A method of designing a spherical flexible rotor

Abstract

The application relates to the technical field of helicopter rotor design optimization, and discloses a ball flexible rotor design method, which comprises the following steps: inputting ball flexible rotor design initial parameters, solving aerodynamic basic parameters based on a free wake and an L-B dynamic stall model; establishing a three-way coupled dynamics equation of blade flapping, lagging and torsion, substituting dynamics input stiffness parameters, solving inherent frequencies and displacement responses of blade first-order flapping, lagging and torsion; combining aerodynamic, centrifugal and deformation loads and a pre-cone angle additional centrifugal bending moment, calculating blade dynamic stress and bearing contact stress and performing strength checking; based on the deviation of actual frequencies and target frequencies, iteratively correcting dynamics input stiffness parameters, and outputting ball flexible rotor design final parameters after meeting a preset convergence threshold. The application can effectively avoid resonance, flutter and lagging instability risks, improve rotor structure reliability and dynamics stability, and is suitable for ball flexible rotor design of light helicopters and unmanned helicopters.

Description

Technical Field

[0001] This invention relates to the field of helicopter rotor design optimization technology, and in particular to a spherical flexible rotor design method. Background Technology

[0002] As a core functional component of helicopters, the rotor system plays a crucial role in providing lift, propulsion, and flight control. Its design directly determines the helicopter's flight quality, operational reliability, maintenance costs, and overall life-cycle performance. The spherical flexible rotor system, through an integrated spherical flexible elastic bearing, achieves three-dimensional hinged motion of blade flapping, oscillation, and torsion. This completely eliminates the mechanical hinges, lubrication systems, and sealing mechanisms required by traditional articulated rotors. Compared to traditional rotor configurations, it boasts significant advantages such as high structural integration, fewer parts, lower maintenance workload, superior vibration and noise levels, and longer fatigue life. Currently, it is widely used in general aviation fields such as civilian light helicopters and industrial-grade heavy-lift unmanned helicopters, representing a key direction for upgrading the core technology and improving the performance of small and medium-sized rotorcraft.

[0003] Currently, the design of spherical flexible rotor systems in the industry mostly adopts a serial, step-by-step design mode, which involves aerodynamic load calculation, rotor dynamics analysis, and structural strength verification. Each design stage is independent, and data is transferred unidirectionally, failing to form a closed-loop coupled full-process design system. This exposes a series of intractable technical bottlenecks in refined engineering applications. Existing design schemes generally do not fully consider the multi-dimensional mechanical effects caused by the blade pre-cone angle, including key influencing factors such as centrifugal stiffening effect, effective span correction, centrifugal load attenuation, and additional bending moment at the blade root. This results in the design of the aerodynamic field, dynamic field, and structural strength field being disconnected from each other, failing to establish a theoretical model and solution system for deep coupling of the three fields. Under unsteady conditions such as high-maneuver flight and dynamic stall, the design calculation results deviate significantly from the actual flight characteristics, making it difficult to meet the requirements of high-precision engineering design.

[0004] Meanwhile, existing design methods fail to establish a quantitative coupling relationship between rotor aerodynamic loads and blade triaxial natural frequencies. Frequency avoidance designs for rotor systems largely rely on engineering experience and historical data, lacking precise theoretical calculation support. This makes it impossible to accurately predict and avoid dynamic risks such as ground resonance, air resonance, and wobbling instability during the design phase, easily leading to dynamic instability issues in the completed rotor system during flight testing and verification, requiring repeated iterative modifications, significantly increasing R&D costs and timelines. For the core components of spherical flexible rotors, namely spherical flexible elastic bearings and dampers, existing designs lack complete system-level dynamic input basis. The triaxial stiffness design of bearings and the stiffness and damping matching of dampers are mostly carried out separately at the component level, failing to achieve full-system coordinated matching with blade structural characteristics and rotor aerodynamic loads under all operating conditions. This easily leads to incompatibility issues between component parameters and system dynamic characteristics, failing to fully utilize the configuration advantages of spherical flexible rotors.

[0005] Furthermore, existing design methods lack a unified iterative convergence threshold and closed-loop optimization design mechanism. The convergence criteria for each design stage are inconsistent, requiring repeated testing, debugging, and parameter correction based on the designer's experience during engineering implementation. This results in low design efficiency and makes it difficult to simultaneously ensure the dynamic stability and structural reliability of the rotor system, hindering the synergistic optimization of multiple design objectives. Among the existing publicly available related technical solutions, some dynamic modeling methods for rotor hubs, such as the multi-path force-transmitting rotor hub structure dynamic modeling method disclosed in CN104951591B, only use aerodynamic loads as fixed boundary condition inputs. They do not establish a two-way coupling response mechanism between aerodynamic loads and blade elastic deformation, and cannot accurately capture the true dynamic characteristics of the rotor system under unsteady conditions. At the same time, these solutions only focus on the dynamic modeling of the hub structure, and the design parameters of core components such as blades, bearings, and dampers still need to rely on empirical formulas or external inputs. They cannot achieve coordinated matching of parameters across the entire system, nor do they possess the reverse design capability from target performance to design parameters.

[0006] Other optimization methods for structural stress calculation, such as the structural characteristic stress optimization calculation method disclosed in CN118862298B under multi-load and out-of-phase loading, can only handle stress analysis under a fixed load spectrum. It cannot respond to structural deformation to realize dynamic load updates, and it is only a single post-processing calculation link. It does not establish a mapping relationship between stress level and dynamic design parameters, and cannot optimize dynamic design parameters in reverse through stress verification results. It is difficult to integrate into the closed-loop design system of the whole process. Summary of the Invention

[0007] To address the aforementioned issues, this invention proposes a spherical flexible rotor design method. Specifically, by establishing a design theory that deeply couples the rotor's aerodynamic load, dynamic characteristics, and structural strength across three fields, the method clarifies the quantitative mapping relationship between load, frequency, blade profile characteristics, bearing stiffness, and damper frequency. It also constructs a closed-loop collaborative iteration mechanism and a quantitative convergence threshold, thereby enabling multi-objective collaborative optimization of the rotor system and improving design accuracy, dynamic stability, and structural reliability.

[0008] The technical solution adopted in this invention is as follows: A spherical flexible rotor design method includes: Input the initial parameters for the design of the flexible spherical rotor, and solve the basic aerodynamic parameters based on the free wake and LB dynamic stall model; Establish the three-dimensional coupled dynamic equations of blade flapping-shielding-torsion, substitute the dynamic input stiffness parameters, and solve for the first-order natural frequencies and displacement responses of blade flapping, shielding and torsion. By combining aerodynamic, centrifugal, and deformation loads with the additional centrifugal bending moment from the pre-cone angle, the dynamic stress of the blades and the contact stress of the bearings are calculated and the strength is checked. Based on the deviation between the actual frequency and the target frequency, the dynamic input stiffness parameters are iteratively corrected, and the final parameters of the spherical flexible rotor design are output after satisfying the preset convergence threshold.

[0009] Furthermore, the initial design parameters of the input spherical flexible rotor are obtained by solving the basic aerodynamic parameters based on the free wake and LB dynamic stall model, including: Input the initial design parameters for the spherical flexible rotor, including rotor system configuration, blade geometry, material constitutive properties, initial characteristics of the spherical flexible bearing, initial characteristics of the damper, and initial parameters for the design flight conditions; The induced inflow distribution characteristics of the rotor across the entire span are solved based on the free wake model, capturing the real-time impact of the rotor wake on the aerodynamic environment of the blade profile; the unsteady aerodynamic characteristics of the blade profile are solved by combining the LB dynamic stall model, covering extreme conditions of high-maneuver flight and dynamic stall; the aerodynamic basic parameters are solved simultaneously, including rotor aerodynamic load, induced inflow coefficient, flapping, flaring and torsional aerodynamic stiffness, and can be dynamically updated with real-time blade deformation.

[0010] Furthermore, the establishment of the three-dimensional coupled dynamic equations for blade flapping-shielding-torsion includes: Using the blade flapping, oscillation, and torsion as generalized displacements, a fully coupled dynamic equation framework is constructed, incorporating the blade coupling mass matrix, the system integrated damping matrix, the blade structural stiffness matrix, the rotor aerodynamic stiffness matrix, and the combined stiffness matrix of the ball flexible bearing and damper. Aerodynamic loads are incorporated into the equations as endogenous variables related to the blade's generalized displacement, rotational angular velocity, and induced inflow coefficient, rather than as fixed exogenous boundary conditions. At the same time, the centrifugal stiffening effect caused by the blade's pre-cone angle is taken into account, forming a dynamic solution equation that deeply couples the aerodynamic field, structural field, and dynamic field, namely, the three-dimensional coupled dynamic equation of blade flapping-shielding-torsion.

[0011] Furthermore, the dynamic input stiffness parameters include: the structural stiffness of the blade itself corresponding to the flapping, oscillation, and torsional degrees of freedom; the initial stiffness of the ball flexible bearing corresponding to the three degrees of freedom; the equivalent stiffness of the anti-oscillation device; and the solved blade flapping, oscillation, and torsional aerodynamic stiffness. All kinds of stiffness parameters together constitute the stiffness term of the dynamic equation, which forms a quantitative mapping relationship with the blade's three-dimensional natural frequencies.

[0012] Furthermore, the solution for the first-order flapping, teetering, and torsional natural frequencies and displacement responses of the blade includes: Substituting the dynamic input stiffness parameters into the three-dimensional coupled dynamic equations, the effect of centrifugal load generated by rotor rotation on stiffness characteristics is taken into account; the first-order flapping, tessellation and torsion natural frequencies of the blade are obtained by solving, and a quantitative coupling mapping relationship between the natural frequencies and the blade structural stiffness, bearing stiffness, damper stiffness and aerodynamic stiffness is established; the real-time deformation displacement response of the blade in the full spanwise direction under the corresponding flight conditions is obtained by solving simultaneously.

[0013] Furthermore, the aerodynamic, centrifugal, and deformation loads, as well as the additional centrifugal bending moment due to the pre-cone angle, include: The aerodynamic loads generated by the unsteady aerodynamic characteristics of the blade profile, the full spanwise centrifugal loads generated by the rotor rotation, the deformation loads generated by the elastic deformation of the blades, and the additional centrifugal bending moment caused by the pre-cone angle of the blades are all dynamically updated based on the real-time deformation displacement response of the blades, while also taking into account the centrifugal load attenuation caused by the pre-cone angle, the effective span correction, and the influence of the additional bending moment at the root.

[0014] Furthermore, the calculation of blade dynamic stress and bearing contact stress, and the subsequent strength verification, include: Based on the real-time deformation displacement response of the blade obtained by the solution, the deformation curvature of each section along the full span of the blade is calculated; combined with the full load input, the comprehensive dynamic stress of the blade section and the contact stress of the ball flexible bearing are calculated. Static strength verification is performed based on the allowable stress of blade and bearing materials, and fatigue life verification is performed based on alternating stress amplitude and material fatigue limit. The strength verification results are used as one of the core criteria for iterative convergence judgment, realizing the synergistic coupling of dynamic design and strength design.

[0015] Furthermore, the iterative correction of the dynamic input stiffness parameters based on the deviation between the actual frequency and the target frequency includes: By comparing the actual three-dimensional natural frequencies of the blade obtained from the solution with the preset target natural frequencies, the deviation of each frequency is calculated. Based on the frequency deviation and combined with the quantitative mapping relationship between natural frequency and stiffness, the correction of the three-dimensional stiffness of the ball flexible bearing and the equivalent stiffness of the damper is obtained by inverse solution. The change of aerodynamic stiffness during the iteration process is simultaneously taken into account to perform coordinated correction of bearing stiffness and damper stiffness. The corrected stiffness parameters are substituted into the aerodynamic basic parameter solution process to start a new round of full-process coupled calculations, forming a closed-loop iterative mechanism. At the same time, based on the oscillation frequency design criteria, the stiffness and damping characteristics of the oscillation damper are matched to avoid the rotor excitation frequency and suppress the risks of ground resonance, air resonance and oscillation instability.

[0016] Furthermore, the preset convergence threshold includes: preset multi-dimensional collaborative convergence criteria covering aerodynamic characteristics, dynamic characteristics, structural strength, and design parameter stability; the multi-dimensional collaborative convergence criteria include the deviation between the blade's three-dimensional natural frequency and the target frequency, the fluctuation of aerodynamic load before and after iteration, the deviation of blade dynamic stress before and after iteration, and the correction of stiffness parameters before and after iteration; iteration convergence is determined only when all convergence criteria simultaneously meet the preset requirements, and the loop calculation is terminated; if any criterion does not meet the requirements, full-process coupled iterative calculation continues to ensure that the design results simultaneously meet the multi-dimensional design requirements of aerodynamics, dynamics, and strength.

[0017] Furthermore, the final design parameters of the spherical flexible rotor include: blade spanwise stiffness distribution parameters matching the target frequency, stiffness parameters of the spherical flexible bearing corresponding to the flapping, oscillation, and torsional degrees of freedom, equivalent stiffness and damping parameters of the oscillation damper, dynamic characteristic parameters of the rotor system under all operating conditions, and structural design parameters verified by static strength and fatigue life.

[0018] The beneficial effects of this invention are as follows: 1. This invention constructs a design system that deeply couples aerodynamic load, rotor dynamics, and structural strength across three fields. By establishing a three-dimensional coupled dynamic equation of blade flapping-shielding-torsion, aerodynamic load is incorporated into the solution system as an endogenous variable that dynamically updates with blade deformation. At the same time, the multi-dimensional mechanical effects brought about by the blade pre-cone angle are fully considered, realizing a two-way coupled solution of the aerodynamic field and the structural field. This can accurately capture the real dynamic characteristics of the rotor system under unsteady flight conditions, significantly improve the calculation accuracy of spherical flexible rotor design, and solve the core problem of large deviations between design results and actual operating conditions caused by the separation of the three fields in traditional design.

[0019] 2. This invention establishes a full-chain quantitative mapping relationship between rotor load, blade three-dimensional natural frequency, blade profile stiffness characteristics, bearing three-dimensional stiffness, and damper parameters. It clarifies the coupling relationship between each design parameter and the system dynamic characteristics, and realizes reverse design from target performance to design parameters. It can accurately solve for core design indicators such as bearing stiffness and damper parameters based on the preset target frequency, eliminating the subjectivity and uncertainty of traditional experience-based design. It achieves parameter coordination and matching of all core components of the rotor system, such as blades, bearings, and dampers, and ensures the dynamic compatibility between components from the design source.

[0020] 3. This invention constructs a standardized closed-loop collaborative iterative design mechanism and a multi-dimensional quantitative convergence threshold, embedding the strength verification step into the iterative design process rather than an independent post-processing step. This achieves collaborative optimization of dynamic design and structural strength design. At the same time, it clarifies the frequency design criteria of the oscillation system and the damping matching method of the oscillation damper, which can accurately avoid the rotor excitation frequency during the design stage, effectively suppress dynamic risks such as ground resonance, air resonance and oscillation instability, significantly improve the dynamic stability and structural reliability of the rotor system, reduce the vibration level and fatigue damage risk of the rotor system, and extend the service life of the product.

[0021] 4. This invention forms a complete, standardized, and engineering-applicable full-process design method for spherical flexible rotors, covering the entire design chain from initial parameter input to final design parameter output. It eliminates the need for extensive experience-based trial and error and experimental debugging, significantly improving design efficiency, shortening the R&D cycle, and reducing R&D and manufacturing costs. It is particularly suitable for the refined design of spherical flexible rotors for light helicopters and industrial-grade heavy-duty unmanned helicopters, and has good engineering application value and promotion prospects. Attached Figure Description

[0022] Figure 1 This is a flowchart of a spherical flexible rotor design method according to Embodiment 1 of the present invention. Detailed Implementation

[0023] To provide a clearer understanding of the technical features, objectives, and effects of the present invention, specific embodiments are now described. It should be understood that the specific embodiments described herein are merely illustrative of the invention and are not intended to limit the invention; that is, the described embodiments are only a part of the embodiments of the invention, not all of them. All other embodiments obtained by those skilled in the art based on the embodiments of the present invention without inventive effort are within the scope of protection of the present invention.

[0024] Example 1 like Figure 1 As shown, this embodiment provides a spherical flexible rotor design method, including: Input the initial parameters for the design of the flexible spherical rotor, and solve the basic aerodynamic parameters based on the free wake and LB dynamic stall model; Establish the three-dimensional coupled dynamic equations of blade flapping-shielding-torsion, substitute the dynamic input stiffness parameters, and solve for the first-order natural frequencies and displacement responses of blade flapping, shielding and torsion. By combining aerodynamic, centrifugal, and deformation loads with the additional centrifugal bending moment from the pre-cone angle, the dynamic stress of the blades and the contact stress of the bearings are calculated and the strength is checked. Based on the deviation between the actual frequency and the target frequency, the dynamic input stiffness parameters are iteratively corrected, and the final parameters of the spherical flexible rotor design are output after satisfying the preset convergence threshold.

[0025] Preferably, the initial design parameters of the input spherical flexible rotor are obtained by solving the aerodynamic fundamental parameters based on the free wake and LB dynamic stall model. This includes: inputting the initial design parameters of the input spherical flexible rotor, including rotor system configuration, blade geometry, material constitutive properties, initial characteristics of the spherical flexible bearing, initial characteristics of the damper, and initial parameters for the design flight conditions; solving the induced inflow distribution characteristics of the rotor across its entire span based on the free wake model to capture the real-time influence of the rotor wake on the aerodynamic environment of the blade profile; combining the LB dynamic stall model to solve the unsteady aerodynamic characteristics of the blade profile, covering extreme conditions of high-maneuverability flight and dynamic stall; and simultaneously solving for the aerodynamic fundamental parameters, which include rotor aerodynamic loads, induced inflow coefficients, and flapping, yaw, and torsional aerodynamic stiffness, and can be dynamically updated with real-time blade deformation.

[0026] Specifically, during implementation, the initial design parameters of all elements of the rotor system are first input, covering the overall rotor configuration, the spanwise and chordwise geometric parameters of the blades, the mechanical performance parameters of the materials used for the blades and bearings, the initial design parameters of the ball bearings and dampers, and the parameters of all flight conditions covered by the design. Based on the free wake model, the rotor flow field is solved to accurately capture the distortion and interference characteristics of the wake during rotor rotation, and to obtain the induced inflow distribution of each section of the blade spanwise. Combined with the LB dynamic stall model, the unsteady aerodynamic characteristics of the blade section are solved, covering the dynamic stall conditions under conventional level flight and high-maneuver flight. The aerodynamic loads, induced inflow coefficients, and aerodynamic stiffness corresponding to the three degrees of freedom of the rotor spanwise are calculated simultaneously, and all aerodynamic parameters can be dynamically updated with the real-time deformation of the blades.

[0027] It should be noted that this step achieves accurate solution of the unsteady aerodynamic characteristics of the rotor by combining the free wake with the LB dynamic stall model. At the same time, the aerodynamic parameters are set as variables that can be dynamically updated with blade deformation, rather than fixed exogenous boundary conditions. This lays the foundation for the subsequent bidirectional coupling solution of the aerodynamic field and the structural field, and effectively solves the problem of insufficient design accuracy under unsteady conditions caused by fixed aerodynamic load input in traditional design.

[0028] Preferably, the establishment of the three-dimensional coupled dynamic equations of blade flapping-shielding-torsion includes: constructing a fully coupled dynamic equation framework with the blade flapping, shielding, and torsion degrees of freedom as generalized displacements, incorporating the blade coupled mass matrix, the system integrated damping matrix, the blade structural stiffness matrix, the rotor aerodynamic stiffness matrix, and the combined stiffness matrix of the ball flexible bearing and damper into the equations; incorporating aerodynamic loads as endogenous variables strongly correlated with the blade generalized displacement, rotational angular velocity, and induced inflow coefficient into the equations, rather than fixing exogenous boundary conditions; and simultaneously taking into account the centrifugal stiffening effect brought about by the blade pre-cone angle, forming a dynamic solution equation that deeply couples the aerodynamic field, structural field, and dynamic field, i.e., the three-dimensional coupled dynamic equations of blade flapping-shielding-torsion.

[0029] Specifically, in the implementation process, the overall framework of the rotor system's dynamic equations is constructed using the three core degrees of freedom of the blades—flapping, oscillation, and torsion—as generalized displacements. Based on the blade's geometric and material parameters, a mass matrix coupling the three degrees of freedom of the blades is constructed. A comprehensive damping matrix is ​​built by integrating aerodynamic damping, structural damping, bearing damping, and damper damping. At the same time, a system stiffness matrix is ​​built by integrating the blade's structural stiffness, aerodynamic stiffness, and the combined stiffness of the bearings and dampers. Aerodynamic loads are included as endogenous variables strongly correlated with the blade's generalized displacement, rotor rotational angular velocity, and induced inflow coefficient in the right-hand side of the equations. The centrifugal stiffening effect brought about by the blade pre-cone angle is also fully considered. Finally, a dynamic solution equation with full coupling of three degrees of freedom and deep correlation across multiple fields is formed.

[0030] It should be noted that the three-dimensional coupled dynamic equations constructed in this step fully realize the deep coupling of the aerodynamic field, structural field and dynamic field, breaking the barrier of unidirectional input of aerodynamics and dynamics in traditional design. At the same time, it fully considers the centrifugal stiffening effect brought about by the pre-cone angle, and can accurately capture the real dynamic characteristics of the spherical flexible rotor system. It effectively solves the problem that traditional dynamic modeling cannot reflect the two-way coupling effect of aerodynamics and structure, and greatly improves the accuracy of dynamic solution.

[0031] Preferably, the dynamic input stiffness parameters include: the structural stiffness of the blade itself corresponding to the flapping, oscillation and torsional degrees of freedom, the initial stiffness of the ball flexible bearing corresponding to the three degrees of freedom, the equivalent stiffness of the anti-oscillation device, and the solved blade flapping, oscillation and torsional aerodynamic stiffness; all kinds of stiffness parameters together constitute the stiffness term of the dynamic equation, and form a quantitative mapping relationship with the blade's three-dimensional natural frequencies.

[0032] Specifically, during implementation, the dynamic input stiffness parameters cover four main categories: structural stiffness based on the blade material and profile characteristics, corresponding to the three-dimensional degrees of freedom of flapping, yaw, and torsion; three-dimensional corresponding stiffness based on the initial design parameters of the ball bearing; equivalent stiffness based on the initial design parameters of the damper; and three-dimensional aerodynamic stiffness obtained from the aforementioned steps. The four types of stiffness parameters are simultaneously substituted into the stiffness matrix of the dynamic equation to form a complete system stiffness term. At the same time, a quantitative mapping relationship between each type of stiffness parameter and the three-dimensional natural frequency of the blade is established to clarify the influence law of each type of stiffness parameter on the natural frequency.

[0033] It should be noted that this step incorporates the blade structure stiffness, bearing stiffness, damper stiffness, and aerodynamic stiffness into the dynamic input stiffness parameters, fully covering all stiffness elements of the rotor system that affect its dynamic characteristics. At the same time, it establishes a quantitative mapping relationship between stiffness parameters and natural frequencies, providing a theoretical basis for subsequent iterative stiffness correction based on frequency deviations. This solves the problem of fragmented stiffness parameters of various components and the inability to coordinate and match them in traditional designs.

[0034] Preferably, the process of solving for the first-order flapping, flaring, and torsional natural frequencies and displacement responses of the blade includes: substituting the dynamic input stiffness parameters into the three-dimensional coupled dynamic equations, taking into account the stiffening effect of the centrifugal load generated by the rotor rotation on the stiffness characteristics; solving for the first-order flapping, flaring, and torsional natural frequencies of the blade, and simultaneously establishing a quantitative coupling mapping relationship between the natural frequencies and the blade structural stiffness, bearing stiffness, damper stiffness, and aerodynamic stiffness; and simultaneously solving for the real-time deformation and displacement responses of the blade across the entire span under the corresponding flight conditions.

[0035] Specifically, during implementation, the integrated dynamic input stiffness parameters are substituted into the three-dimensional coupled dynamic equations. The stiffening effect of the centrifugal load generated by the high-speed rotation of the rotor on the blade stiffness characteristics is fully considered during the solution process. The first-order flapping, flaring, and torsional natural frequencies of the blade are obtained through eigenvalue solving. At the same time, based on the solution results, the quantitative coupling mapping relationship between the natural frequencies and various input stiffness parameters is further improved, and the influence weight of each stiffness parameter on the three-dimensional natural frequencies is clarified. Simultaneously, the real-time deformation displacement response of flapping, flaring, and torsional sections of the blade in the full spanwise direction under the corresponding flight conditions is obtained through time-domain solving, providing accurate deformation input for subsequent load calculation and strength verification.

[0036] It should be noted that this step fully considers the influence of centrifugal stiffening effect in the process of solving for frequency and displacement. At the same time, it realizes the synchronous construction of the quantitative coupling mapping relationship between natural frequency and stiffness parameters. This not only obtains accurate dynamic characteristic solution results, but also provides a clear reverse design basis for subsequent parameter iteration correction. This solves the problem that in traditional design, frequency is only an analysis result and cannot support reverse design.

[0037] Preferably, the aerodynamic, centrifugal, and deformation loads, as well as the additional centrifugal bending moment due to the pre-cone angle, include: aerodynamic loads generated by the unsteady aerodynamic characteristics of the blade profile, full spanwise centrifugal loads generated by rotor rotation, deformation loads generated by the elastic deformation of the blades, and additional centrifugal bending moments caused by the blade pre-cone angle; the aerodynamic loads, centrifugal loads, and deformation loads are all dynamically updated based on the real-time deformation displacement response of the blades, while also taking into account the effects of centrifugal load attenuation due to the pre-cone angle, effective span correction, and additional bending moment at the root.

[0038] Specifically, during implementation, the total loads used for strength calculations cover four main categories: aerodynamic loads calculated based on the unsteady aerodynamic characteristics of the blade profile, centrifugal loads generated throughout the span during rotor rotation, deformation loads generated by the elastic deformation of the blades, and additional centrifugal bending moments caused by the blade pre-cone angle. All loads are dynamically updated based on the real-time deformation displacement response of the blades obtained from the aforementioned steps, rather than using a fixed preset load spectrum. At the same time, during the load calculation process, the effects of centrifugal load attenuation caused by the pre-cone angle, effective span correction, and additional bending moments at the blade root are fully taken into account, forming a real-time load input covering all operating conditions and all dimensions.

[0039] It should be noted that this step realizes the dynamic updating of the load as the blade deforms, and at the same time fully considers the multi-dimensional influence of the pre-cone angle on the load characteristics. It can accurately reflect the true load level of the rotor under nonlinear aerodynamic-structural coupling state, solve the problem that the fixed load spectrum in traditional design cannot reflect the true stress level under coupled conditions, and greatly improve the accuracy of subsequent strength calculations.

[0040] Preferably, the calculation of blade dynamic stress and bearing contact stress and the strength verification include: calculating the deformation curvature of each section of the blade along its entire span based on the real-time deformation displacement response of the blade obtained by the solution; calculating the comprehensive dynamic stress of the blade section and the contact stress of the ball flexible bearing in combination with the full load input; completing the static strength verification based on the allowable stress of the blade material and the bearing material; completing the fatigue life verification based on the alternating stress amplitude and the material fatigue limit; and using the strength verification results as one of the core bases for judging iterative convergence, so as to realize the synergistic coupling of dynamic design and strength design.

[0041] Specifically, during implementation, the deformation curvature of each section is first calculated based on the real-time deformation displacement response of the blades across the entire span. Then, combined with the full real-time load input obtained in the preceding steps, the comprehensive dynamic stress of each blade section and the contact stress of the ball bearing are calculated. Based on the allowable stress of the materials used in the blades and bearings, static strength verification is performed to verify the structure's load-bearing capacity under ultimate loads. Based on the alternating stress amplitude of the blade sections and the symmetrical cyclic fatigue limit of the materials, fatigue life verification is performed to verify the long-term service reliability of the structure. Simultaneously, the results of the static strength and fatigue life verifications are used as one of the core bases for subsequent iterative convergence judgments, embedding the strength verification process within the entire design iteration flow.

[0042] It should be noted that this step embeds the strength verification process into the design iteration process, rather than as an independent post-processing step. This achieves synergistic coupling between dynamic design and strength design, avoiding the problem of repeated revisions of design results caused by the separation of dynamic design and strength design in traditional design. At the same time, through multi-dimensional strength verification, the structural reliability and service safety of the rotor system are fully guaranteed.

[0043] Preferably, the iterative correction of the dynamic input stiffness parameters based on the deviation between the actual frequency and the target frequency includes: comparing the solved blade three-dimensional actual natural frequencies with the preset target natural frequencies, and calculating the frequency deviation in each direction; based on the frequency deviation, and combined with the quantitative mapping relationship between natural frequency and stiffness, inversely solving for the correction of the three-dimensional stiffness of the ball flexible bearing and the equivalent stiffness of the damper; simultaneously taking into account the change in aerodynamic stiffness during the iteration process, and coordinating the correction of the bearing stiffness and the damper stiffness; substituting the corrected stiffness parameters into the aerodynamic basic parameter solution stage, starting a new round of full-process coupled calculation, forming a closed-loop iterative mechanism; and simultaneously, based on the oscillation frequency design criteria, matching the damper stiffness and damping characteristics, avoiding the rotor excitation frequency, and suppressing the risks of ground resonance, air resonance, and oscillation instability.

[0044] Specifically, in the implementation process, the actual three-dimensional natural frequencies of the blades obtained by the solution are first compared with the preset design target natural frequencies to calculate the deviation of each frequency. Based on the frequency deviation, combined with the quantitative mapping relationship between natural frequencies and stiffness parameters established in the previous steps, the correction amounts of the three-dimensional stiffness of the ball flexible bearing and the equivalent stiffness of the damper are obtained through algebraic inverse solution. During the correction process, the change in aerodynamic stiffness during the iteration process is simultaneously taken into account to coordinate the correction of bearing stiffness and damper stiffness, ensuring the consistency of parameter matching. The corrected stiffness parameters are substituted into the aerodynamic basic parameter solution process to start a new round of full-process coupled calculation, forming a complete closed-loop iteration mechanism. At the same time, during the iteration process, based on the oscillation frequency design criteria, the stiffness and damping characteristics of the damper are matched and optimized to ensure that the oscillation system frequency avoids the rotor excitation frequency, suppressing the risks of ground resonance, air resonance and oscillation instability from the design source.

[0045] It should be noted that this step constructs a closed-loop iterative mechanism for stiffness collaborative correction based on frequency deviation, realizing reverse design from design objectives to design parameters. This solves the problem of repeated trial and error caused by the traditional design method which can only perform forward analysis and cannot perform reverse design, thus greatly improving design efficiency. At the same time, by optimizing the matching of the oscillation frequency and the damper parameters, the risk of dynamic instability of the rotor system is avoided from the design source, greatly improving the dynamic stability of the rotor system.

[0046] Preferably, the preset convergence threshold includes: a preset multi-dimensional collaborative convergence criterion covering aerodynamic characteristics, dynamic characteristics, structural strength, and design parameter stability; the multi-dimensional collaborative convergence criterion includes the deviation between the blade's three-dimensional natural frequency and the target frequency, the fluctuation of aerodynamic load before and after iteration, the deviation of blade dynamic stress before and after iteration, and the correction of stiffness parameters before and after iteration; the iteration is determined to be converged and the loop calculation is terminated only when all convergence criteria simultaneously meet the preset requirements; if any criterion does not meet the requirements, the full-process coupled iterative calculation continues to ensure that the design results simultaneously meet the multi-dimensional design requirements of aerodynamics, dynamics, and strength.

[0047] Specifically, during implementation, multi-dimensional collaborative convergence criteria are first preset, covering four core dimensions: aerodynamic characteristics, dynamic characteristics, structural strength, and design parameter stability. These criteria include four main categories: deviation of the blade's three-dimensional natural frequency from the target frequency, fluctuation of aerodynamic load between adjacent iteration steps, deviation of blade dynamic stress between adjacent iteration steps, and correction of stiffness parameters between adjacent iteration steps. After each iteration, all convergence criteria are synchronously verified. Iteration convergence is determined and the loop calculation is terminated only when all criteria simultaneously meet the preset control requirements. If any criterion does not meet the preset requirements, the full-process coupled iterative calculation continues until all criteria meet the requirements.

[0048] It should be noted that this step sets up a multi-dimensional collaborative convergence criterion, rather than a single frequency convergence criterion. This ensures that the design results simultaneously meet the multi-dimensional design requirements of aerodynamics, dynamics, and strength, avoiding the problem of other design dimensions failing to meet the standards due to convergence of a single criterion. At the same time, the quantified convergence threshold achieves the standardization and normalization of the design process, greatly improving the consistency and reliability of the design results.

[0049] Preferably, the final design parameters of the spherical flexible rotor include: blade spanwise stiffness distribution parameters matching the target frequency, stiffness parameters of the spherical flexible bearing corresponding to the flapping, oscillation, and torsional degrees of freedom, equivalent stiffness and damping parameters of the oscillation damper, dynamic characteristic parameters of the rotor system under all operating conditions, and structural design parameters verified by static strength and fatigue life.

[0050] Specifically, after the iterative convergence terminates the loop, the final parameters of the complete spherical flexible rotor design are output. These parameters include the stiffness distribution parameters of each section of the blade spanwise that are precisely matched with the preset target frequency, the stiffness parameters of the three degrees of freedom of the spherical flexible bearing that are matched with the blade characteristics and rotor load, the equivalent stiffness and damping parameters of the damper that meet the oscillation frequency design criteria and damping requirements, the dynamic characteristic parameters of the rotor system under all flight conditions, and all structural design parameters that have passed the static strength and fatigue life verification. This forms a complete design result covering all core components and all design dimensions of the rotor system.

[0051] It should be noted that the final design parameters output in this step are the complete design results after the whole system is coordinated and matched and optimized by multiple fields. They cover all the core aspects of the spherical flexible rotor design and can be directly used for the fine engineering design and manufacturing of spherical flexible rotors without the need for additional parameter debugging and matching, which greatly shortens the engineering implementation cycle of the rotor system.

[0052] Example 2 This embodiment provides a spherical flexible rotor design method, including the following steps: S1: Input parameters; S2: Based on the free wake and LB dynamic stall model, solve the rotor aerodynamic load and induced inflow coefficient, and calculate the flapping, flaring and torsional aerodynamic stiffness; S3: Establish the three-dimensional coupled dynamic equations of the rotor, substitute the blade structural stiffness, initial ball flexible bearing stiffness, damper equivalent stiffness and aerodynamic stiffness, and solve for the first-order flapping, oscillation, torsional frequency and displacement response of the blade. S4: Combine aerodynamic, centrifugal, and deformation loads with the additional centrifugal bending moment at the pre-cone angle to calculate the dynamic stress of the blades and the contact stress of the bearings, and complete the strength check. S5: Based on the deviation between the actual frequency and the target frequency, the triaxial stiffness of the bearing and the equivalent stiffness of the damper are iteratively modified. S6: Determine whether the iteration convergence simultaneously meets the following thresholds: three-dimensional frequency deviation rate ≤ 0.5%, aerodynamic load fluctuation ≤ 2%, dynamic stress deviation ≤ 3%, stiffness correction ≤ 0.3% (reference value); if satisfied, output the design parameters; otherwise, return to step S2 for iterative calculation.

[0053] Preferably, the rotor three-dimensional coupled dynamic equation in step S3 is:

[0054] In the formula: —The blade flapping-shielding-torsional coupling mass matrix, in units of ; —The system damping matrix includes aerodynamic damping, structural damping, bearing damping, and oscillation damper damping, with units of _____. ; —Strength matrix of the blade structure, in units of ; — Rotor aerodynamic stiffness matrix, in units of ; —A matrix consisting of the triaxial stiffness of the ball bearing and the equivalent stiffness of the damper, in units of ; —Generalized displacement vector, where To wave the horn, For the pendulum angle, The angle of twist is expressed in units of 1 / 2. ; The first derivative of the generalized displacement vector. The second derivative of the generalized displacement vector; —Aerodynamic load moment vector, solved based on the free wake model and the LB dynamic stall model, in units of ; — Rotor rotational angular velocity, in units of ; — Rotor-induced inflow coefficient, dimensionless.

[0055] In this embodiment, step S2 includes calculating the dynamic aerodynamic load and centrifugal load of the rotor.

[0056] Optionally, this embodiment provides a blade profile aerodynamic lift. Calculation formula:

[0057] Optionally, this embodiment provides an aerodynamic torsional moment in the blade profile. Calculation formula:

[0058] Optionally, this embodiment provides a blade spanwise centrifugal force load. Calculation formula:

[0059] In the formula: —Standard atmospheric density; —blade span The combined velocity of the cross section is expressed in units of ; — Blade chord length, in units of ; —The slope of the lift line in the blade profile, in units of ; —Zero-lift angle of attack of the propeller blade profile, in units of ; — Blade spanwise linear density, in units of ; — Blade pre-cone angle, in degrees; —The zero-lift moment coefficient of the blade profile is dimensionless; —The derivative of the blade profile moment coefficient with respect to the angle of attack, in units of ; —Real-time angle of attack at blade span r, in units of ; —The derivative of the blade spanwise flapping angle distribution is dimensionless; , —Time derivatives of the swing angle and twist angle, in units of .

[0060] In this embodiment, step S3 includes load-triaxial natural frequency coupling calculation.

[0061] Optionally, this embodiment provides a formula for the first-order waving natural frequency:

[0062] Optionally, this embodiment provides a formula for the natural frequency of a first-order oscillation:

[0063] Optionally, this embodiment provides a formula for the first-order torsional natural frequency:

[0064] In the formula: , , —First-order flapping, oscillating, and torsional natural angular frequencies of the propeller blades, in units of ; , , —Strength of the propeller blade itself during flapping, oscillation, and torsion, in units of ; , , —The blade flapping, oscillating, and torsional aerodynamic stiffness are respectively related to the induced inflow coefficient. Speed ​​ratio Real-time angle of attack Related, unit is ; , , —Swing, oscillation, and torsional stiffness of ball bearings, in units of ; , , —The flapping, oscillating, and torsional moments of inertia of the blades about the hinge point at the root of the blade hub, in units of ; — Rotor tip speed ratio, dimensionless; —The stiffness derivative of the aerodynamic torsional moment with respect to the torsional angle, in units of .

[0065] In this embodiment, step S3 includes frequency-blade profile characteristic mapping calculation. Optionally, this embodiment provides a method for establishing a quantitative mapping relationship between the three-dimensional natural frequencies of the blade and its profile stiffness:

[0066]

[0067]

[0068] In the formula: , —Flapping and oscillating bending stiffness of the blade section at spanwise r, in units of ; — Torsional stiffness of the blade section at spanwise r, in units of ; , , —Dimensionless spanwise correction factor, determined based on the spanwise stiffness distribution characteristics of the blade; , , —The dimensionless normalized stiffness distribution function in the spanwise direction of the blade, with a value range of [0,1].

[0069] In this embodiment, step S4 includes structural strength verification. Optionally, this embodiment provides a total dynamic stress of the blade profile. Calculation formula:

[0070] Optionally, this embodiment provides a strength and fatigue verification criterion:

[0071] In the formula: —Sectional modulus at the blade spanwise r, in units of ( (Corresponding to the directions of waving, swinging, and twisting, respectively). —The cross-sectional deformation curvature at the spanwise r of the blade, in units of ; —Sectional stress caused by aerodynamic loads, in units of ; —The normal stress in the cross section caused by centrifugal load, in units of ; —Additional stress caused by centrifugal bending due to pre-cone angle, in units of ; —Allowable yield stress of blade material, in units of ; —Fatigue safety factor, dimensionless; —The symmetrical cyclic fatigue limit of the blade material, in units of ; —Amplitude of alternating stress in the blade profile, in units of ; —Contact stress of a flexible ball bearing, in units of ; —Maximum stress, in units of .

[0072] In this embodiment, step S5 includes iterative calculations of the ball bearing stiffness and damper frequency design. Preferably, this embodiment provides a three-dimensional stiffness inverse formula for the ball bearing, based on a rigorous algebraic inverse solution of the load-frequency coupling formula, fully encompassing the blade structure stiffness and aerodynamic stiffness, ensuring the accuracy of stiffness calculations.

[0073]

[0074]

[0075] In the formula: , , —The three-dimensional natural angular frequency of the propeller blade, in units of .

[0076] Specifically, in this embodiment, the following is set , ; , ; , Preferably, The three-dimensional stiffness coupling ratio of the ball flexible bearing satisfies... ;in ∈[3,8], ∈[3,8], ∈[0.5,1]. Preferably, =5:5:1. Further, the coupling relationship between aerodynamic stiffness and ball bearing stiffness is derived as follows: , , .

[0077] In this embodiment, step S5 includes calculating the frequency and stiffness of the damper. Preferably, to suppress ground resonance, air resonance, and oscillation instability, the frequency of the oscillation system must avoid the rotor excitation frequency, i.e.: ,in ,and ∈[0.7Ω, 0.85Ω]. The optimal target oscillation frequency for the project is 0.75Ω. Further, preferably, the following calculation is performed:

[0078] In the formula: —Equivalent damping ratio of the oscillation system; —Combined equivalent damping coefficient of the oscillation damper and bearing; —Equivalent stiffness of the gyroscope damper; —Moment of inertia of blade oscillation.

[0079] Specifically, the damping ratio under ground resonance conditions ∈[6%,12%], damping ratio under aerial resonance conditions ∈[4%,8%], the random parameters are adaptively determined.

[0080] In this embodiment, step S5 includes a collaborative iterative correction calculation of the ball bearing stiffness and the damper frequency. Preferably, based on the deviation between the actual frequency and the target frequency, the triaxial stiffness of the bearing and the equivalent stiffness of the damper are iteratively corrected, and the iterative formula is as follows:

[0081]

[0082] In the formula: —The stiffness correction amount in the nth iteration, in units of ; , —Stiffness values ​​for the nth and (n+1)th iterations, in units of ; —The actual angular frequency of the nth iteration, in units of ; —The aerodynamic stiffness correction for the nth iteration, in units of ( These correspond to the directions of waving, swinging, and twisting, respectively.

[0083] In this embodiment, step S6 includes setting an iteration convergence threshold. To ensure design accuracy and engineering reliability, preferably, the following iteration convergence threshold is set: 1. Frequency convergence: The three-dimensional natural frequency deviation rate of the blade is ≤0.5%, and the frequency resonance avoidance criterion is satisfied. ,in ; 2. Load convergence: Rotor aerodynamic load fluctuation ≤2%; 3. Strength convergence: dynamic stress deviation of blade profile ≤3%, fatigue safety factor ≥1.5; 4. Design deviation of oscillation frequency ≤ 0.5%; correction of equivalent stiffness of oscillation damper ≤ 0.3%; damping margin deviation ≤ 10%.

[0084] 5. Stiffness convergence: The correction amount for the triaxial stiffness of the ball flexible bearing and the equivalent stiffness of the damper is ≤0.3%.

[0085] Example 3 This embodiment is based on embodiment 2: This embodiment provides a spherical flexible rotor design method, specifically taking a spherical flexible rotor with a three-bladed layout as the design object, and the implementation steps are as follows.

[0086] Step 1: Input parameters.

[0087] First, import the specific parameters of the blades, as shown in the table below: Table 1 - Blade Material and Structural Parameters

[0088] Furthermore, the parameters for the ball flexible bearing are shown in the table below: Table 2 - Parameters of Ball Flexible Bearings

[0089] Furthermore, through precise calculations, the following dynamic parameters were obtained: Table 3 Dynamic Reference Parameters

[0090] Step 2: Importing Aerodynamic Loads and Induced Inflow Solving Based on the free wake model and LB dynamic stall model, substitute the above-mentioned reference parameters to solve for the resultant velocity of the blade spanwise profile. Aerodynamic lift Aerodynamic torsional torque ; Calculate the centrifugal force load after pre-cone angle correction With a three-bladed symmetrical layout, the aerodynamic loads are superimposed and balanced, and the initial value of the overall aerodynamic load fluctuation is 1.7%, which meets the load convergence threshold requirement.

[0091] Step 3: Initial frequency solution. Substitute the initial structural stiffness of the blade, the initial bearing stiffness, and the aerodynamic stiffness coupling relationship to calculate the three-dimensional initial natural frequencies.

[0092] 1. Initial swing frequency:

[0093] Substitute the initial swing bearing stiffness Calculated If the relative deviation from the target frequency is greater than 0.5%, proceed to iterative optimization.

[0094] 2. Initial frequency of oscillation:

[0095] Substitute the initial bearing stiffness Calculated If the relative deviation from the target frequency is greater than 0.5%, proceed to iterative optimization.

[0096] 3. Initial frequency of torsion:

[0097] Substitute the initial torsional bearing stiffness Calculated If the relative deviation from the target frequency is greater than 0.5%, proceed to iterative optimization.

[0098] Step 4: Structural strength and fatigue check.

[0099] 1. Calculate the total dynamic stress of the blade profile by superimposing aerodynamic loads, structural bending deformation stress, centrifugal normal stress, and additional centrifugal bending stress caused by the pre-cone angle; 2. Precise calculation results of maximum dynamic stress in the blade profile: The allowable limit for materials is ,satisfy ; 3. Fatigue safety factor calculation: It meets the fatigue strength requirements; 4. Calculation of contact stress in ball bearings: This meets the bearing strength requirements.

[0100] Step 5: Based on the deviation between the actual frequency and the target frequency, correct the three-dimensional stiffness of the bearing and the equivalent stiffness of the damper, and optimize the bending and torsional stiffness of the blade profile in reverse.

[0101] 1. Centrifugal force of a single blade Based on engineering experience, axial variation is allowed. =1mm, the axial stiffness of the elastic bearing can be obtained from the shape. =144KN / mm. Furthermore, combining the aerodynamic stiffness coupling relationship, the three-dimensional stiffness of the bearing is solved simultaneously, and the stiffness coupling ratio satisfies... In this embodiment, it is set , ; , ; , Preferably, Furthermore, the coupling relationship between aerodynamic stiffness and ball bearing stiffness is derived as follows: , , By combining the above formulas, the stiffness of the ball bearing can be accurately calculated, i.e. , =1145.92N m / rad =229.18N m / rad.

[0102] 2. Precise calculation of oscillation damper parameters. Target oscillation frequency. , in Within the safe range, the equivalent stiffness of the inverse damper is:

[0103] Substituting the parameters and calculating precisely yields: .

[0104] Step 6: Determine the iterative convergence threshold 1. Three-dimensional stiffness of ball bearing: 1145.92N in the swing direction. m / rad, oscillation direction 1145.92N m / rad, torsional (pitch) direction 229.18N m / rad, stiffness coupling ratio is Furthermore, the stiffness correction amounts are all ≤0.3%, satisfying the convergence threshold.

[0105] 2. Damper parameters: Damper equivalent stiffness Matching the target frequency of the oscillation ; Swing damping ratio It falls within the optimal damping range of ground resonance (6%–12%) and air resonance (7%–8%).

[0106] 3. Maximum dynamic stress in blade profile ≤0.8 Fatigue safety factor ≥1.5; Contact stress of ball bearing ≤1200MPa.

[0107] 4. Three-dimensional natural frequencies of the propeller blades: flapping natural frequency natural frequency of oscillation Twist natural frequency All of them meet the frequency convergence threshold ≤ 0.5% and avoid the rotor excitation frequency, with no risk of resonance.

[0108] 5. Verify convergence: 1. Three-dimensional frequency deviation: waving frequency deviation 0.28%, oscillation frequency deviation 0.25%, and torsional frequency deviation 0.3%, all ≤0.5%; 2. Stiffness correction: The three-dimensional stiffness correction for bearings is ≤0.3%, and the stiffness correction for dampers is ≤0.3%, both ≤0.3%; 3. Load fluctuation: aerodynamic load fluctuation ≤ 2% (1.7%); 4. Dynamic stress deviation: The dynamic stress deviation of the blade profile is 1.8% ≤ 3%.

[0109] All convergence metrics meet the threshold requirements, and the iterative calculation is completely closed-loop.

[0110] In summary, this embodiment, through complete three-field coupling calculation and closed-loop iteration, outputs all design parameters that meet the preset convergence threshold and engineering development requirements, verifying the accuracy, repeatability, and engineering feasibility of the design method of this invention. It can be applied to the design of flexible rotor systems for light helicopters and unmanned helicopters.

[0111] The above description is merely a preferred embodiment of the present invention. It should be understood that the present invention is not limited to the forms disclosed herein and should not be construed as excluding other embodiments. It can be used in various other combinations, modifications, and environments, and can be altered within the scope of the concept described herein through the above teachings or related technologies or knowledge. Modifications and variations made by those skilled in the art that do not depart from the spirit and scope of the present invention should be within the protection scope of the appended claims.

[0112] It should be noted that, for the sake of simplicity, the foregoing method embodiments are described as a series of actions. However, those skilled in the art should understand that this application is not limited to the described order of actions, as some steps may be performed in other orders or simultaneously according to this application. Furthermore, those skilled in the art should also understand that the embodiments described in the specification are preferred embodiments, and the actions and modules involved are not necessarily essential to this application.

Claims

1. A spherical flexible rotor design method, characterized in that, include: Input the initial parameters for the design of the flexible spherical rotor, and solve the basic aerodynamic parameters based on the free wake and LB dynamic stall model; Establish the three-dimensional coupled dynamic equations of blade flapping-shielding-torsion, substitute the dynamic input stiffness parameters, and solve for the first-order natural frequencies and displacement responses of blade flapping, shielding and torsion. By combining aerodynamic, centrifugal, and deformation loads with the additional centrifugal bending moment from the pre-cone angle, the dynamic stress of the blades and the contact stress of the bearings are calculated and the strength is checked. Based on the deviation between the actual frequency and the target frequency, the dynamic input stiffness parameters are iteratively corrected, and the final parameters of the spherical flexible rotor design are output after satisfying the preset convergence threshold. The initial design parameters of the input spherical flexible rotor are obtained by solving the basic aerodynamic parameters based on the free wake and LB dynamic stall model, including: Input the initial design parameters for the spherical flexible rotor, including rotor system configuration, blade geometry, material constitutive properties, initial characteristics of the spherical flexible bearing, initial characteristics of the damper, and initial parameters for the design flight conditions; The induced inflow distribution characteristics of the rotor across the entire span are solved based on the free wake model, capturing the real-time impact of the rotor wake on the aerodynamic environment of the blade profile; the unsteady aerodynamic characteristics of the blade profile are solved by combining the LB dynamic stall model, covering extreme conditions of high-maneuver flight and dynamic stall; the basic aerodynamic parameters are solved simultaneously, including rotor aerodynamic load, induced inflow coefficient, flapping, flaring and torsional aerodynamic stiffness, and can be dynamically updated with real-time blade deformation; The preset convergence threshold includes: The system includes a multi-dimensional collaborative convergence criterion that covers aerodynamic characteristics, dynamic characteristics, structural strength, and design parameter stability. The multi-dimensional collaborative convergence criterion includes the deviation between the blade's three-dimensional natural frequency and the target frequency, the fluctuation of aerodynamic load before and after iteration, the deviation of blade dynamic stress before and after iteration, and the correction of stiffness parameters before and after iteration. The iteration is considered to be converged and the loop calculation is terminated only when all convergence criteria simultaneously meet the preset requirements; if any criterion does not meet the requirements, the full-process coupled iterative calculation continues to ensure that the design results simultaneously meet the multi-dimensional design requirements of aerodynamics, dynamics and strength.

2. The spherical flexible rotor design method according to claim 1, characterized in that, The establishment of the three-dimensional coupled dynamic equations for blade flapping, oscillation, and torsion includes: Using the blade flapping, oscillation, and torsion as generalized displacements, a fully coupled dynamic equation framework is constructed, incorporating the blade coupling mass matrix, the system integrated damping matrix, the blade structural stiffness matrix, the rotor aerodynamic stiffness matrix, and the combined stiffness matrix of the ball flexible bearing and damper. Aerodynamic loads are incorporated into the equations as endogenous variables related to the blade's generalized displacement, rotational angular velocity, and induced inflow coefficient, rather than as fixed exogenous boundary conditions. At the same time, the centrifugal stiffening effect caused by the blade's pre-cone angle is taken into account, forming a dynamic solution equation that deeply couples the aerodynamic field, structural field, and dynamic field, namely, the three-dimensional coupled dynamic equation of blade flapping-shielding-torsion.

3. The spherical flexible rotor design method according to claim 1, characterized in that, The dynamic input stiffness parameters include: The structural stiffness of the blade itself corresponding to the flapping, oscillation, and torsion three-dimensional degrees of freedom, the initial stiffness of the ball flexible bearing corresponding to the three-dimensional degrees of freedom, the equivalent stiffness of the anti-oscillation device, and the blade flapping, oscillation, and torsion aerodynamic stiffness obtained by solution; all kinds of stiffness parameters together constitute the stiffness term of the dynamic equation, which forms a quantitative mapping relationship with the blade's three-dimensional natural frequencies.

4. The spherical flexible rotor design method according to claim 1, characterized in that, The solution for the first-order flapping, teetering, and torsional natural frequencies and displacement responses of the blades includes: Substituting the dynamic input stiffness parameters into the three-dimensional coupled dynamic equations, the effect of centrifugal load generated by rotor rotation on stiffness characteristics is taken into account; the first-order flapping, tessellation and torsion natural frequencies of the blade are obtained by solving, and a quantitative coupling mapping relationship between the natural frequencies and the blade structural stiffness, bearing stiffness, damper stiffness and aerodynamic stiffness is established; the real-time deformation displacement response of the blade in the full spanwise direction under the corresponding flight conditions is obtained by solving simultaneously.

5. The spherical flexible rotor design method according to claim 1, characterized in that, The aerodynamic, centrifugal, and deformation loads, as well as the additional centrifugal bending moment due to the pre-cone angle, include: The aerodynamic loads generated by the unsteady aerodynamic characteristics of the blade profile, the full spanwise centrifugal loads generated by the rotor rotation, the deformation loads generated by the elastic deformation of the blades, and the additional centrifugal bending moment caused by the pre-cone angle of the blades are all dynamically updated based on the real-time deformation displacement response of the blades, while also taking into account the centrifugal load attenuation caused by the pre-cone angle, the effective span correction, and the influence of the additional bending moment at the root.

6. The spherical flexible rotor design method according to claim 1, characterized in that, The calculation of blade dynamic stress and bearing contact stress, and the subsequent strength verification, include: Based on the real-time deformation displacement response of the blade obtained by the solution, the deformation curvature of each section along the full span of the blade is calculated; combined with the full load input, the comprehensive dynamic stress of the blade section and the contact stress of the ball flexible bearing are calculated. Static strength verification is performed based on the allowable stress of blade and bearing materials, and fatigue life verification is performed based on alternating stress amplitude and material fatigue limit. The strength verification results are used as one of the core criteria for iterative convergence judgment, realizing the synergistic coupling of dynamic design and strength design.

7. The spherical flexible rotor design method according to claim 1, characterized in that, The iterative correction of the dynamic input stiffness parameters based on the deviation between the actual frequency and the target frequency includes: By comparing the actual three-dimensional natural frequencies of the blade obtained from the solution with the preset target natural frequencies, the deviation of each frequency is calculated. Based on the frequency deviation and combined with the quantitative mapping relationship between natural frequency and stiffness, the correction of the three-dimensional stiffness of the ball flexible bearing and the equivalent stiffness of the damper is obtained by inverse solution. The change of aerodynamic stiffness during the iteration process is simultaneously taken into account to perform coordinated correction of bearing stiffness and damper stiffness. The corrected stiffness parameters are substituted into the aerodynamic basic parameter solution process to start a new round of full-process coupled calculations, forming a closed-loop iterative mechanism. At the same time, based on the oscillation frequency design criteria, the stiffness and damping characteristics of the oscillation damper are matched to avoid the rotor excitation frequency and suppress the risks of ground resonance, air resonance and oscillation instability.

8. The spherical flexible rotor design method according to claim 1, characterized in that, The final design parameters for the spherical flexible rotor include: The blade spanwise stiffness distribution parameters matched to the target frequency, the stiffness parameters of the ball bearing corresponding to the flapping, oscillation and torsional degrees of freedom, the equivalent stiffness and damping parameters of the damper, the dynamic characteristic parameters of the rotor system under all working conditions, and the structural design parameters verified by static strength and fatigue life.

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