Design method for free wake model of rotor wing in state space form

Abstract

The invention provides a state space type rotor free wake model design method, which comprises the following steps: firstly, establishing a blade motion model, a rotor aerodynamic force model and a rotor free wake nonlinear model, then establishing a rotor free wake linear time-varying model, and finally designing a rotor free wake model through a harmonic decomposition method and a singular perturbation theory. And finally, establishing a rotor wing free wake state space model, and verifying the established rotor wing free wake state space model. According to the method, the state space form representation of the rotor wing free wake is realized, the calculation precision of the rotor wing wake and the aerodynamic force is ensured, and a universal solution is provided for multi-rotor wing aerodynamic interference analysis and aerodynamic force response prediction.

Description

Technical Field

[0001] This invention relates to the field of rotor aerodynamic performance design, specifically a state-space form rotor free wake model design method. Background Technology

[0002] The aerodynamic characteristics of a rotor are closely related to its wake. The calculation of rotor wakes and aerodynamic forces plays a crucial role in the study of helicopter aerodynamics and flight dynamics. While computational fluid dynamics (CFD) methods based on the Euler perspective can capture detailed flow characteristics such as blade airflow separation, stall, and shock waves, they suffer from numerical vortex dissipation problems, cannot accurately predict the influence of tip vortices, and are computationally time-consuming. Vorticity transport methods (VTM) and viscous vortex particle methods (VVPM) can effectively solve the numerical dissipation problem of CFD methods and are suitable for high-precision wake characteristic studies, but still require significant computational resources. The efficiency issues of these high-precision methods limit their application in flight dynamics modeling. In contrast, free wake models based on linear vortex discretization can achieve relatively high computational efficiency while maintaining a certain level of computational accuracy.

[0003] Multi-rotor electric vertical takeoff and landing (eVTOL) aircraft have seen rapid development in recent years due to their advantages such as environmental friendliness, high efficiency, and flexible takeoff and landing. While the computational complexity of free wake calculations for single-rotor aircraft is acceptable, it is significantly higher for multi-rotor eVTOLs. Figure 1 As shown, the increased number of rotors leads to a sharp increase in computation, which seriously affects the efficiency of solving rotor wakes and aerodynamic forces. Therefore, it is necessary to improve the existing free wake model.

[0004] To improve the computational efficiency of free wake models, scholars both domestically and internationally have conducted extensive research on improvements to single-rotor free wake models. (Reference: BHAGWAT MJ, LEISHMAN J G. Stability Analysis of Helicopter RotorWakes in Axial Flight[J]. Journal of the American Helicopter...) Compared to the predictive corrected second-order backward difference (PC2B) algorithm proposed in Society, 2000, 45(3):165, the computational efficiency of the second-order explicit algorithm (CB2D) proposed by Li Pan (Li Pan. Research on unsteady free trails of rotors and high-confidence helicopter flight mechanics modeling [D]. Nanjing: Nanjing University of Aeronautics and Astronautics, 2010.) and the explicit correction PC2B algorithm developed by Liu Yong (Liu Yong, Shao Song, Zhu Qinghua, et al. Analysis of unsteady aerodynamic characteristics of rotors based on time-accurate free trail method [J]. Acta Aeronautica Sinica, 2012, 33(4):607-616.) is improved by about 1 times. However, these methods are essentially improvements to the difference scheme of the trail control equations, without changing the highly nonlinear nature of the free trail model, and it is difficult to significantly improve the computational efficiency of the free trail model. Therefore, linearization of the free trail model has become an important research direction. Rand (RAND O, KHROMOV V. Free-Wake–Based Dynamic Inflow Model for Hover, Forward, and Maneuvering Flight[J]. Journal of the American Helicopter Society, 2017.) et al. established a low-order linearized finite-state inflow model based on a time-stepped free wake model, utilizing frequency sweeping and frequency domain system identification techniques with finite perturbation input. However, this method is cumbersome and sensitive to the magnitude of the perturbation.Celi (CELIR. State-Space Representation of Vortex Wakes by the Method of Lines[J]. Journal of the American Helicopter Society, 2005, 50(2): 195-205.) based on the University of Maryland free wake model (ASHISH B. Contributions to the mathematical modeling of rotor flow fields using apseudo-implicit free-wake analysis.[D]. University of Maryland, College Park., 1995.), expressed the wake control equations as a system of first-order ordinary differential equations, laying the foundation for the state-space representation of the free wake model. Summary of the Invention

[0005] To address the problems of existing technologies, this invention provides a state-space form rotor free wake model design method, which realizes the state-space form representation of rotor free wake, ensures the calculation accuracy of rotor wake and aerodynamic forces, and provides a general solution for multi-rotor aerodynamic interference analysis and aerodynamic response prediction.

[0006] This invention provides a method for designing a state-space form rotor free wake model, comprising the following steps:

[0007] 1) Establish a nonlinear model of the rotor's free wake:

[0008] 1.1) Establish the blade motion model and rotor aerodynamic model. Assume that the blade only has flapping motion and the blade is rigid. Discretize the blade using blade element theory. Obtain the two-dimensional lift and drag coefficients based on the airfoil aerodynamic data table, and then obtain the blade micro-segment aerodynamic force and attached circulation.

[0009] 1.2) Considering the blade motion equations and the wake control equations, establish a nonlinear model of the rotor's free wake and calculate the rotor aerodynamic forces and wake.

[0010] 2) Establish a linear time-varying model of the rotor free wake: Based on steady flight conditions, calculate the periodic equilibrium state solution of the nonlinear model, and use linearization processing at each discrete azimuth angle to obtain a small disturbance linearized model, and construct a linear time-period LTP model near the periodic equilibrium solution.

[0011] 3) Establish a state-space model of the rotor's free wake:

[0012] 3.1) The harmonic decomposition method is used to expand the state variables, input variables, and output variables of the LTP model into finite-order harmonics at integer multiples of the fundamental frequency of the rotor speed, thereby converting the LTP model into a higher-order linear time-invariant LTI model.

[0013] 3.2) Based on the singular perturbation theory, the zeroth harmonic component of the state variables of the high-order LTI model is selected as the slowly changing state variable, and the non-zero harmonic component is selected as the rapidly changing state variable. The high-order LTI model is reduced in order to establish the rotor free wake state space model.

[0014] 4) Verify the state-space model of the rotor's free wake:

[0015] 4.1) By comparing the induced velocity calculation results with the wind tunnel test results, the accuracy of the steady-state solution of the nonlinear model of the rotor free wake is verified;

[0016] 4.2) Verify the dynamic characteristics of the nonlinear model based on the rotor aerodynamic response results under collective pitch control;

[0017] 4.3) Based on the calculation results of the nonlinear model, compare the accuracy and computational efficiency of the high-order LTI model and the state-space model of the rotor free wake, and verify the accuracy and efficiency of the state-space model of the rotor free wake.

[0018] Further improvements are made, and the process of establishing the blade motion model and rotor aerodynamic model described in step 1.1) is as follows:

[0019] 1.11) Blade Motion Model:

[0020] Assuming the propeller blades only undergo flapping motion and are rigid, then the following equations for flapping motion exist:

[0021]

[0022] In the formula, β is the waving angle, and υ β M is the dimensionless waving frequency. β I is the aerodynamic torque of the propeller blade. β Ω represents the flapping moment of inertia, and Ω represents the rotor speed.

[0023] Considering dimensionless angular velocity The waving equations can be restated as the following system of first-order ordinary differential equations:

[0024]

[0025] 1.12) Rotor aerodynamic model:

[0026] The blades were discretized using blade element theory, and the induced velocities v of each blade profile were calculated using a free wake model. iCalculate the angle of attack α of the blade element profile, and then find the corresponding two-dimensional airfoil lift-drag coefficient C at that angle of attack from a table. l and C d This leads to the calculation of lift resistance and adhesion circulation Γ of the leaf element profile. B :

[0027]

[0028] The aerodynamic force of the rotor is finally obtained by integration.

[0029] Further improvements are made, and the process of establishing the nonlinear model of the rotor free wake in step 1.2) is as follows:

[0030] The free wake is divided into two parts: the near wake and the far wake. The near wake is represented by a vortex lattice model, including the detached vortices and the following vortices inside the blade. The far wake is represented by a single tip vortex filament with concentrated vorticity. The governing equations for the free wake nodes are written in the following partial differential equation form:

[0031]

[0032] Where r is the wake node coordinate vector, ζ is the wake node lifetime angle, and the velocity term v consists of the free flow velocity, the wake-induced velocity, and the wake entrainment velocity caused by the rotor shaft or airframe motion. v is the most computationally expensive part of the free wake model.

[0033] The change in the wake circulation Γ can be expressed in the form of the following partial differential equation:

[0034]

[0035] Using the "straight line method" and a five-point upwind difference scheme, the partial derivatives of the above equations in the lifetime angle direction are discretized:

[0036]

[0037] Moving the lifetime angle partial derivative term, which is replaced by a difference in the left side of the equation, to the right side of the equation, we obtain a set of first-order ordinary differential equations concerning the node coordinates and circulation of the free wake.

[0038] For the tip vortex filament, with respect to its nodal coordinates r TV (ψ,ζ) and circulation Γ TV The equation for (ψ,ζ) is expressed as:

[0039]

[0040] Where, matrix A ζ For finite difference matrices:

[0041]

[0042] The near-wake vortex lattice uses quadrilateral elements, and its boundary conditions are determined by the attached circulation Γ of the rotor blades. B Determine the nodal position r of the near-wake vortex lattice. NW (ψ,ζ) and circulation Γ NW The equation for (ψ,ζ) is also expressed as:

[0043]

[0044] Simultaneously considering the rotor blade motion equation (2) and the wake control equations (7) and (9), a first-order nonlinear model of the rotor free wake is established:

[0045]

[0046] The nonlinear model of the rotor free wake is a nonlinear time-periodic system, and the system state variables are given by the rotor state variable x. R and the tail state quantity x W Common components:

[0047] x=(x R ,x W (11)

[0048] rotor state variables Where, β k It is the flapping angle of the kth blade. It is the flapping angular velocity of the k-th blade, and the wake state variable x. W =(r TV ,Γ TV ,r NW ,Γ NW ,Γ B The circulation is composed of the node positions and circulation of the near-wake vortex grid and the tip vortex filaments, as well as the attached circulation of the blade.

[0049] Control quantity Where, θ0,θ 1c ,θ 1s These are the collective pitch and cyclic pitch of the rotor, u, v, w are the longitudinal, lateral, and vertical velocities defined in the body coordinate system, and p, q, r are... These are the roll, pitch, and yaw angular velocities and angular accelerations defined in the body coordinate system.

[0050] Further improvements are made, and the method for establishing the linear time-varying model of the rotor free wake in step 2) is as follows:

[0051] The nonlinear model of the rotor free wake is expressed as a first-order nonlinear time-periodic NLTP system:

[0052]

[0053] Where x is the system state variable, u is the system control variable, and t is time. In the nonlinear model of the rotor free wake, the reference blade azimuth angle ψ, which is normalized to the rotor speed Ω, is used instead of t.

[0054] During stable flight, the blade motion, rotor aerodynamic forces, and rotor wake are in an approximately periodic equilibrium state. Let x... * (ψ) and u * (ψ) represents the periodic equilibrium solution and control input of the system at the azimuth angle ψ. The original NLTP system is linearized at each discrete azimuth angle:

[0055]

[0056] The coefficient matrix is ​​as follows:

[0057]

[0058] In the formula, the coefficient matrix F∈R n× n, G∈R n×m , P∈R l× n, Q∈R l×m The dimension is determined by the number of state variables n, the number of input variables m, and the number of output variables l.

[0059] Further improvements, the specific process of converting the LTP model into a higher-order linear time-invariant LTI model in step 3.1) is as follows:

[0060] Using the harmonic decomposition method, the state variables x, input variables u, and output variables y of the LTP model are expanded into finite-order harmonic forms at the fundamental frequency:

[0061]

[0062] Differentiating the harmonic expansion of the state variables with respect to time, we get:

[0063]

[0064] Substituting equations (16) and (17) into equation (13), we can obtain the state equations in the harmonic expansion form through the corresponding trigonometric integrals:

[0065]

[0066] Similarly, the output equation for the harmonic expansion form can be used to transform the original LTP model into an approximate higher-order LTI model:

[0067]

[0068] In the higher-order LTI model, the state variables X∈R n(2N+1)Control quantity U∈R m(2M+1) and output quantity Y∈R l(2L+1) for:

[0069]

[0070] This refers to the harmonic components of the state variable x, input variable u, and output variable y of the original LTP model at integer multiples of the rotor speed Ω. The coefficient matrix A∈R n(2N+1)*n(2N+1) , B∈R n(2N+1)*m(2M+1) , C∈R l(2L+1)*n(2N+1) and D∈R l(2L+1)*m(2M+1) Given a constant matrix, solve using the Fast Fourier Transform.

[0071] Further improvements are made, and the specific process for establishing the rotor free wake state space model in step 3.2) is as follows:

[0072] Based on singular perturbation theory, the order of the high-order LTI model of the rotor free wake is reduced to obtain the state space model of the rotor free wake.

[0073] Decompose the state variable X into slowly varying state components Xs. s and fast-changing state components X f :

[0074] X = (X s ,X f )(twenty three)

[0075] The zeroth harmonic component of the state variables in the higher-order LTI model is selected as the slowly varying state variable, and the non-zero harmonic components are selected as the rapidly varying state variables, that is:

[0076]

[0077] Accordingly, the state equations of the higher-order LTI model are rewritten in block form:

[0078]

[0079] By neglecting the dynamic response of the rapidly changing state components, i.e., X f = 0 and perform algebraic operations to derive only from the slowly varying state X s The state-space model is constructed, and its state equations are as follows:

[0080]

[0081] in:

[0082]

[0083] The output equation of the higher-order LTI model in equation (21) is divided into the following blocks:

[0084]

[0085] The derived output equation after residualization is as follows:

[0086]

[0087] in:

[0088]

[0089] Finally, a state-space model of the rotor's free wake is established, and its state-space equations are as follows:

[0090]

[0091] If we now choose to match the output variables with the state variables of the original higher-order LTI model, that is:

[0092] Y = [x0 x] 1c x 1s …x Nc x Ns (32)

[0093] This model will be able to predict the impact of the reduced-order state variables on the zeroth and higher-order harmonic state variables in the original LTI model.

[0094] The beneficial effects of this invention are as follows:

[0095] 1. A state-space representation of the rotor free wake has been implemented. Compared with the traditional time-stepped free wake model, the rotor free wake state-space model significantly improves the solution efficiency of rotor wake and aerodynamic forces, and has high engineering practical value.

[0096] 2. The calculation results of the established rotor free wake state space model are in good agreement with the rotor free wake nonlinear model, ensuring the calculation accuracy of rotor wake and aerodynamic forces.

[0097] 3. The proposed state-space representation method for rotor free wakes is also applicable to multi-rotor aircraft such as eVTOLs. Compared to single-rotor free wakes, multi-rotor free wakes have significantly more state variables, but the state-space representation process for multi-rotor free wakes (linearization, harmonic decomposition, model order reduction) is consistent with that of single-rotor free wakes. The state-space representation method for rotor free wakes proposed in this invention provides a general solution for multi-rotor aerodynamic disturbance analysis and aerodynamic response prediction. Attached Figure Description

[0098] To more clearly illustrate the technical solutions in the embodiments of the present invention, the accompanying drawings used in the embodiments will be briefly introduced below. Obviously, the drawings described below are only some embodiments of the present invention. For those skilled in the art, other drawings can be obtained based on these drawings without creative effort.

[0099] Figure 1 Image of Joby S4 multi-rotor eVTOL;

[0100] Figure 2 Flowchart for simulation calculation of nonlinear model of rotor free wake;

[0101] Figure 3 Flowchart for simulation calculation of rotor free wake state space model;

[0102] Figure 4 This is a radial distribution diagram of the rotor induced velocity in hovering mode.

[0103] Figure 5 The time-averaged induced inflow distribution diagram for a rotor in forward flight mode;

[0104] Figure 6 Time history graph of rotor thrust coefficient;

[0105] Figure 7 A time history diagram of the rotor flapping angle;

[0106] Figure 8 Diagram of total pitch disturbance input;

[0107] Figure 9 A comparison chart of rotor hovering thrust coefficient response;

[0108] Figure 10 A comparison chart of the rotor's forward thrust coefficient response;

[0109] Figure 11 Comparison of rotor hovering induced inflow response;

[0110] Figure 12 Comparison of rotor forward flight induced inflow response;

[0111] Figure 13 A comparison chart showing the time taken by different models with different numbers of wake loops. Detailed Implementation

[0112] The technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of the present invention, and not all embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of the present invention.

[0113] I. Nonlinear Model of Rotor Free Wake:

[0114] The rotor wake is influenced by the blade motion and aerodynamic forces, forming a highly coupled complex nonlinear dynamic system with the blade motion and aerodynamic forces.

[0115] 1.1 Blade Motion Model:

[0116] Assuming the propeller blades only undergo flapping motion and are rigid, then the following equations for flapping motion exist:

[0117]

[0118] In the formula, β is the waving angle, and υ β M is the dimensionless waving frequency. β I is the aerodynamic torque of the propeller blade. β Ω represents the flapping moment of inertia, and Ω represents the rotor speed.

[0119] Considering dimensionless angular velocity The waving equation can be reformulated as the following system of first-order ordinary differential equations:

[0120]

[0121] 1.2 Rotor Aerodynamic Model:

[0122] The blades were discretized using blade element theory, and the induced velocities v of each blade profile were calculated using a free wake model. i The angle of attack α of the blade element profile is calculated accordingly. The corresponding two-dimensional airfoil lift-drag coefficient C at this angle of attack is then obtained from a table. l and C d This leads to the calculation of the lift resistance and adhesion circulation Γ of the leaf element profile. B :

[0123]

[0124] The aerodynamic force of the rotor is finally obtained by integration.

[0125] 1.3 Free Trail Model:

[0126] To accurately simulate the flow field near the blades and improve the accuracy of rotor aerodynamic calculations, this invention divides the free wake into two parts: the near wake and the far wake. The near wake is represented by a vortex lattice model, including the detached vortices and the following vortices inside the blades; the far wake is represented by a single tip vortex filament with concentrated vorticity. The governing equations for the free wake nodes can be written in the following partial differential equation form:

[0127]

[0128] Where r is the wake node coordinate vector, ζ is the wake node lifetime angle, and the velocity term v consists of the free flow velocity, the wake-induced velocity, and the wake entrainment velocity caused by the rotor shaft (or airframe) motion. v is the most computationally expensive part of the free wake model.

[0129] The change in the wake circulation Γ can be written in the form of the following partial differential equation:

[0130]

[0131] Using the "straight line method" and the five-point upwind difference (5PBU4) scheme, the partial derivatives of the above equations in the lifetime angle direction are discretized:

[0132]

[0133] Moving the lifetime angle partial derivative, which is replaced by a difference term on the left side of the equation, to the right side, we obtain a system of first-order ordinary differential equations concerning the node coordinates and circulation of the free wake.

[0134] For the tip vortex filament, with respect to its nodal coordinates r TV (ψ,ζ) and circulation Γ TV The equation for (ψ,ζ) can be expressed as:

[0135]

[0136] Where, matrix A ζ For finite difference matrices:

[0137]

[0138] Near-wake vortex lattices use quadrilateral elements, and their basic equations are similar to those of tip vortices. The boundary conditions for near-wake vortex lattices are determined by the attached circulation Γ of the rotor blades. B Therefore, the nodal position r of the near-wake vortex lattice is determined. NW (ψ,ζ) and circulation Γ NW The equation for (ψ,ζ) can also be expressed as:

[0139]

[0140] Simultaneously considering the rotor blade motion equation (2) and the wake control equations (7) and (9), a first-order nonlinear model of the rotor free wake is established:

[0141]

[0142] It is a non-linear time-varying periodic (NLTP) system. The system's state variables consist of the rotor state variable x. Rand the tail state quantity x W Common components:

[0143] x=(x R ,x W (11)

[0144] rotor state variables Where, β k It is the flapping angle of the kth blade. It is the flapping angular velocity of the k-th blade. The wake state variable x W =(r TV ,Γ TV ,r NW ,Γ NW ,Γ B The vortex filament is composed of the node position and circulation of the near-wake vortex filament and the tip vortex filament, as well as the attached circulation of the blade.

[0145] Control quantity Where, θ0,θ 1c ,θ 1s These are the collective pitch and cyclic pitch of the rotor. u, v, w are the longitudinal, lateral, and vertical velocities defined in the body coordinate system. p, q, r and... These are the roll, pitch, and yaw angular velocities and angular accelerations defined in the body coordinate system.

[0146] Solving the nonlinear model of a rotor free wake requires considering the call sequence problem. This is because the induced velocity at the blade control point determines the local aerodynamic distribution and the attached circulation Γ. B ; and the amount of attached rings Γ B This also serves as a boundary condition for wake generation. To avoid algebraic coupling and improve computational efficiency, the solution process is as follows: First, based on the wake state x at the current time step... W The induced velocity v at the blade control point is solved using the Biot-Savart law. i Subsequently, the derivatives of the rotor state variables are solved using the blade motion equation (2). Attachment Circulation Γ B and rotor aerodynamics; finally, with Γ B As the boundary conditions for the wake model, based on equations (6) and (7), the fourth-order Runge-Kutta method is used to update the wake state variables for the next time step, thus completing the calculation of the rotor free wake for the current time step. The specific process for calculating the nonlinear model of the rotor free wake is as follows: Figure 2 As shown.

[0147] II. Model linearization:

[0148] Nonlinear models of rotor free wakes can be used for rotor wake and aerodynamic calculations, but they are computationally expensive. Therefore, linearization of the free wake model is necessary to improve computational efficiency. The free wake model uses Biot-Savart's law to calculate the induced velocities at all wake nodes and blade control points, achieving a computational complexity of O(N^2) within a single integration step. 2 (N is the number of wake nodes); To ensure the stability of the wake solution, it is usually necessary to use fourth-order Runge-Kutta method or predictor-corrector algorithms, and select a small azimuth step size (generally 5°-15°). The computational cost of the free wake model is acceptable for a single rotor, but for eVTOLs with multi-rotor configurations, the increase in the number of rotors leads to a sharp increase in computational cost. To improve computational efficiency, the nonlinear model needs to be linearized.

[0149] The nonlinear model of a rotor free wake can be expressed as a first-order nonlinear time-varying periodic (NLTP) system. Consider the general form of the NLTP system:

[0150]

[0151] Where x is the system state variable, u is the system control variable, and t is time. In the nonlinear model of the rotor free wake, the reference blade azimuth angle ψ, which is normalized to the rotor speed Ω, is used instead of t.

[0152] During stable flight, the blade motion, rotor aerodynamic forces, and rotor wake are in an approximately periodic equilibrium state. Let x... * (ψ) and u * (ψ) represents the periodic equilibrium solution and control input of the system at the azimuth angle ψ. The original NLTP system is linearized at each discrete azimuth angle:

[0153]

[0154] The coefficient matrix is ​​as follows:

[0155]

[0156] In the formula, the coefficient matrix F∈R n× n, G∈R n×m , P∈R l× n, Q∈R l×mThe dimensions are determined by the number of state variables n, the number of input variables m, and the number of output variables l. The coefficient matrix mentioned above is an explicit function dependent on the discrete azimuth angle ψ and is periodic. Therefore, the above equation provides a linear time-varying periodic (LTP) model of the original NLTP system near its periodic equilibrium solution. In this model, the state variables Δx, input variables Δu, and output variables Δy all represent the perturbations relative to their respective periodic equilibrium solutions. Hereafter, x, u, and y are used to replace Δx, Δu, and Δy.

[0157] III. State-space form of rotor free wake model:

[0158] While the aforementioned LTP system provides a fast method for calculating rotor free wakes, its practical application is limited due to its time-varying nature. Therefore, it is necessary to establish a state-space model of the rotor free wake that not only satisfies the characteristics of LTI systems but also possesses high computational efficiency.

[0159] 3.1 High-order LTI model of rotor free wake:

[0160] Traditional averaging methods neglect harmonic components that reflect higher-order dynamic characteristics of the rotor and wake. To effectively convert the LTP model to the LTI model, this invention employs a harmonic decomposition method, expanding the state variables x, input u, and output y of the LTP model into finite-order harmonic forms at the fundamental frequency (an integer multiple of the rotor speed Ω):

[0161]

[0162] Differentiating the harmonic expansion of the state variables with respect to time, we get:

[0163]

[0164] Substituting equations (16) and (17) into equation (13), we can obtain the state equations in the harmonic expansion form through the corresponding trigonometric integrals:

[0165]

[0166] Similarly, the output equation for the harmonic expansion can also be obtained. Therefore, the original LTP model can be transformed into an approximate higher-order LTI model:

[0167]

[0168] In the higher-order LTI model, the state variables X∈R n(2N+1) Control quantity U∈R m(2M+1) and output quantity Y∈R l(2L+1) for:

[0169]

[0170] This refers to the harmonic components of the state variable x, input variable u, and output variable y of the original LTP model at integer multiples of the rotor speed Ω. The coefficient matrix A∈R n(2N+1)*n(2N+1) , B∈R n(2N+1)*m(2M+1) , C∈R l(2L+1)*n(2N+1) and D∈R l(2L+1)*m(2M+1) Since it is a constant matrix, it can be solved using the Fast Fourier Transform.

[0171] To effectively simulate the dynamic characteristics of the rotor and wake, the state variables need to retain higher-order harmonic components. Taking a four-bladed single rotor as an example, if the blades are discretized into 8 segments, the wake is discretized with a 10° lifespan angle, a 50° near-wake and 3 far-wakes are selected, and the state variables retain the fourth-order harmonic components, the number of state variables in the rotor free wake model reaches as high as 22212. As the harmonic order increases, the number of state variables in the higher-order LTI model increases significantly. An excessively high number of state variables will seriously affect the computational efficiency of the LTI model.

[0172] 3.2 State-space form of rotor free wake model

[0173] The number of state variables in the LTI model depends on the number of state variables in the LTP model and the harmonic order of the retained state variables. The number of state variables in the high-order LTI model of the rotor free wake is too high. In order to improve computational efficiency, it is necessary to reasonably reduce the harmonic order of the state variables.

[0174] This invention, based on singular perturbation theory, reduces the order of a high-order LTI model of a rotor free wake to obtain a state-space model of the rotor free wake. This state-space model not only improves computational efficiency but also avoids predicting high-order harmonic state variables. Furthermore, while eliminating high-order harmonic state variables, it still preserves some of the system's high-order harmonic characteristics. The basic principle of the order reduction method is: when the dynamic response speed of some states of the system is significantly faster than that of other states, the state variable X can be decomposed into slowly varying state components Xi. s and fast-changing state components X f :

[0175] X = (X s ,X f )(twenty three)

[0176] In this invention, the zeroth harmonic component of the state variables in the higher-order LTI model is selected as the slowly varying state variable, and the non-zero harmonic components are selected as the rapidly varying state variables, that is:

[0177]

[0178] Accordingly, the state equations of higher-order LTI models can be rewritten in block form:

[0179]

[0180] By neglecting the dynamic response of the rapidly changing state components (i.e., X) f =0) and perform algebraic operations to derive only from the slowly varying state X s The state-space model is constructed, and its state equations are as follows:

[0181]

[0182] in:

[0183]

[0184] To ensure that the model can reflect not only the influence of the reduced-order state variables on the zeroth harmonic of the output, but also its partial influence on the higher-order harmonics of the output, the higher-order LTI model output equation in equation (21) is divided into the following blocks:

[0185]

[0186] The derived output equation after residualization is as follows:

[0187]

[0188] in:

[0189]

[0190] Finally, a state-space model of the rotor's free wake is established, and its state-space equations are as follows:

[0191]

[0192] Specifically, if we now choose the output variables to be consistent with the state variables of the original higher-order LTI model, that is:

[0193] Y = [x0 x] 1c x 1s …x Nc x Ns (32)

[0194] This model will be able to predict the impact of the reduced-order state variables on the zeroth and higher-order harmonic state variables in the original LTI model.

[0195] The specific process for calculating the rotor wake and aerodynamic forces using the rotor free wake state-space model is as follows: Figure 3 As shown.

[0196] IV. Method Validation:

[0197] The state-space representation method for rotor free wake is validated: First, the accuracy of the nonlinear model of rotor free wake is verified, providing a reliable basis for the validation of the state-space model of rotor free wake. On this basis, the accuracy of the state-space representation method is verified by comparing the rotor aerodynamic response results calculated by the high-order LTI model and the state-space model with the nonlinear model. At the same time, the efficiency of the rotor free wake state-space model established in this invention is verified by comparing the computational efficiency of the nonlinear model, the high-order LTI model and the state-space model.

[0198] 4.1 Verification of the nonlinear model of the rotor's free wake:

[0199] First, the calculated rotor induced velocities in hovering and forward flight states were compared with the wind tunnel test results. The rotor parameters for the hovering wind tunnel test were as follows: blade radius 94.5 cm, chord length 7.6 cm, airfoil NACA0012, blades with no negative torque, and rotor speed 1200 rpm. Five free wakes were recorded. Figure 4 The calculated and experimental results of the radial distribution of the axially induced velocity below the propeller disk are presented. Figure 4 (a) shows z = -0.1R, and (b) shows the longitudinal periodic pitch. It can be seen that the model calculation results agree well with the experimental data. In forward flight mode, rotor wind tunnel test data from the Langley Research Center in the United States were used for verification. The model rotor radius was 2m, the thrust coefficient was 0.0064, the advance ratio was 0.15, and the rotor shaft tilt angle was -3°. Three free wakes were recorded. Figure 5 The calculated and experimental results of the time-averaged induced inflow at a position one chord length above the paddle disk plane are presented. Figure 5 (a) represents the rotor collective pitch, and (b) represents z = -0.3R. It can be seen that the calculated results agree well with the experimental values.

[0200] To verify that the nonlinear model of the rotor free wake can effectively simulate the aerodynamic response of the rotor, the calculated and experimental results of the rotor aerodynamic response under hovering collective pitch step were compared. The experimental model was a three-bladed articulated rotor with a flapping hinge offset of 0m, no pre-torsion, a rotor radius of 5.79m, a blade chord length of 0.2551m, and a rotor speed of 220rpm. The rotor collective pitch varied from 0° to 12° at rates of 200° / s, 48° / s, and 20° / s. Five free wakes were recorded, and the time history of the rotor thrust coefficient and flapping angle under the three states was calculated and compared with the experimental results. Figure 6 The time history of the rotor thrust coefficient is given. Figure 7The time history of the blade flapping angle is shown. The results show that the nonlinear model of rotor free wake established in this invention can effectively simulate the rotor aerodynamic response when the collective pitch increases suddenly. The calculated response change trend is in good agreement with the experimental values. The model can reflect the overshoot of the thrust coefficient and the response time lag when the collective pitch increases suddenly.

[0201] 4.2 Verification of the state-space form rotor free tail rotor model:

[0202] Taking the UH-60 rotor as an example, under steady-state conditions with the thrust coefficient trimmed to 0.006, the aerodynamic response of the rotor to collective pitch input in hover and forward flight states (μ = 0.15, rotor shaft tilt angle -3°) is calculated. This input is a double-step disturbance with an amplitude of 1°, and its waveform is shown below. Figure 8 As shown.

[0203] Using the state-space representation method proposed in this invention, a high-order LTI model of the rotor free wake after harmonic decomposition is obtained, wherein the state and output quantities of the high-order LTI model are retained down to the fourth-order harmonic components. The state-space model of the rotor free wake is obtained after order reduction of the high-order LTI model. The accuracy of the state-space representation method of this invention is verified by comparing the calculation results of the high-order LTI model, the state-space model, and the nonlinear model.

[0204] Figure 9 , Figure 10 The time histories of rotor thrust coefficients are presented for hovering and forward flight states, respectively, using the nonlinear model, the higher-order LTI model, and the state-space model. The results show that the thrust response trends predicted by the higher-order LTI model and the state-space model are in good agreement with the results of the nonlinear model, accurately capturing the thrust changes caused by the double-step input: in hovering state, the dynamic response of the wake geometry does not end after the collective pitch disturbance input ends, and the rotor thrust coefficient shows significant overshoot; in forward flight, the rotor thrust coefficient change amplitude is 0.001, approximately 17% of the thrust coefficient in the trim state, representing a large disturbance. Figure 11 , Figure 12 The induced inflow response histories of the nonlinear model, the higher-order LTI model, and the state-space model are presented for hovering and forward flight states, respectively. Figure 12 (a) represents the average induced inflow, (b) represents the longitudinal average induced inflow, and (c) represents the transverse average induced inflow. As can be seen from the figures, for the first-order induced inflow component, the prediction results of both the higher-order LTI model and the state-space model are in good agreement with the nonlinear model, effectively capturing the hysteresis phenomena in the longitudinal and transverse induced inflow responses during forward flight. For the average induced inflow component, the state-space model response shows some deviation from the nonlinear model results, but it remains within an acceptable range.

[0205] 4.3 Computational Efficiency Analysis:

[0206] Taking the dynamic response calculation of the UH-60 rotor in forward flight as an example, simulation time comparison tests were conducted on a personal computer equipped with an Intel Core™ i7-14650HX and 32GB of memory, comparing the simulation time of nonlinear models, high-order LTI models, and state-space models. The above calculation results are summarized in Table 1.

[0207] Table 1. Computational Efficiency Analysis

[0208]

[0209] The computational complexity of the high-order LTI model and state-space model is approximately O(N²), where N is the number of model states. The calculation method for the free wake of a multi-rotor system is similar to that of a single-rotor system, requiring coupled calculation of the mutual induced velocities between the wakes of each rotor. Compared to a single-rotor free wake, the number of state variables in a multi-rotor free wake is significantly increased. To verify the efficiency of the rotor free wake state-space model established in this invention, the forward flight dynamic response under different rotor wake rotation numbers was calculated. The time consumption comparison results for each model are as follows: Figure 13 As shown.

[0210] In summary, the rotor free wake state-space model significantly improves computational efficiency (approximately 30 times faster than the nonlinear model). The state-space representation process for multi-rotor free wakes (linearization, harmonic decomposition, and model order reduction) is consistent with that for single-rotor models. Therefore, the rotor free wake state-space representation method proposed in this invention significantly improves computational efficiency while maintaining computational accuracy, and is applicable to flight dynamics modeling tasks for multi-rotor aircraft such as eVTOL.

[0211] The various embodiments in this specification are described in a progressive manner. Similar or identical parts between embodiments can be referred to interchangeably. Each embodiment focuses on its differences from other embodiments. In particular, for the device embodiments, the above descriptions are merely preferred embodiments of the present invention. Since they are fundamentally similar to the method embodiments, the descriptions are relatively simple, and relevant parts can be referred to the descriptions of the method embodiments. The above descriptions are merely specific embodiments of the present invention, but the scope of protection of the present invention is not limited thereto. Any variations or substitutions that can be easily conceived by those skilled in the art within the scope of the technology disclosed in the present invention, without departing from the principle of the present invention, should be included within the scope of protection of the present invention. Therefore, the scope of protection of the present invention should be determined by the scope of the claims.

Claims

1. A method for designing a state-space form rotor free wake model, characterized in that... Includes the following steps: 1) Establish a nonlinear model of the rotor's free wake: 1.1) Establish the blade motion model and rotor aerodynamic model. Assume that the blade only has flapping motion and the blade is rigid. Discretize the blade using blade element theory. Obtain the two-dimensional lift and drag coefficients based on the airfoil aerodynamic data table, and then obtain the blade micro-segment aerodynamic force and attached circulation. 1.2) Considering the blade motion equations and the wake control equations, establish a nonlinear model of the rotor's free wake and calculate the rotor aerodynamic forces and wake. 2) Establish a linear time-varying model of the rotor free wake: Based on steady flight conditions, calculate the periodic equilibrium state solution of the nonlinear model, and use linearization processing at each discrete azimuth angle to obtain a small disturbance linearized model, and construct a linear time-period LTP model near the periodic equilibrium solution. 3) Establish a state-space model of the rotor's free wake: 3.1) The harmonic decomposition method is used to expand the state variables, input variables, and output variables of the LTP model into finite-order harmonics at integer multiples of the fundamental frequency of the rotor speed, thereby converting the LTP model into a higher-order linear time-invariant LTI model. 3.2) Based on the singular perturbation theory, the zeroth harmonic component of the state variables of the high-order LTI model is selected as the slowly changing state variable, and the non-zero harmonic component is selected as the rapidly changing state variable. The high-order LTI model is reduced in order to establish the rotor free wake state space model. 4) Verify the state-space model of the rotor's free wake: 4.1) By comparing the induced velocity calculation results with the wind tunnel test results, the accuracy of the steady-state solution of the nonlinear model of the rotor free wake is verified; 4.2) Verify the dynamic characteristics of the nonlinear model based on the rotor aerodynamic response results under collective pitch control; 4.3) Based on the calculation results of the nonlinear model, compare the accuracy and computational efficiency of the high-order LTI model and the state-space model of the rotor free wake, and verify the accuracy and efficiency of the state-space model of the rotor free wake.

2. The method for designing a state-space form rotor free wake model according to claim 1, characterized in that: The process of establishing the blade motion model and rotor aerodynamic model in step 1.1) is as follows: 1.11) Blade Motion Model: Assuming the propeller blades only undergo flapping motion and are rigid, then the following equations for flapping motion exist: In the formula, β is the waving angle, and υ β M is the dimensionless waving frequency. β I is the aerodynamic torque of the propeller blade. β Ω represents the flapping moment of inertia, and Ω represents the rotor speed. Considering dimensionless angular velocity of the swing The waving equations can be restated as the following system of first-order ordinary differential equations: 1.12) Rotor aerodynamic model: The blades were discretized using blade element theory, and the induced velocities v of each blade profile were calculated using a free wake model. i Calculate the angle of attack α of the blade element profile, and then find the corresponding two-dimensional airfoil lift-drag coefficient C at that angle of attack from a table. l and C d This leads to the calculation of lift resistance and adhesion circulation Γ of the leaf element profile. B : The aerodynamic force of the rotor is finally obtained by integration.

3. The method for designing a state-space form rotor free wake model according to claim 1 or 2, characterized in that: The process of establishing the nonlinear model of the rotor free wake in step 1.2) is as follows: The free wake is divided into two parts: the near wake and the far wake. The near wake is represented by a vortex lattice model, including the detached vortices and the following vortices inside the blade. The far wake is represented by a single tip vortex filament with concentrated vorticity. The governing equations for the free wake nodes are written in the following partial differential equation form: Where r is the wake node coordinate vector, ζ is the wake node lifetime angle, and the velocity term v consists of the free flow velocity, the wake-induced velocity, and the wake entrainment velocity caused by the rotor shaft or airframe motion. v is the most computationally expensive part of the free wake model. The change in the wake circulation Γ can be expressed in the form of the following partial differential equation: Using the "straight line method" and a five-point upwind difference scheme, the partial derivatives of the above equations in the lifetime angle direction are discretized: Moving the lifetime angle partial derivative term, which is replaced by a difference in the left side of the equation, to the right side of the equation, we obtain a set of first-order ordinary differential equations concerning the node coordinates and circulation of the free wake. For the tip vortex filament, with respect to its nodal coordinates r TV (ψ,ζ) and circulation Γ TV The equation for (ψ,ζ) is expressed as: Where, matrix A ζ For finite difference matrices: The near-wake vortex lattice uses quadrilateral elements, and its boundary conditions are determined by the attached circulation Γ of the rotor blades. B Confirm the nodal position r of the near-wake vortex lattice. NW (ψ,ζ) and circulation Γ NW The equation for (ψ,ζ) is also expressed as: Simultaneously considering the rotor blade motion equation (2) and the wake control equations (7) and (9), a first-order nonlinear model of the rotor free wake is established: The nonlinear model of the rotor free wake is a nonlinear time-periodic system, and the system state variables are given by the rotor state variable x. R and the tail state quantity x W Common components: x=(x R ,x W )(11) rotor state variables Where, β k It is the flapping angle of the kth blade. It is the flapping angular velocity of the k-th blade, and the wake state variable x. W =(r TV ,Γ TV ,r NW ,Γ NW ,Γ B The circulation is composed of the node positions and circulation of the near-wake vortex grid and the tip vortex filaments, as well as the attached circulation of the blade. Control quantity Where, θ0,θ 1c ,θ 1s These are the collective pitch and cyclic pitch of the rotor, u, v, w are the longitudinal, lateral, and vertical velocities defined in the body coordinate system, and p, q, r are... These are the roll, pitch, and yaw angular velocities and angular accelerations defined in the body coordinate system.

4. The method for designing a state-space form rotor free wake model according to claim 1, characterized in that: The method for establishing the linear time-varying model of the rotor free wake in step 2) is as follows: The nonlinear model of the rotor free wake is expressed as a first-order nonlinear time-periodic NLTP system: Where x is the system state variable, u is the system control variable, and t is time. In the nonlinear model of the rotor free wake, the reference blade azimuth angle ψ, which is normalized to the rotor speed Ω, is used instead of t. During stable flight, the blade motion, rotor aerodynamic forces, and rotor wake are in an approximately periodic equilibrium state. Let x... * (ψ) and u * (ψ) represents the periodic equilibrium solution and control input of the system at the azimuth angle ψ. The original NLTP system is linearized at each discrete azimuth angle: The coefficient matrix is ​​as follows: In the formula, the coefficient matrix F∈R n× n, G∈R n×m , P∈R l× n, Q∈R l×m The dimension is determined by the number of state variables n, the number of input variables m, and the number of output variables l.

5. The method for designing a state-space form rotor free wake model according to claim 4, characterized in that: The specific process of converting the LTP model into a higher-order linear time-invariant LTI model as described in step 3.1) is as follows: Using the harmonic decomposition method, the state variables x, input variables u, and output variables y of the LTP model are expanded into finite-order harmonic forms at the fundamental frequency: Differentiating the harmonic expansion of the state variables with respect to time, we get: Substituting equations (16) and (17) into equation (13), we can obtain the state equations in the harmonic expansion form through the corresponding trigonometric integrals: Similarly, the output equation for the harmonic expansion form can be used to transform the original LTP model into an approximate higher-order LTI model: In the higher-order LTI model, the state variables X∈R n(2N+1) Control quantity U∈R m(2M+1) and output quantity Y∈R l(2L+1) for: This refers to the harmonic components of the state variable x, input variable u, and output variable y of the original LTP model at integer multiples of the rotor speed Ω. The coefficient matrix A∈R n(2N+1)*n(2N+1) , B∈R n(2N+1)*m(2M+1) , C∈R l(2L+1)*n(2N+1) and D∈R l(2L+1)*m(2M+1) Given a constant matrix, solve using the Fast Fourier Transform.

6. The method for designing a state-space form rotor free wake model according to claim 5, characterized in that: The specific process for establishing the rotor free wake state space model in step 3.2) is as follows: Based on singular perturbation theory, the order of the high-order LTI model of the rotor free wake is reduced to obtain the state space model of the rotor free wake. Decompose the state variable X into slowly varying state components Xs. s and fast-changing state components X f : X=(X s ,X f )(23) The zeroth harmonic component of the state variables in the higher-order LTI model is selected as the slowly varying state variable, and the non-zero harmonic components are selected as the rapidly varying state variables, that is: Accordingly, the state equations of the higher-order LTI model are rewritten in block form: By neglecting the dynamic response of the rapidly changing state components, i.e., X f = 0 and perform algebraic operations to derive only from the slowly varying state X s The state-space model is constructed, and its state equations are as follows: in: The output equation of the higher-order LTI model in equation (21) is divided into the following blocks: The derived output equation after residualization is as follows: in: Finally, a state-space model of the rotor's free wake is established, and its state-space equations are as follows:

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