A method for topological optimization of aeroelastic stability of a blade row of a turbomachine

Abstract

The application discloses a kind of impeller mechanical blade row aeroelastic stability topology optimization method, including step one: given the blade mounting angle θ in blade row, blade spacing s, incoming flow velocity U ∞ ;Step two: calculate the mass of blade unit blade height m, mass static moment S α Around elastic axis and mass inertia moment I α , obtain mass matrix step three: the density field ρ of blade two-dimensional section is mapped to the model of three-dimensional entity blade;Step four: establish a cycle for phase angle For each cycle: the dimensionless aerodynamic force coefficient (C Fq ) η ,(C Fα ) η And moment coefficient (C Mq ) η ,(C Mα ) η Corresponding to the current phase angle are calculated by iterative method, obtain aerodynamic matrix and step five: solving objective function and constraint function and its sensitivity;Step six: obtain updated density field ρ k . The technical effect of the application is to optimize the material arrangement of the bending-torsion coupling hollow blade hollow area in the blade row of the impeller machine, so that the blade row can maintain the aeroelastic stability state while being lightweight designed.

Description

Technical Field

[0001] The present invention relates to the technical field of structural topology optimization, and in particular to a topology optimization method for aeroelastic stability of a turbine blade cascade. Background Art

[0002] Turbomachinery cascades are key components of aircraft engines. With the continuous improvement of aircraft engine performance, aerodynamic limit loads are increasing. Under these aerodynamic loads, the hollow blades within the turbomachinery cascade become thinner and longer, reducing their stiffness and increasing the incidence of aeroelastic instability. Aeroelastic instability can quickly damage the cascade, resulting in serious consequences. Therefore, the aeroelastic stability of hollow blades in turbomachinery cascades has become a critical issue in the design and operation of advanced aircraft engines.

[0003] Scholars have conducted extensive research on blade optimization design. Strength, frequency, and aerodynamic efficiency are the main mechanical properties that these studies focus on. These studies use parameter optimization and topology optimization, taking into account mechanical properties such as stress, frequency, and weight, to optimize the material layout of the hollow area of ​​the blade. Most of the above studies did not consider aeroelastic stability during optimization. In terms of aeroelastic stability, structural detuning and aerodynamic detuning are common means to prevent aeroelastic instability of the blade cascade. Some researchers have also improved the aeroelastic stability of the blade through other methods, such as modifying the blade shape, modifying the casing profile, and changing the radial distribution of the installation angle. However, optimization of the hollow structure inside the blade is rarely used to suppress flutter.

[0004] Extensive research has been conducted on aeroelastic stability optimization, but these studies primarily focus on wings and flat panels. Compared to these two research subjects, the aeroelastic stability of turbomachinery blade cascades is more complex. While wings and flat panels are independent entities, a turbomachinery cascade may contain dozens or even tens of blades. Blade movement constantly influences the unsteady aerodynamic forces of adjacent blades and the blade itself, creating aerodynamic coupling between the individual blades in the cascade. This significantly impacts cascade flutter, which can occur even when a single, isolated blade is vibrating stably.

[0005] Based on a two-degree-of-freedom oscillator, REKielb proposed a method to effectively alleviate the flutter of hollow fan blades by adjusting the position, size, and density of the balancing mass in the hollow area. By designing the mass balance structure in the hollow area of ​​the blade, the blade maintains aeroelastic stability within the operating range. This study shows that different material layouts in the hollow area provide more possibilities for improving the aeroelastic stability of the blade. This design method is similar to parameter optimization, and the optimization effect is significant, but the design space is limited and the structural form is single. In order to better utilize the hollow area and obtain innovative designs with better performance, it is necessary to develop a more advanced optimization method with a larger design space.

[0006] In summary, it is necessary to consider the aerodynamic coupling between blades in the optimization design process and form a topology optimization design method for the aeroelastic stability of the impeller machinery cascade. Summary of the Invention

[0007] The object of the present invention is to provide a topology optimization method for aeroelastic stability of a turbine blade cascade to solve the problems raised in the above background technology.

[0008] To achieve the above objectives, the invention provides the following technical solutions: A method for topological optimization of the aeroelastic stability of a turbine blade cascade, the flow chart of which is as follows: Figure 3 As shown, the specific steps include:

[0009] Step 1: Given the blade installation angle θ, blade spacing s, and incoming flow velocity U in the cascade ∞ , and the initial density field ρ = ρ0 in the design domain of the blade section;

[0010] Step 2: Calculate the blade unit mass m and the mass moment S around the elastic axis based on the design domain density field ρ α and mass moment of inertia I α , and get the mass matrix

[0011] Step 3: Map the density field ρ of the two-dimensional blade section to the three-dimensional solid blade model. Based on the current density field, calculate the bending frequency ω of the blade B and torsional frequency ω T , and obtain the stiffness matrix

[0012] Step 4: Create a loop for the phase angle For each loop:

[0013] The dimensionless aerodynamic coefficient force (C Fq ) η ,(C Fα ) η and moment coefficient (C Mq ) η ,(C Mα ) η , and the aerodynamic matrix is ​​obtained and

[0014] Based on the mass matrix obtained in the above steps Stiffness Matrix and aerodynamic array Solve the eigenvalue equation shown in formula (6) to obtain the eigenvalue λ of the cascade at the current phase angle jl =p jl +iωjl , which is the eigenvalue of the lth mode under the jth node diameter. When the real part of the eigenvalue p jl When it is greater than 0, the cascade is in a state of aeroelastic instability, ω jl is the imaginary part of the eigenvalue, which represents the vibration frequency of the blade;

[0015] Step 5: Based on the eigenvalue of the cascade at the node diameter of interest, solve the objective function and constraint function and their sensitivity;

[0016] Step 6: Use the moving asymptote method (MMA) to solve the optimization problem and obtain the updated density field ρ k If the convergence condition is met, the optimization ends; otherwise, let ρ = ρ k And return to step 2.

[0017] Preferably, step one also includes an aeroelastic stability equivalent analysis, specifically:

[0018] A real turbomachinery cascade is modeled as an infinite two-dimensional cascade of identical planar airfoils in a uniform upstream flow: U ∞ is the incoming flow velocity, b R is the blade half-chord length, c is the blade chord length, θ is the installation angle, and s is the blade spacing: Assuming that the motion of the blades is simple harmonic motion with the same amplitude and the phase angle between adjacent blades is constant, the phase angle σ can be expressed as:

[0019]

[0020] Where N is the number of blades in the turbomachinery cascade;

[0021] The blade is modeled as a two-degree-of-freedom oscillator. The heaving and pitching motions are inertially coupled. The bending and torsion coupling blade characteristic section model is used. The meanings of the symbols are shown in the following table:

[0022]

[0023] The heaving motion represents the bending mode, and the pitching motion represents the torsion mode. The blade section at 75% of the blade height is taken as the research object. The characteristic section is composed of the bending and torsion springs K h and K α , and are hung and fixed respectively.

[0024] Preferably, for the bending-torsion coupling blade characteristic cross-section model, the blade motion equation can be expressed as:

[0025]

[0026] Assume that the blade is in simple harmonic oscillation:

[0027]

[0028] in, is the bending amplitude, is the torsional amplitude, p is the real part of the eigenvalue, and ω is the imaginary part of the eigenvalue;

[0029] The Whitehead unsteady aerodynamic theory is used to calculate the aerodynamic forces and moments acting on the blades. In this theory, it is assumed that the fluid is incompressible and inviscid, that the blades do not stall, and therefore the flow always flows along the blade surface. The effects of the blade curvature and thickness are ignored, and that the blades are flat. It is assumed that the blades operate at zero mean angle of incidence, so the mean deflection angle is zero. Based on the above assumptions, the aerodynamic lift and moment can be expressed as:

[0030]

[0031] in is the blade translation velocity caused by vibration, (C Fq ) η and (C Fα ) η is the dimensionless lift coefficient, (C Mq ) η and (C Mα ) η is the dimensionless moment coefficient;

[0032] Substituting equations (3) and (4) into (2), we can obtain the following eigenvalue equation:

[0033]

[0034] The above formula can be written in matrix form:

[0035]

[0036] where x α is the dimensionless static imbalance, r α is the radius of gyration, μ is the mass ratio:

[0037]

[0038] λ jl =p jl +iω jl is the eigenvalue of the lth mode under the jth node diameter. When the real part of the eigenvalue p jl When it is greater than 0, the cascade is in a state of aeroelastic instability, ω jl is the imaginary part of the eigenvalue, which represents the vibration frequency of the blade.

[0039] Preferably, the optimization formula of the aeroelastic stability topology optimization method is also included:

[0040] The topology optimization problem with aeroelastic stability as a constraint can be expressed as:

[0041]

[0042] in is the cell density of the i-th cell in the design domain, ω jl is the frequency of the lth mode at the jth node diameter, N is the number of elements in the design domain, V i is the unit volume of the i-th unit in the design domain, V* is the volume of the solid blade cross section, V f is the body weight ratio, p jl is the real part of the eigenvalue of the lth mode under the jth node diameter; the objective function of the optimization problem is to maximize the blade frequency, h(ρ) is the volume constraint function, and g(ρ) is the aeroelastic stability constraint function, so that the structure will not flutter at the current flow rate. However, the maximization-minimization objective function and the aeroelastic stability constraint function g(ρ) are not differentiable. Therefore, the KS aggregation function is used to approximate the objective function and the constraint function g(ρ). For a set of frequencies {ω jl |j=1,…,J;l=1,2}, the minimum value can be expressed as:

[0043]

[0044] The positive parameter ξ determines the degree of approximation of the approximation function to the minimum value of the original value set. For the set of real parts of eigenvalues ​​{p j |j=1,...,J;l=1,2}, the maximum value can be described as

[0045]

[0046] Preferably, the sensitivity of the objective function and the constraint function is also included:

[0047] Perform sensitivity analysis on the objective function and constraint function. The sensitivity of the objective function is obtained by deriving the KS function in the formula:

[0048] Similarly, the sensitivity of the constraint function g(ρ) can be obtained by differentiating the KS function in the formula:

[0049]

[0050] It can be seen that the key point of these two sensitivities is the frequency ω j and the real part of the eigenvalue These two sensitivities can be obtained from the complex eigenvalue expression shown in formula (5):

[0051]

[0052] and The sensitivity of the objective function can be obtained by taking the derivative of equation (6), which can be simplified as:

[0053]

[0054] The derivative of equation (15) can be expressed as:

[0055]

[0056] is the left eigenvector of matrix R:

[0057] v jl T R = 0 (17)

[0058] Combining equation (16) and equation (17), the following expression can be obtained:

[0059]

[0060] The detailed form of equation (18) is:

[0061]

[0062] The above equation is a complex scalar equation, where is the only unknown complex number. The equation can be split into two real number equations, from which and can be solved respectively. and Substituting into equation (13) and equation (14), the sensitivity of the objective function and the constraint function can be obtained.

[0063] Compared with the prior art, the beneficial effects of the application are: the aeroelastic stability topology optimization method for turbomachinery cascade can optimize the material arrangement of the hollow region of the bend-twist coupled hollow blade in the turbomachinery cascade, so that the cascade can maintain the aeroelastic stability state while being lightweight designed. BRIEF DESCRIPTION OF DRAWINGS

[0064] Figure 1 is a schematic diagram of the cascade model in the embodiment;

[0065] Figure 2 is a schematic diagram of the bend-twist coupled blade feature section in the embodiment;

[0066] Figure 3 is a flow chart of the aeroelastic stability topology optimization in the embodiment;

[0067] Figure 4 is a schematic diagram of the hollow region and design domain of the blade in the embodiment;

[0068] Figure 5 Schematic diagram of the iterative process of the gas-elastic stability optimization problem in the embodiment;

[0069] Figure 6 Schematic diagram of a hollow blade designed by aeroelastic stability topology optimization method in an embodiment;

[0070] Figure 7 Schematic diagram of the aeroelastic stability of the optimized cascade and the cascade at various nodal diameters under the initial density field in the embodiment. DETAILED DESCRIPTION

[0071] The following will clearly and completely describe the technical solutions in the embodiments of the present invention in conjunction with the accompanying drawings. Obviously, the described embodiments are only part of the embodiments of the present invention, not all of the embodiments. Based on the embodiments of the present invention, all other embodiments obtained by ordinary technicians in this field without making creative efforts are within the scope of protection of the present invention.

[0072] See also Figure 1-7 The invention provides a technical solution: a method for topological optimization of the aeroelastic stability of a turbine blade cascade, which specifically includes the following steps:

[0073] Step 1: Given the blade installation angle θ, blade spacing s, and incoming flow velocity U in the cascade ∞ , and the initial density field ρ = ρ0 in the design domain of the blade section;

[0074] Step 2: Calculate the blade unit mass m and the mass moment S around the elastic axis based on the design domain density field ρ α and mass moment of inertia I α , and get the mass matrix

[0075] Step 3: Map the density field ρ of the two-dimensional blade section to the three-dimensional solid blade model. Based on the current density field, calculate the bending frequency ω of the blade B and torsional frequency ω T , and obtain the stiffness matrix

[0076] Step 4: Create a loop for the phase angle Where N is the number of blades in the turbomachinery cascade, for each cycle:

[0077] The dimensionless aerodynamic coefficient force (C Fq ) η ,(C Fα ) η and moment coefficient (C Mq ) η ,(CMα ) η , and the aerodynamic matrix is ​​obtained and

[0078] Based on the mass matrix obtained in the above steps Stiffness Matrix and aerodynamic array Solve the eigenvalue equation shown in formula (6) to obtain the eigenvalue λ of the cascade at the current phase angle jl =p jl +iω jl , which is the eigenvalue of the lth mode under the jth node diameter. When the real part of the eigenvalue p jl When it is greater than 0, the cascade is in a state of aeroelastic instability, ω jl is the imaginary part of the eigenvalue, which represents the vibration frequency of the blade;

[0079] Step 5: Based on the eigenvalue of the cascade at the node diameter of interest, solve the objective function and constraint function and their sensitivity;

[0080] Step 6: Use the moving asymptote method (MMA) to solve the optimization problem and obtain the updated density field ρ k If the convergence condition is met, the optimization ends; otherwise, let ρ = ρ k And return to step 2.

[0081] Preferably, step one also includes an aeroelastic stability equivalent analysis, specifically:

[0082] A real turbomachinery cascade is modeled as an infinite two-dimensional cascade of identical planar airfoils in an upstream uniform flow as Figure 1 Cascade model shown: U ∞ is the incoming flow velocity, b R is the blade half-chord length, c is the blade chord length, θ is the installation angle, and s is the blade spacing: Assuming that the motion of the blades is simple harmonic motion with the same amplitude and the phase angle between adjacent blades is constant, the phase angle σ can be expressed as:

[0083]

[0084] Where N is the number of blades in the turbomachinery cascade;

[0085] The blade is modeled as a two-degree-of-freedom oscillator, and the heaving and pitching motions are inertially coupled, as shown in Figure 2 The characteristic cross-section model of the bending-torsion coupling blade is shown in the following table:

[0086]

[0087] The heaving motion represents the bending mode, and the pitching motion represents the torsion mode. The blade section at 75% of the blade height is taken as the research object. The characteristic section is composed of the bending and torsion springs K h and K α , and are hung and fixed respectively.

[0088] Preferably, for the bending-torsion coupling blade characteristic cross-section model, the blade motion equation can be expressed as:

[0089]

[0090] Assume that the blade is in simple harmonic oscillation:

[0091]

[0092] in is the bending amplitude, is the torsional amplitude, p is the real part of the eigenvalue, and ω is the imaginary part of the eigenvalue.

[0093] The Whitehead unsteady aerodynamic theory is used to calculate the aerodynamic forces and moments on the blades. In this theory, the fluid is assumed to be incompressible and inviscid, the blade is assumed to not stall, so the flow always flows along the blade surface, the influence of the blade curvature and thickness is ignored, the blade is assumed to be a flat plate, and the blade is assumed to operate under zero mean incidence, so the average deflection is zero. Based on the above assumptions, the aerodynamic lift and moment can be expressed as:

[0094]

[0095] in is the blade translation velocity caused by vibration, (C Fq ) η and (C Fα ) η is the dimensionless lift coefficient, (C Mq ) η and (C Mα ) η is the dimensionless moment coefficient;

[0096] Substituting equations (3) and (4) into equation (2), we can obtain the following eigenvalue equation:

[0097]

[0098] The formula can be written in matrix form:

[0099]

[0100] where x α is the dimensionless static imbalance, r α is the radius of gyration, μ is the mass ratio:

[0101]

[0102] λ jl =p jl +iω jl is the eigenvalue of the lth mode under the jth node diameter. When the real part of the eigenvalue p jl When it is greater than 0, the cascade is in a state of aeroelastic instability, ω jl is the imaginary part of the eigenvalue, which represents the vibration frequency of the blade.

[0103] Preferably, the optimization formula of the aeroelastic stability topology optimization method is also included:

[0104] The topology optimization problem with aeroelastic stability as a constraint can be expressed as:

[0105]

[0106] in is the cell density of the i-th cell in the design domain, ω jl is the frequency of the lth mode at the jth node diameter, N is the number of elements in the design domain, V i is the unit volume of the i-th unit in the design domain, V* is the volume of the solid blade cross section, V f is the body weight ratio, p jl is the real part of the eigenvalue of the lth mode under the jth node diameter; the objective function of the optimization problem is to maximize the blade frequency, h(ρ) is the volume constraint function, and g(ρ) is the aeroelastic stability constraint function, so that the structure will not flutter at the current flow rate. However, the maximization-minimization objective function and the aeroelastic stability constraint function g(ρ) are not differentiable. Therefore, the KS aggregation function is used to approximate the objective function and the constraint function g(ρ). For a set of frequencies {ω jl |j=1,...,J;l=1,2}, the minimum value can be expressed as:

[0107]

[0108] The positive parameter ξ determines the degree of approximation of the approximation function to the minimum value of the original value set. For the set of real parts of eigenvalues ​​{p j |j=1,...,J;l=1,2}, the maximum value can be described as

[0109]

[0110] Preferably, the sensitivity of the objective function and the constraint function is also included:

[0111] Perform sensitivity analysis on the objective function and constraint function. The sensitivity of the objective function is obtained by deriving the KS function in the formula:

[0112] Similarly, the sensitivity of the constraint function g(ρ) can be obtained by differentiating the KS function in the formula:

[0113]

[0114] It can be seen that the key point of these two sensitivities is the frequency ω j and the real part of the eigenvalue These two sensitivities can be obtained from the complex eigenvalue expression shown in formula (5):

[0115]

[0116] and The sensitivity of can be obtained by taking the derivative of formula (6), which can be simplified as:

[0117]

[0118] The derivative of formula (15) can be expressed as:

[0119]

[0120] is the left eigenvector of the matrix R:

[0121] v jl T R=0 (17)

[0122] Combining equations (16) and (17), we can obtain the following expression:

[0123]

[0124] The detailed form of formula (18) is:

[0125]

[0126] The above formula is a complex scalar equation, where is the only unknown complex quantity. The equation can be split into two real equations, which can be solved separately. and Will and Substituting into equations (13) and (14), we can obtain the sensitivity of the objective function and constraint function.

[0127] Test verification: Taking unit density as the design variable, the material layout design of the blade hollow area is carried out. Figure 4As shown in the figure, the design domain of the blade is located between 28% and 88% of the blade height and 10% and 80% of the chord length. The thickness of the hollow blade skin is 20% of the blade thickness. The blade cross section is modeled using 1mm quadrilateral elements. The solid blade is modeled using 1mm hexahedral elements.

[0128] The aeroelastic stability topology optimization method is used to optimize the material layout of the hollow area of ​​the blade. The iterative process of the objective function and constraint function of the optimization problem is as follows: Figure 5 As the optimization process progresses, the topology optimization converges, achieving the goal of maximizing the structural frequency while satisfying aeroelastic stability and volume constraints. The iterative process shows that, under the initial uniform density field, the cascade is aeroelastically unstable at the 4th, 5th, and 6th nodal diameters. With the gradual optimization of the material layout, the material layout in the hollow area is redesigned, and the cascade is aeroelastically stable at all nodal diameters.

[0129] The optimization results are as follows Figure 6 As shown in Figure 2, the material distribution in the hollow area of ​​the blade shows that most of the material is distributed in the hollow area in the form of ribs to resist shear and torsional deformation, while a small amount of material is distributed on the suction and pressure sides of the blade to resist bending deformation.

[0130] Figure 7 The aeroelastic stability results of the initial density field and the optimized design are shown. The flutter problem mainly occurs in the torsional branch, while the curved branch is always in an aeroelastic stable state. For the torsional branch, in the initial density field, the blade is in an aeroelastic instability state at the 4th, 5th, and 6th node diameters. The blade designed using aeroelastic stability topology optimization is in an aeroelastic stable state at all node diameters. These results verify the effectiveness of the aeroelastic stability topology optimization method proposed in this paper. The aeroelastic stability topology optimization method proposed in this paper can be used to find a material layout that is conducive to the aeroelastic stability of the blade.

[0131] The technical effect is: the impeller machinery blade cascade aeroelastic stability topology optimization method can optimize the material arrangement of the hollow area of ​​the bending-torsion coupled hollow blades in the impeller machinery blade cascade, so that the blade cascade can maintain an aeroelastic stable state while being lightweight in design.

[0132] Obviously, those skilled in the art may make various changes and modifications to the present invention without departing from the spirit and scope of the present invention. Thus, if such changes and modifications fall within the scope of the claims and their equivalents, the present invention is intended to include such changes and modifications.

Claims

1. A topology optimization method for aeroelastic stability of a turbomachinery cascade, characterized by: The specific steps include: Step 1: Given the blade installation angle in the cascade , blade spacing , incoming flow velocity , and the initial density field in the blade cross-section design domain ; Step 2: Based on the design domain density field Calculate the unit leaf mass , mass moment of static about the elastic axis and mass moment of inertia , and get the mass matrix ; Step 3: Density field of the two-dimensional section of the blade Mapped onto a 3D solid blade model; based on the current density field, calculate the blade's bending frequency and torsional frequency , and obtain the stiffness matrix ; Step 4: Create a loop for the phase angle , where N is the number of blades in the turbomachinery cascade, for each cycle: Calculate the dimensionless aerodynamic coefficient force corresponding to the current phase angle through the iterative method , and moment coefficient , , and the aerodynamic matrix is ​​obtained and , specifically: ； Based on the mass matrix obtained in the above steps , stiffness matrix and aerodynamic array , , the solution Eigenvalue equation, get the eigenvalue of the cascade at the current phase angle , which is the Section diameter The eigenvalue of the order mode, when the real part of the eigenvalue When it is greater than 0, the cascade is in a state of aeroelastic instability. is the imaginary part of the eigenvalue, which represents the vibration frequency of the blade; Step 5: Based on the eigenvalue of the cascade at the node diameter of interest, solve the objective function and constraint function and their sensitivity; Step 6: Use the moving asymptote method (MMA) to solve the optimization problem and obtain the updated density field ; If the convergence condition is met, the optimization ends; otherwise, And return to step 2.

2. The method for topological optimization of aeroelastic stability of a turbomachinery cascade according to claim 1, characterized in that: Step 1 also includes an equivalent analysis of aeroelastic stability, specifically: A real turbomachinery cascade is modeled as an infinite 2D cascade of identical planar airfoils in an upstream uniform flow: is the incoming flow velocity, is the half chord length of the blade, is the blade chord length, is the installation angle, is the blade spacing: Assuming that the motion of the blades is simple harmonic motion with the same amplitude, the phase angle between adjacent blades is constant, and the phase angle It can be expressed as: （1） in is the number of blades in the turbomachinery cascade; The blade is modeled as a two-degree-of-freedom oscillator, the heaving and pitching motions are inertially coupled, and the bending and torsion coupling blade characteristic section model; The heaving motion represents the bending mode, and the pitching motion represents the torsional mode; The blade section at 75% of the blade height is taken as the research object. The characteristic section is composed of bending and torsion springs. and , and hung and fixed respectively.

3. The method for topological optimization of aeroelastic stability of a turbomachinery cascade according to claim 2, characterized in that: The characteristic cross-section model of the bending-torsion coupling blade, the motion equation of the blade can be expressed as: （2） Assume that the blade is in simple harmonic oscillation: （3） The Whitehead unsteady aerodynamic theory is used to calculate the aerodynamic forces and moments on the blades. In this theory, the fluid is assumed to be incompressible and inviscid, the blade is assumed to not stall, so the flow always flows along the blade surface, the influence of the blade curvature and thickness is ignored, the blade is assumed to be a flat plate, and the blade is assumed to operate under zero mean incidence, so the average deflection is zero. Based on the above assumptions, the aerodynamic lift and moment can be expressed as: （4） in is the blade translational velocity caused by vibration, and is the dimensionless lift coefficient, and is the dimensionless moment coefficient; Substituting equations (3) and (4) into equation (2), we can obtain the following eigenvalue equation: （5） The formula can be written in matrix form: (6) in is the dimensionless static imbalance quantity, is the radius of rotation, is the mass ratio: （7）。 4. The method for topological optimization of aeroelastic stability of a turbomachinery cascade according to claim 1, characterized in that: It also includes the optimization formula of aeroelastic stability topology optimization method: The topology optimization problem with aeroelastic stability as a constraint can be expressed as: （8） in is the cell density of the ith cell in the design domain, It is Section diameter The frequency of the first mode, is the number of elements in the design domain, is the cell volume of the i-th cell in the design domain, is the volume of the solid blade cross section, is the body ratio, It is Section diameter The real part of the eigenvalue of the order mode; the objective function of the optimization problem is to maximize the blade frequency, is the volume constraint function, is the aeroelastic stability constraint function, which prevents the structure from fluttering at the current flow rate. However, the maximization-minimization objective function and the aeroelastic stability constraint function It is not differentiable, so the KS aggregation function is used to approximate the objective function and constraint function. , for a set of frequencies , the minimum value can be expressed as: （9） The positive parameter Determines the degree of approximation of the approximate function to the minimum value of the original value set, and the set of real parts of the eigenvalues , the maximum value can be described as (10).

5. The method for topological optimization of aeroelastic stability of a turbomachinery cascade according to claim 1, characterized in that: It also includes sensitivity solutions for objective and constraint functions: Perform sensitivity analysis on the objective function and constraint function. The sensitivity of the objective function is obtained by deriving the KS function in the formula: （11） Similarly, for the constraint function The sensitivity can be obtained by taking the derivative of the KS function in the formula: （12) It can be seen that the key point of these two sensitivities is frequency and the real part of the eigenvalue These two sensitivities can be obtained from the complex eigenvalue expression shown in formula (5): (13) (14) and The sensitivity of can be obtained by taking the derivative of formula (6), which can be simplified as: (15) The derivative of formula (15) can be expressed as: (16) is a matrix The left eigenvector of : (17) Combining equations (16) and (17), we can obtain the following expression: (18) The detailed form of formula (18) is: (19) The above formula is a complex scalar equation, where is the only unknown complex quantity. Equation (19) can be split into two real equations, from which we can solve and ,Will and Substituting into equations (13) and (14), we can obtain the sensitivity of the objective function and constraint function.

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