system reliability, the objective of maximizing system reliability is transformed into a minimization objective; weighted composite objective function. Through quality target weighting coefficients And system reliability target weighting coefficient It expresses the target preferences under different design conditions. Through selection, crossover, mutation, and non-dominated sorting, the candidate structural parameter combinations are updated generation by generation, and finally the Pareto non-dominated solution front of system quality and system reliability is obtained.

[0058] After obtaining the Pareto non-dominated solution front, the TOPSIS approximation ideal solution ranking method is further used to determine the final scheme. First, the quality objective and system reliability objective of the Pareto candidate schemes are normalized; second, the weight coefficients of the quality objective are set according to the weights of actual operating condition indicators. And system reliability target weighting coefficient ,in Next, the positive and negative ideal solutions are determined. The positive ideal solution corresponds to an ideal state with lower mass and higher system reliability, while the negative ideal solution corresponds to an unfavorable state with higher mass and lower system reliability. Finally, the distance between each candidate structural parameter combination and the positive and negative ideal solutions is calculated, and the relative closeness is also calculated. The candidate structural parameter combination with the highest relative closeness is the optimal structural parameter combination that balances lightweight design and high reliability.

[0059] In an optional embodiment, the initial population size of the non-dominated sorting genetic algorithm can be set to 300, the number of iterations can be set to 200, the crossover probability can be set to 0.8, and the mutation probability can be set to 0.05. The target running time can be determined based on the mission profile of the aerospace planetary gear mechanism, for example, using the continuous running time under cruise conditions as the reliability assessment time. In TOPSIS sorting, if the reliability objective is more important than the quality objective, a larger system reliability objective weight coefficient can be set. If a higher level of lightweighting is required, the weighting coefficient of the quality target can be increased. By adjusting the weights, design preferences for different aviation equipment mission scenarios can be met.

[0060] Furthermore, this embodiment may include the following more specific operational procedures during implementation: First, establish a basic parameter library for the target aerospace planetary gear mechanism. This library includes gear geometry parameters, transmission component parameters, material parameters, support boundary parameters, lubrication temperature, input torque data, input speed data, machining error parameters, and assembly error parameters. The basic parameter library can be constructed using design drawings, 3D models, experimental data, and operating condition spectra.

[0061] Second, a global three-dimensional dynamic rigid-flexible coupling model considering the flexibility of the thin-walled internal gear ring and the planetary carrier side plate is established. In the model, the sun gear and planetary gears can be regarded as rigid components, while the thin-walled internal gear ring and the planetary carrier side plate are regarded as flexible components. The flexible bodies of the internal gear ring and the planetary carrier are condensed and then participate in the multibody dynamics solution as condensed flexible bodies after being condensed by the free interface modal synthesis method. The bearings are connected to the corresponding components in the form of a six-degree-of-freedom stiffness matrix. The gear pair meshing is described by a nonlinear multibody contact dynamics model to capture edge contact, meshing off-center loading, and tooth root stress concentration.

[0062] Third, dynamic load history and tooth root bending stress data extraction were performed. The input torque data was discretized into multiple typical load levels, and transient dynamic simulations were conducted at each load level to obtain the dynamic load history of the gear pair. The dynamic load was discretized into multiple equivalent impact loads along the tooth surface contact line, which were applied sequentially to the corresponding contact line, and the tooth root bending stress time history was extracted at the critical section.

[0063] Fourth, an equivalent tooth root alternating bending stress dataset was established. Rainflow counting was performed on the time history of tooth root bending stress under various typical load levels to obtain stress amplitude, mean stress, and number of cycles. The Goodman method was used to correct the mean stress of the stress amplitude and mean stress, obtaining the corrected stress cycles. Using Miner's linear cumulative damage theory and the median SN curve, the corrected stress cycles were converted into constant amplitude equivalent bending stress. The mean and standard deviation of the constant amplitude equivalent bending stress were statistically analyzed to establish a normally distributed equivalent tooth root alternating bending stress dataset.

[0064] Fifth, conduct overall gear fatigue life tests. Based on the tooth structure parameters of the gear to be evaluated, design the main test gear. Install the main test gear on a power-flow enclosed gear rotary testing machine and conduct accelerated bending fatigue life tests under preset oil bath lubrication temperatures and multiple preset stress levels. The failure criterion is the occurrence of a visible crack at the root of the first tooth, which develops into fatigue fracture. Vibration signals are collected using an accelerometer to trigger a machine shutdown, and the cumulative number of cycles of the main test gear is read to obtain the overall gear fatigue life data.

[0065] Sixth, perform probabilistic life conversion for gear tooth elements. A two-parameter Weibull distribution is used to fit the overall gear fatigue life data to obtain the overall gear life distribution. Using the minimum order statistic theory, the overall gear life distribution is converted into a single gear tooth element life distribution. Further, maximum likelihood estimation correction is performed using the random truncated life information of unbroken teeth. Finally, the gear tooth PSN curve is fitted based on the corrected single gear tooth element life distribution to obtain the probabilistic life distribution of the gear tooth elements.

[0066] Seventh, construct a system-level dynamic fatigue reliability assessment model. Each gear tooth unit in the target aerospace planetary gear mechanism is considered a potential failure functional unit, and the life probability density function of each gear tooth unit under random load conditions is established based on the probability life distribution of the gear tooth units and the equivalent tooth root alternating bending stress dataset. The stress intensity interference model is extended based on the conditional expectation algorithm, incorporating the load correlation between gear teeth and the alternating meshing sequence into the system survival probability model. The cumulative number of constant amplitude alternating load applications for each gear tooth unit within the target operating time is calculated based on the absolute angular velocities of the sun gear, planet gears, internal gear ring, and planet carrier, and then substituted into the system survival probability model.

[0067] Eighth, construct the Kriging reliability surrogate model. Multiple sample points are selected within the flexible structure design parameter space. The system-level dynamic fatigue reliability assessment model is invoked to obtain the system fatigue reliability response values ​​corresponding to these sample points, thus constructing a training sample set. After training the Kriging model, a continuous mathematical mapping from the flexibility coefficients of the thin-walled internal gear ring rim and planetary carrier side plate to the system fatigue reliability response values ​​is obtained. Therefore, during the optimization process, it is unnecessary to perform costly simulation calculations for each candidate solution; the system reliability can be quickly predicted using the Kriging reliability surrogate model.

[0068] Ninth, perform coordinated optimization and screening of structural parameters. Using the flexibility coefficient of the thin-walled internal gear ring rim and the flexibility coefficient of the planetary carrier side plate as design variables, the optimization objective is formed by minimizing the system quality normalization objective and the system reliability maximization objective after negative sign transformation. This optimization is then achieved through a weighting coefficient for the quality objective. And system reliability target weighting coefficient A weighted composite objective function is established, and a non-dominated sorting genetic algorithm is used for co-evolutionary search to obtain the Pareto non-dominated solution front for system quality and system reliability. Then, the ideal solution is approximated and sorted using TOPSIS. Based on the weights of the quality and system reliability objectives, the relative closeness of each candidate structural parameter combination is calculated, and the candidate structural parameter combination with the highest relative closeness is output as the optimal structural parameter combination.

[0069] The beneficial effects of this embodiment are as follows: Firstly, by utilizing a global three-dimensional dynamic rigid-flexible coupling model and nonlinear multibody contact dynamics, the influence of the macroscopic flexibility of the thin-walled internal gear ring and planetary carrier side plate on the dynamic load and tooth root bending stress of the gear pair can be reflected. Secondly, through the overall fatigue life test of the gear, the minimum order statistic theory, and the random truncated maximum likelihood estimation, the overall gear life data can be converted into the probability life distribution of gear tooth elements, thereby improving the statistical basis for system-level fatigue reliability assessment. Thirdly, by extending the stress intensity interference model through the conditional expectation algorithm, the load correlation between gear teeth and the alternating meshing sequence can be considered simultaneously, avoiding the simplistic treatment of multi-tooth system failure as independent failure. Finally, through the Kriging reliability surrogate model, the non-dominated sorting genetic algorithm, and TOPSIS sorting, the synergistic optimization of the rim flexibility coefficient of the thin-walled internal gear ring and the planetary carrier side plate flexibility coefficient can be achieved with low computational cost.

[0070] In summary, the dynamic fatigue reliability collaborative design method for aerospace planetary gear mechanisms provided in this embodiment can integrate dynamic load history, tooth root bending stress, tooth probability life distribution, system-level dynamic fatigue reliability, and structural quality into a unified collaborative design framework. It is applicable to the forward design, reliability verification, and lightweight optimization of aerospace planetary gear mechanisms.

[0071] The above are merely preferred embodiments of the present invention and are not intended to limit the present invention in any other way. Any person skilled in the art may make changes or modifications to the above-disclosed technical content to create equivalent embodiments that can be applied to other fields. However, any simple modifications, equivalent changes, and modifications made to the above embodiments based on the technical essence of the present invention without departing from the scope of the present invention shall still fall within the protection scope of the present invention.

Claims

1. A collaborative design method for dynamic fatigue reliability of aerospace planetary gear mechanisms, characterized in that, Includes the following steps: Acquire the dynamic load history and tooth root bending stress data of the target aerospace planetary gear mechanism under all working conditions; Establish an equivalent tooth root alternating bending stress dataset; Obtain overall fatigue life data for gears; The conversion yields the probability lifetime distribution of the gear tooth element; Determine the flexibility coefficient of the thin-walled internal gear ring rim and the flexibility coefficient of the planetary carrier side plate; Construct a system-level dynamic fatigue reliability assessment model; Construct a Kriging reliability proxy model; Solve and select the optimal combination of structural parameters.

2. The collaborative design method for dynamic fatigue reliability of aerospace planetary gear mechanisms according to claim 1, characterized in that, The specific steps for obtaining the full-condition dynamic load history and tooth root bending stress data of the target aerospace planetary gear mechanism are as follows: Select the target aerospace planetary gear mechanism and obtain its geometric parameters, transmission component parameters, meshing pair parameters, support boundary parameters, input torque data, and input speed data. The thin-walled internal gear ring and the planet carrier side plate in the target aerospace planetary gear mechanism are discretized in three dimensions using finite element method to obtain the internal gear ring flexible body and the planet carrier flexible body. The internal gear ring flexible body and the planet carrier flexible body are condensed using the free interface modal synthesis method to obtain the internal gear ring condensed flexible body and the planet carrier condensed flexible body. The internal gear ring condensed flexible body, the planet carrier condensed flexible body, the sun gear rigid body, the planet gear rigid body and the bearing six-degree-of-freedom stiffness matrix are coupled to establish a global three-dimensional dynamic rigid-flexible coupling model. Based on the global three-dimensional dynamic rigid-flexible coupling model and nonlinear multibody contact dynamics, the edge contact, stress concentration and dynamic load of the gear pair of the target aerospace planetary gear mechanism in the dynamic alternating meshing process are solved to obtain the dynamic load history under all working conditions. The dynamic load history under all working conditions is discretized into multiple equivalent impact loads along the tooth surface contact line. The multiple equivalent impact loads are applied sequentially to the corresponding contact lines, and the tooth root bending stress time history of the critical section is extracted to obtain the tooth root bending stress data.

3. The collaborative design method for dynamic fatigue reliability of aerospace planetary gear mechanisms according to claim 2, characterized in that, The specific steps for establishing an equivalent tooth root alternating bending stress dataset are as follows: The input torque data, which follows a normal distribution, is obtained and then discretized into multiple typical load levels. The multiple typical load levels are respectively input into the global three-dimensional dynamic rigid-flexible coupling model to obtain the transient tooth root bending stress time history corresponding to each typical load level. Rainflow counting was performed on the transient tooth root bending stress time history to obtain several sets of stress cycle parameters, including stress amplitude, average stress and number of cycles. Goodman mean stress correction is applied to the stress amplitude and mean stress in the stress cycle parameters to obtain the corrected stress cycle; Based on Miner's linear cumulative damage theory and the median SN curve under the stress ratio of -1, the modified stress cycle is equivalently converted into a constant amplitude equivalent bending stress, which satisfies the following: ； In the formula, This represents the constant amplitude equivalent bending stress. This indicates the stress level of the modified stress cycle. Indicates the stress level number. Indicates the first The number of cycles corresponding to each stress level Indicates the first The Goodman corrected stress amplitude corresponding to each stress level The slope exponent of the median SN curve; The mean and standard deviation of the constant amplitude equivalent bending stress are statistically analyzed to establish an equivalent tooth root alternating bending stress dataset that conforms to a normal distribution.

4. The collaborative design method for dynamic fatigue reliability of aerospace planetary gear mechanisms according to claim 1, characterized in that, The specific steps for obtaining overall fatigue life data of gears are as follows: Based on the gear tooth structure parameters of the target aerospace planetary gear mechanism, a corresponding main test gear is designed. The gear to be evaluated includes at least one of a sun gear, a planet gear, and an internal gear ring, so that the material, heat treatment process, tooth profile parameters, and tooth root stress state of each main test gear are matched with the corresponding gear tooth structure parameters of the gear to be evaluated. Each of the main test gears was installed on a power flow closed gear rotation tester, and bending fatigue accelerated life test was carried out under preset oil bath lubrication temperature and multiple preset stress levels. The failure criterion was to use the appearance of a visible crack at the root of the first tooth in each of the main test gears and its development into fatigue fracture. Vibration signals were collected by an accelerometer placed near the corresponding bearing housing. The power flow closed gear rotation testing machine is stopped based on the vibration signal. The cumulative number of rotations of each main test gear is read and used as the overall fatigue life data of the corresponding gear to be evaluated.

5. The collaborative design method for dynamic fatigue reliability of aerospace planetary gear mechanisms according to claim 4, characterized in that, The specific steps for converting the gear tooth element probability lifetime distribution are as follows: The overall fatigue life data of the gear was fitted using a two-parameter Weibull distribution to obtain the overall gear life distribution; Using the minimum order statistic theory, the overall gear life distribution is transformed into a single gear tooth unit life distribution. The probability density function of the extreme value distribution transformation model is: ； In the formula, The probability density function representing the overall lifespan distribution of the gear. Represents a random variable representing lifespan. This indicates the number of teeth on the main test gear. The cumulative distribution function represents the population probability lifetime distribution of a single gear tooth element. The probability density function represents the population distribution of the probability lifetime of a single gear tooth element; When the probability lifetime distribution of a single gear tooth unit follows a two-parameter Weibull distribution, the shape and scale parameters of the overall gear lifetime distribution and the probability lifetime distribution of a single gear tooth unit satisfy the following: ； The parameters of the probability life distribution of the single gear tooth unit, calculated from the overall gear life distribution, are as follows: ； In the formula, Shape parameters representing the overall life distribution of the gear. A scale parameter representing the overall life distribution of the gear. The shape parameter represents the probability lifetime distribution of a single gear tooth element. The scale parameter representing the probability lifetime distribution of a single gear tooth element; The maximum likelihood estimation method using random truncated data is employed to analyze the truncated life information of the unbroken teeth in the main test gear, and the shape and scale parameters of the probability life distribution of the single gear tooth unit are corrected and estimated. The joint maximum likelihood estimation function is: ； In the formula, Denotes the joint maximum likelihood estimation function. This represents the total observed sample size. Indicates the gear tooth unit number. Indicates the first The number of test cycles for each gear tooth unit Represents a state indicator variable. This represents the probability density function value of the probability lifetime distribution of a single gear tooth unit at the number of test cycles. This represents the cumulative distribution function value of the probability lifetime distribution of a single gear tooth unit at the number of test cycles. When the gear tooth unit fails due to bending fatigue fracture, the state indicator variable is set to 1. When the gear tooth unit does not fracture and the test stops with the whole machine, the state indicator variable is set to 0. The gear tooth PSN curve is fitted according to the corrected estimated single gear tooth unit life distribution to obtain the gear tooth unit probability life distribution.

6. The collaborative design method for dynamic fatigue reliability of aerospace planetary gear mechanisms according to claim 1, characterized in that, The specific steps for determining the flexibility coefficient of the thin-walled internal gear ring rim and the flexibility coefficient of the planetary carrier side plate are as follows: Extract the outer radius of the inner gear ring rim, the root radius of the inner gear ring, the thickness of the planetary carrier side plate, and the radius of the planetary carrier side plate from the geometric parameters of the target aerospace planetary gear mechanism; The flexibility coefficient of the thin-walled internal gear ring, characterized by the relative thickness of the rim, is calculated based on the outer radius of the rim and the root radius of the internal gear ring. The formula is as follows: ； In the formula, This represents the rim flexibility coefficient of a thin-walled internal gear ring. Indicates the outer radius of the rim of the internal gear ring. Indicates the radius of the root circle of the internal gear ring; The planetary carrier side plate flexibility coefficient, characterized by the relative thickness of the side plates, is calculated based on the thickness and radius of the planetary carrier side plate. The formula is as follows: ； In the formula, This indicates the flexibility coefficient of the planetary carrier side plate. Indicates the thickness of the planetary carrier side plates. Indicates the radius of the planetary carrier side plate circle; The range of the flexibility coefficient of the thin-walled internal gear ring rim is set to 0.01 to 0.05, and the range of the flexibility coefficient of the planetary carrier side plate is set to 0.02 to 0.06, thus obtaining the flexible structure design parameter space.

7. The collaborative design method for dynamic fatigue reliability of aerospace planetary gear mechanisms according to claim 1, characterized in that, The specific steps for constructing a system-level dynamic fatigue reliability assessment model are as follows: Each gear tooth unit in the target aerospace planetary gear mechanism is taken as a potential failure functional unit. Based on the probability lifetime distribution of the gear tooth unit and the equivalent tooth root alternating bending stress dataset, the lifetime probability density function of each gear tooth unit under random load conditions is obtained. The stress intensity interference model is extended based on the conditional expectation algorithm, and the inter-tooth load correlation and alternating meshing timing are incorporated into the stress intensity interference model to obtain the system survival probability model, the formula of which is: ； In the formula, This represents the system survival probability under the target runtime. Indicates the target runtime. Represents a standard normal random variable. This represents the probability density function of the standard normal distribution. This indicates the total number of gear tooth functional units within the system. Indicates the gear tooth functional unit number. Represents the lifespan integral variable. Indicates the first The cumulative number of constant amplitude alternating loads applied to each gear tooth functional unit within the target operating time. Indicates the first The lifetime probability density function of each gear tooth functional unit under random load conditions Indicates the first The average value of the equivalent stress of discrete loads corresponding to each gear tooth functional unit. Indicates the first The standard deviation of the discrete load equivalent stress corresponding to each gear tooth functional unit; The cumulative number of constant-amplitude alternating load applications on representative teeth of each meshing pair within a specified operating time is calculated based on the absolute angular velocities of the sun gear, planet gears, internal gear ring, and planet carrier. This cumulative number of constant-amplitude alternating load applications is then used as a factor in the system survival probability model. A system-level dynamic fatigue reliability assessment model was obtained.

8. The collaborative design method for dynamic fatigue reliability of aerospace planetary gear mechanisms according to claim 6, characterized in that, The specific steps for constructing the Kriging reliability proxy model are as follows: Multiple sample points are selected in the flexible structure design parameter space, and the flexible coefficient of the thin-walled internal gear ring rim and the flexible coefficient of the planetary carrier side plate corresponding to each sample point are combined to form a multidimensional flexible structure design variable space vector. The system-level dynamic fatigue reliability assessment model is called for each sample point to obtain the corresponding system fatigue reliability response value, and the training sample set is composed of the multiple sample points and the system fatigue reliability response value. When constructing the training sample set, the mean of the equivalent tooth root alternating bending stress dataset is kept constant, so that the stress amplitude characteristics of the equivalent tooth root alternating bending stress dataset exhibit a gradient distribution characteristic with increasing dispersion as the flexibility coefficient of the thin-walled inner gear ring rim and the flexibility coefficient of the planetary carrier side plate decreases. The Kriging model is trained using regression basis functions and correlation function hyperparameters. The fatigue reliability of the system with unknown design is represented as a linear weighted combination of known sample responses, and the prediction expression is: ； In the formula, This represents the reliability prediction value at the space vector of design variables for a multidimensional flexible structure. Represents the space vector of design variables for multidimensional flexible structures. Represents the total number of observed samples. Indicates the observation sample number, Indicates the first The spatial correlation weighting coefficients of each observed sample relative to the spatial vector of the multidimensional flexible structure design variables. Indicates the first The system fatigue reliability response value corresponding to each observation sample.

9. The collaborative design method for dynamic fatigue reliability of aerospace planetary gear mechanisms according to claim 8, characterized in that, The specific steps for solving and selecting the optimal combination of structural parameters are as follows: Using the Kriging reliability surrogate model as the fitness calculation model, a non-dominated sorting genetic algorithm is used to perform a co-evolutionary search on the rim flexibility coefficient of the thin-walled internal gear ring and the side plate flexibility coefficient of the planetary carrier, so as to obtain the Pareto non-dominated solution front of system quality and system reliability. The quality objective function, reliability objective function, weighted composite objective function, and constraints of the co-evolutionary search are as follows: ； ； ； In the formula, This represents the rim flexibility coefficient of a thin-walled internal gear ring. This indicates the flexibility coefficient of the planetary carrier side plate. This represents the system quality normalization objective function. This represents the minimized objective function formed by transforming the objective function for maximizing system reliability with a negative sign. This represents the weighted composite objective function. This represents the total structural mass of the planetary transmission system. This represents the maximum system quality among the candidate solutions. This represents the system reliability at the target runtime predicted by the Kriging reliability surrogate model. Indicates the target runtime. This represents the weighting coefficient of the quality target. This represents the target weighting coefficient for system reliability; The candidate structural parameter combinations in the Pareto non-dominated solution front are sorted by TOPSIS approximation of the ideal solution, and the candidate structural parameter combination with the highest relative closeness is taken as the optimal structural parameter combination and output.

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