Method and system for simulating dynamic response of three-dimensional tip clearance with cracked blade

Abstract

The application discloses a kind of three-dimensional tip clearance dynamic response simulation analysis method and system of crack-containing blade, comprising: for blade radial vibration, crack is equivalent to blade radial tensile spring, the natural frequency of blade radial vibration and mode equation are calculated;For blade bending vibration, crack is equivalent to torsional spring and blade width / thickness direction tensile spring, the natural frequency of blade bending vibration and the mode equation of displacement and cross-section rotation angle caused by bending are calculated;With equal-section torsional blade with through crack at trailing edge as research object, according to Hamilton principle, combined with Galerkin method, the dynamic model of crack-containing blade is established, the dynamic model is solved and the dynamic response of crack-containing blade tip surface is obtained;According to the geometric relationship between the dynamic response of blade tip surface and the deformation of blade, the three-dimensional tip clearance dynamic response of crack-containing blade is calculated.The method thus provides guidance for aero-engine blade crack fault diagnosis based on three-dimensional tip clearance.

Description

Technical Field

[0001] This invention relates to the technology for analyzing the failure mechanism of cracks in aero-engine blades, specifically to a method and system for simulating and analyzing the dynamic response of three-dimensional tip clearance of a cracked blade. Background Technology

[0002] The aircraft engine is the most critical component in an aircraft structure, serving as the source of its flight power and often referred to as the "heart" of the aircraft. As a core part of the engine's functional transition, the health of the engine blades directly affects its safety and stability. The operating environment of aircraft engine blades is extremely harsh, often subjected to enormous centrifugal loads, aerodynamic loads, thermal loads, and alternating loads caused by vibration. This makes the blades highly susceptible to fatigue cracks and even fracture, seriously jeopardizing the operational safety of the aircraft engine.

[0003] Tip clearance is one of the key parameters of aero-engines. Its size and variation not only affect engine efficiency but also reflect the health status of the blades. Studying the mechanism of tip clearance variation is crucial for aero-engine blade condition monitoring and fault diagnosis. Currently, research on the mechanism of aero-engine tip clearance variation mainly involves establishing simplified mathematical models of tip clearance or using finite element simulation methods. Furthermore, most studies only consider the radial clearance of the blade, which includes relatively limited information about the blade's condition. In reality, when a blade develops cracks, it undergoes bending and torsional deformation in three-dimensional space, resulting in a three-dimensional tip clearance characteristic.

[0004] To study the dynamic characteristics of three-dimensional tip clearance in a cracked blade from a dynamic perspective, it is first necessary to establish a dynamic model of the cracked blade. Currently, aero-engine blades are typically simplified as cantilever beams or thin shells, and the crack is equated to a spring, thus establishing a dynamic model of the cracked blade based on beam theory or plate and shell theory. Although some literature has studied the dynamic characteristics of the cracked blade itself, it has not linked the blade's dynamic characteristics to the dynamic response of the three-dimensional tip clearance, thus lacking research on the dynamic characteristics of three-dimensional tip clearance in aero-engines. Summary of the Invention

[0005] To address the shortcomings and deficiencies in the dynamics aspects of existing analyses of the three-dimensional tip clearance variation mechanism of aero-engines, and the inability of existing dynamic models of cracked blades to analyze the dynamic characteristics of the three-dimensional tip clearance of aero-engines, the present invention aims to provide a simulation analysis method and system for the dynamic response of three-dimensional tip clearance of cracked blades. This method provides theoretical guidance for the diagnosis of aero-engine blade crack faults based on three-dimensional tip clearance.

[0006] To achieve the above objectives, the present invention adopts the following technical solution:

[0007] A three-dimensional blade tip clearance dynamic response simulation analysis method for a cracked blade includes:

[0008] The cracked blade is simplified as a cracked Timoshenko beam. For radial vibration of the blade, the crack is equivalent to a radial tension spring of the blade, and the natural frequency and mode shape equation of the radial vibration of the blade are calculated. For bending vibration of the blade, the crack is equivalent to a torsional spring and a tension spring in the width / thickness direction of the blade, and the natural frequency of bending vibration of the blade and the mode shape equation of the displacement and cross-sectional rotation caused by bending are calculated.

[0009] Taking a torsion blade with a through crack at the trailing edge as the research object, a dynamic model of the blade with cracks is established based on Hamilton's principle and combined with Galerkin's method. The dynamic model is solved and the dynamic response of the blade tip surface with cracks is obtained.

[0010] The three-dimensional dynamic response of the blade tip gap with cracks is calculated based on the dynamic response of the blade tip surface and the geometric relationship before and after blade deformation.

[0011] As a further improvement of the present invention, the cracked blade is simplified to a cracked Timoshenko beam, and for the radial vibration of the blade, the crack is equivalent to a radial tension spring of the blade, and the natural frequency and mode shape equation of the radial vibration of the blade are calculated; including:

[0012] The cracked blade is simplified into a Timoshenko cantilever beam, and the cantilever beam is divided into two segments at the crack.

[0013] For the radial vibration of the blade, the crack is equivalent to a massless tension spring connecting two beam segments; the natural frequency ω of the blade's radial vibration is calculated based on beam vibration theory. 1i Let i represent the natural frequency order, and solve the mode shape equations of the radial vibration of the blades on both sides of the crack using the transfer matrix method. and

[0014]

[0015] In the formula, ξ = x / L, where x is the distance from any section of the cantilever beam to the fixed end, and L is the blade length. E and ρ are the elastic modulus and density of the blade material, respectively; D 1i ~D 4i It is a constant, calculated based on the boundary conditions of the cantilever beam and the compatibility conditions at the crack.

[0016] As a further improvement of the present invention, the method for addressing blade bending vibration is to treat the crack as an equivalent to a torsional spring and a tension spring in the blade width / thickness direction, and to calculate the natural frequency of the blade bending vibration and the mode shape equations of the displacement and cross-sectional rotation caused by bending; including:

[0017] To address the blade's width-direction displacement and cross-sectional rotation caused by blade bending vibration, the crack is equivalent to a massless torsional spring and a tension spring. Based on beam vibration theory, the natural frequency ω of the blade's width-direction bending vibration is calculated. 3i The displacement mode equations of the bending vibrations of the blades on both sides of the crack along the blade width direction are solved using the transfer matrix method. and and the cross-sectional rotation mode equation and

[0018]

[0019]

[0020] In the formula, β 1i β 2i m 1i and m 2i C is a coefficient related to the blade's natural frequency and blade material parameters. 1i ~C 8i It is a constant, determined based on the boundary conditions of the cantilever beam and the compatibility conditions at the crack;

[0021] For the bending vibration in the thickness direction of the blade, its natural frequency ω is calculated according to the vibration theory of beams. 2i The displacement mode equations of the bending vibrations of the blades on both sides of the crack along the thickness direction are solved using the transfer matrix method. and and the cross-sectional rotation mode equation and

[0022]

[0023]

[0024] In the formula, β 3i β 4i m 3i and m 4i B is a coefficient related to the blade's natural frequency and blade material parameters. 1i ~B 8i It is a constant, determined based on the boundary conditions of the cantilever beam and the compatibility conditions at the crack.

[0025] As a further improvement of the present invention, the study focuses on a torsion blade with a penetrating crack at the trailing edge, including:

[0026] A dynamic model is established using a torsion blade with a through-crack at the trailing edge as the research object. The origin O of the global coordinate system XYZ is located at the rotor center, and the Z-axis coincides with the rotor rotation axis; the rotating coordinate system xr y r z r z r The x-axis coincides with the Z-axis. r The axis is perpendicular to the cross-section of the cracked blade. The rotating coordinate system is rotated by an angle θ = Ωt relative to the global coordinate system, where Ω represents the rotor speed. The local coordinate system xyz at the blade root is located at the center of the blade root cross-section, and the x-axis is perpendicular to the x-axis. r The axes are in the same direction, with the y-axis and z-axis perpendicular to the two sides of the blade root section respectively; the installation angle of the blade root section is γ0, and the torsion angle of any section on the blade is γ(x), where x represents the distance from any section of the blade to the blade root.

[0027] As a further improvement of the present invention, the step of establishing a dynamic model of a cracked blade based on Hamilton's principle and the Galerkin method includes:

[0028] Taking a uniform cross-section torsion blade with a penetrating crack at the trailing edge as the research object, this paper considers the influence of radial deformation, bending deformation, shear deformation, rotational effect, and aerodynamic load on the blade. Based on Hamilton's principle, the differential equations of blade motion are derived. The resulting vibration mode equations of the cracked blade are then used as the basis for further analysis. and The Galerkin method was used to discretize the differential equations of blade motion, resulting in a dynamic model of the cracked aero-engine blade.

[0029]

[0030] Where M is the blade mass matrix, G is the blade Coriolis force matrix, C is the blade damping matrix, and K is the blade overall stiffness matrix. These are the generalized displacement vector, generalized velocity vector, and generalized acceleration vector, respectively; F is the external force vector, including the centrifugal load in the radial direction of the blade and the aerodynamic load along the width and thickness directions of the blade.

[0031] As a further improvement of the present invention, the solution of the dynamic model and the obtaining of the dynamic response of the blade tip surface containing the crack includes:

[0032] The Newmark-β numerical algorithm is used to solve the above dynamic model, and the generalized displacement q of the cracked blade is obtained:

[0033] q = [U i (t),V i (t),W i (t),Ψ i (t),Φ i (t)] T i = 1, 2, ..., N

[0034] In the formula, N is the modal cutoff number, U i (t), Vi (t), W i (t), Ψ i (t) and Φ i (t)(i=1,2,…,N) represent the radial displacement, displacement along the blade thickness direction, displacement along the blade width direction, cross-sectional rotation along the blade thickness direction, and cross-sectional rotation along the blade width direction of the cracked blade, respectively. Based on the modal superposition principle, the dynamic response u(t) of the radial displacement of the cracked blade tip surface and the dynamic response u(t) of the cross-sectional rotation along the blade thickness direction are calculated. The dynamic response φ(t) of the cross-sectional rotation angle in the blade width direction.

[0035]

[0036] As a further improvement of the present invention, the calculation of the three-dimensional blade tip clearance dynamic response of the cracked blade based on the dynamic response of the blade tip surface and the geometric relationship before and after blade deformation includes:

[0037] Under the action of external forces, the blades of an aero-engine will deform, and the blade tip surface will deflect accordingly; a spatial rectangular coordinate system u is established with the center of the blade tip surface before deflection as the origin. L v L w L , where the coordinate axis u L Perpendicular to the blade tip surface, v L and w L Located within the blade tip surface; dynamic response based on the cross-sectional rotation angle along the blade thickness direction. By combining the dynamic response φ(t) of the blade width section rotation angle with the geometric relationship of the blade tip surface before and after deflection, the normal vector n1 of the blade tip surface after deflection is solved.

[0038]

[0039] Based on the geometric relationship of the blade tip surface before and after deflection, combined with the twist angle γ of the blade tip surface... L Further establish the following system of equations

[0040]

[0041] Where α(t) and β(t) are the dynamic responses of the blade's axial deflection angle and circumferential slip angle to be solved; f y and f z R represents the unit vector along the circumferential and axial directions of the aero-engine rotor within the blade tip surface, respectively; u (γ L ) indicates revolving around u L Rotate the axis counterclockwise γ L The rotation matrix of degrees; R(f) z ,-β(t)) represents the orbit around fz The rotation matrix for rotating the axis clockwise by β(t) degrees; R(f) y ,α(t)) represents the orbit around f y The rotation matrix for a counterclockwise rotation of the shaft by α(t) degrees is obtained; solving this matrix yields the dynamic response α(t) of the axial deflection angle and the dynamic response β(t) of the circumferential slip angle of the cracked blade.

[0042]

[0043] Based on the obtained dynamic response u(t) of the radial displacement of the blade tip surface, and combined with the initial radial clearance r0, the dynamic response r(t) of the radial clearance is calculated.

[0044] r(t) = r0 - u(t)

[0045] The three-dimensional tip clearance dynamic response r(t), α(t), and β(t) of the cracked blade were then obtained.

[0046] A three-dimensional blade tip clearance dynamic response simulation analysis system for a cracked blade, comprising:

[0047] The module for calculating natural frequencies and mode shapes is used to simplify cracked blades into cracked Timoshenko beams. For radial vibration of the blade, the crack is equivalent to a radial tension spring, and the module calculates the natural frequencies and mode shapes of the radial vibration. For bending vibration of the blade, the crack is equivalent to a torsional spring and a tension spring in the width / thickness direction of the blade, and the module calculates the natural frequencies of the bending vibration and the mode shapes of the displacement and cross-sectional rotation caused by bending.

[0048] The dynamic model solving module is used to study a uniform cross-section torsion blade with a penetrating crack at the trailing edge. Based on Hamilton's principle and combined with the Galerkin method, a dynamic model of the cracked blade is established, the dynamic model is solved, and the dynamic response of the blade tip surface is obtained.

[0049] The three-dimensional blade tip clearance dynamic response calculation module is used to calculate the three-dimensional blade tip clearance dynamic response of a blade with cracks based on the dynamic response of the blade tip surface and the geometric relationship before and after blade deformation.

[0050] Compared with the prior art, the present invention has the following beneficial effects:

[0051] The advantage of this invention is that it can achieve rapid and accurate calculation of the dynamic response of the three-dimensional tip clearance of a cracked blade, which makes up for the deficiency of existing cracked blade dynamic models that can only analyze the dynamic characteristics of the blade itself and cannot obtain the dynamic response of the three-dimensional tip clearance of the aero-engine. It can provide theoretical guidance for the aero-engine blade crack fault diagnosis method based on the three-dimensional tip clearance. Attached Figure Description

[0052] The present invention will be further described in detail below with reference to the accompanying drawings and specific embodiments.

[0053] Figure 1 This is a flowchart of the method of the present invention;

[0054] Figure 2 This is a simplified mechanical model of the cracked blade and its radial vibration in this invention.

[0055] Figure 3 This is a schematic diagram of the equivalent mechanical model of the bending vibration of the cracked blade in this invention;

[0056] Figure 4 This is a schematic diagram of the dynamic model of the cracked blade in this invention;

[0057] Figure 5 This is a schematic diagram illustrating the calculation of three-dimensional blade tip clearance in an aero-engine according to the present invention;

[0058] Figure 6 This is a schematic diagram of the dynamic response simulation analysis system for the three-dimensional tip clearance of a cracked blade in this invention. Detailed Implementation

[0059] To enable those skilled in the art to better understand the present invention, the technical solutions of the present invention will be clearly and completely described below with reference to the accompanying drawings of the embodiments of the present invention. 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 should fall within the scope of protection of the present invention.

[0060] It should be noted that the terms "first," "second," etc., in the specification, claims, and accompanying drawings of this invention are used to distinguish similar objects and are not necessarily used to describe a specific order or sequence. It should be understood that such data can be interchanged where appropriate so that the embodiments of the invention described herein can be implemented in orders other than those illustrated or described herein. Furthermore, the terms "comprising" and "having," and any variations thereof, are intended to cover a non-exclusive inclusion; for example, a process, method, system, product, or apparatus that comprises a series of steps or units is not necessarily limited to those steps or units explicitly listed, but may include other steps or units not explicitly listed or inherent to such processes, methods, products, or apparatus.

[0061] like Figure 1 As shown, the first objective of this invention is to provide a three-dimensional blade tip clearance dynamic response simulation analysis method for blades with cracks, comprising the following steps:

[0062] S1. Simplify the cracked blade into a cracked Timoshenko beam. For the radial vibration of the blade, the crack is equivalent to a radial tension spring of the blade. Calculate the natural frequency and mode shape equation of the radial vibration of the blade. For the bending vibration of the blade, the crack is equivalent to a torsional spring and a tension spring in the width / thickness direction of the blade. Calculate the natural frequency of the bending vibration of the blade and the mode shape equation of the displacement and cross-sectional rotation caused by bending.

[0063] S2. Taking a uniform cross-section torsion blade with a penetrating crack at the trailing edge as the research object, a dynamic model of the cracked blade is established based on Hamilton's principle and combined with Galerkin's method. The dynamic model is solved and the dynamic response of the blade tip surface is obtained.

[0064] S3. Calculate the three-dimensional tip gap dynamic response of the blade with cracks based on the dynamic response of the blade tip surface and the geometric relationship before and after blade deformation.

[0065] Each step includes the following steps:

[0066] (1) The cracked blade is simplified as a Timoshenko cantilever beam, and the cantilever beam is divided into two segments at the crack. For the radial vibration of the blade, the crack is equivalent to a massless tension spring connecting the two segments of the beam. The natural frequency ω of the radial vibration of the blade is calculated based on the vibration theory of beams. 1i (i represents the natural frequency order), and the mode shape equations of the radial vibration of the blades on both sides of the crack are solved using the transfer matrix method. and

[0067]

[0068] In the formula, ξ = x / L, where x is the distance from any section of the cantilever beam to the fixed end, and L is the blade length. E and ρ are the elastic modulus and density of the blade material, respectively. 1i ~D 4i It is a constant and can be calculated based on the boundary conditions of the cantilever beam and the compatibility conditions at the crack.

[0069] (2) Regarding the blade width-direction displacement and section rotation caused by blade bending vibration, the crack is equivalent to a massless torsional spring and a tension spring. Based on beam vibration theory, the natural frequency ω of the blade width-direction bending vibration is calculated. 3i The displacement mode equations of the bending vibrations of the blades on both sides of the crack along the width direction are solved using the transfer matrix method. and and the cross-sectional rotation mode equation and

[0070]

[0071]

[0072] In the formula, β 1i β 2i m 1i and m 2i C is a coefficient related to the blade's natural frequency and blade material parameters. 1i ~C 8i It is a constant and can be determined based on the boundary conditions of the cantilever beam and the compatibility conditions at the crack.

[0073] For the bending vibration in the thickness direction of the blade, its natural frequency ω is calculated according to the vibration theory of beams. 2i The displacement mode equations of the bending vibrations of the blades on both sides of the crack along the thickness direction are solved using the transfer matrix method. and and the cross-sectional rotation mode equation and

[0074]

[0075]

[0076] In the formula, β 3i β 4i m 3i and m 4i This is a coefficient related to the blade's natural frequency and blade material parameters. B 1i ~B 8i It is a constant and can be determined based on the boundary conditions of the cantilever beam and the compatibility conditions at the crack.

[0077] (3) Taking a uniform cross-section torsion blade with a penetrating crack at the trailing edge as the research object, considering the radial deformation, bending deformation, shear deformation, rotational effect, and the influence of aerodynamic loads on the blade, the differential equations of blade motion are derived based on Hamilton's principle. The vibration mode equations of the cracked blade obtained in steps (1) and (2) are used as a basis. and The Galerkin method was used to discretize the differential equations of blade motion, resulting in a dynamic model of the cracked aero-engine blade.

[0078]

[0079] Where M is the blade mass matrix, G is the blade Coriolis force matrix, C is the blade damping matrix, and K is the blade overall stiffness matrix. Let be the generalized displacement vector, generalized velocity vector, and generalized acceleration vector, respectively; and F be the external force vector, including the centrifugal load in the radial direction of the blade and the aerodynamic loads along the width and thickness directions of the blade. The generalized displacement q of the cracked blade can be obtained by solving the above dynamic model using the Newmark-β numerical algorithm.

[0080] q = [U i (t),V i (t),W i (t),Ψ i (t),Φ i (t)] T ,i=1,2,...,N (11)

[0081] In the formula, N is the modal cutoff number, U i (t), V i (t), W i (t), Ψ i (t) and Φ i (t)(i=1,2,…,N) represent the radial displacement, displacement along the blade thickness direction, displacement along the blade width direction, cross-sectional rotation along the blade thickness direction, and cross-sectional rotation along the blade width direction of the cracked blade, respectively. Based on the modal superposition principle, the dynamic response u(t) of the radial displacement of the cracked blade tip surface and the dynamic response u(t) of the cross-sectional rotation along the blade thickness direction can be calculated. The dynamic response φ(t) of the cross-sectional rotation angle in the blade width direction.

[0082]

[0083] (4) Under the action of external forces, the aero-engine blades will deform, and the blade tip surface will deflect accordingly. Establish a spatial rectangular coordinate system u with the center of the blade tip surface before deflection as the origin. L v L w L , where the coordinate axis u L Perpendicular to the blade tip surface, v L and w L Located within the blade tip surface. Dynamic response based on the cross-sectional rotation angle along the blade thickness direction. By combining the dynamic response φ(t) of the blade width section rotation angle with the geometric relationship of the blade tip surface before and after deflection, the normal vector n1 of the blade tip surface after deflection can be solved.

[0084]

[0085] Then, based on the geometric relationship of the blade tip surface before and after deflection, combined with the twist angle γ of the blade tip surface... L The following system of equations can be further established.

[0086]

[0087] Where α(t) and β(t) are the dynamic responses of the blade's axial deflection angle and circumferential slip angle to be solved; f y and f z R represents the unit vector along the circumferential and axial directions of the aero-engine rotor within the blade tip surface, respectively; u (γ L ) indicates revolving around u L Rotate the axis counterclockwise γ L The rotation matrix of degrees; R(f) z ,-β(t)) represents the orbit around f z The rotation matrix for rotating the axis clockwise by β(t) degrees; R(f) y ,α(t)) represents the orbit around f y The rotation matrix is ​​the resultant of the shaft rotating counterclockwise by α(t). Solving equation (19) yields the dynamic response α(t) of the axial deflection angle and the dynamic response β(t) of the circumferential slip angle of the cracked blade.

[0088]

[0089] Finally, based on the dynamic response u(t) of the radial displacement of the blade tip surface obtained in step (3), and combined with the initial radial clearance r0, the dynamic response r(t) of the radial clearance is calculated.

[0090] r(t)=r0-u(t) (21)

[0091] Thus, based on equations (20) to (21), the three-dimensional tip gap dynamic response r(t), α(t), and β(t) of the blade with cracks are obtained.

[0092] The present invention will now be described in detail with reference to the accompanying drawings and specific embodiments.

[0093] Example

[0094] This invention presents a simulation analysis method for the dynamic response of three-dimensional blade tip clearance in a cracked blade. This method provides an in-depth analysis of the impact of blade cracks on the dynamic characteristics of three-dimensional blade tip clearance in aero-engines. It is beneficial for a deeper understanding of the blade crack failure mechanism and can also provide theoretical guidance for the diagnosis of aero-engine blade crack failure based on three-dimensional blade tip clearance.

[0095] Reference Figure 1 This invention relates to a three-dimensional blade tip clearance dynamic response simulation analysis method for a blade with cracks, comprising the following steps:

[0096] (1) The cracked blade is simplified as a cracked Timoshenko cantilever beam. For the radial vibration of the blade, the crack is equivalent to a massless radial tension spring. An equivalent mechanical model of the radial vibration of the cracked blade is established. Then, the natural frequency ω of the radial vibration of the cracked blade is calculated according to the vibration theory of beams and the transfer matrix method.1i The mode shape equations of the radial displacements of the blades on both sides of the crack and

[0097] (2) The cracked blade is simplified as a cracked Timoshenko cantilever beam. According to the theory of mechanics of materials, the bending deformation of the beam can be expressed as the beam's displacement (deflection) and cross-sectional rotation angle. Therefore, for the bending vibration of the blade, the crack is equivalent to a massless torsional spring and a tension spring in the width / thickness direction of the blade. An equivalent mechanical model of the bending vibration of the cracked blade is established. Based on the vibration theory of beams, the natural frequencies ω of the bending vibration in the thickness and width directions of the cracked blade are calculated respectively. 2i and ω 3i Then, using the transfer matrix method, the displacement mode equations for the bending vibrations of the cracked blade along the thickness direction are calculated. and the cross section rotation mode equation And the displacement mode equations for the bending vibration of the cracked blade along the width direction. and the cross section rotation mode equation

[0098] (3) Taking a uniform cross-section torsion blade with a penetrating crack at the trailing edge as the research object, the kinetic energy, potential energy, and work done by external forces on the blade with cracks are calculated. Based on Hamilton's principle, the set of differential equations of motion for the blade with cracks is obtained. Combining the mode shape equations of the cracked blade obtained in steps (1) and (2), the Galerkin method is used to discretize the above differential equations, thereby establishing a dynamic model of the cracked blade. The Newmark-β algorithm is used to solve the dynamic model, obtaining the dynamic response u(t) of the radial displacement of the blade tip surface and the dynamic response of the cross-sectional rotation angle of the blade tip surface along the thickness direction. And the dynamic response φ(t) of the cross-sectional rotation angle of the blade tip surface along the width direction.

[0099] (4) Based on the geometric relationship before and after the deflection of the blade tip surface, and combined with the dynamic response of the blade tip surface obtained in step (3), calculate the three-dimensional blade tip clearance dynamic response of the blade containing the crack, including the dynamic response of the blade tip radial clearance r(t), axial deflection angle α(t) and circumferential slip angle β(t).

[0100] The specific steps of the plan are explained below:

[0101] Reference Figure 2 Calculate the natural frequency and mode shape equation of the radial vibration of the cracked blade in step (1).

[0102] The cracked blade is simplified to a cracked Timoshenko cantilever beam. Figure 2The left end is the fixed end of the cantilever beam, and the right end is the free end. The x, y, and z directions in the rectangular coordinate system represent the radial, thickness, and width directions of the blade, respectively. Considering the radial vibration of the blade, the beam is divided into left and right segments at the crack, and the crack is equivalent to a tension spring. Based on the vibration theory of beams and the transfer matrix method, the overall transfer matrix T for the radial vibration of the blade is... radial It can be represented as

[0103]

[0104] Where T2(ξ) represents the transfer matrix of the cantilever beam at ξ = x / L, x is the distance from any section of the cantilever beam to the fixed end, and T... c2 Let ξ represent the transfer matrix at the crack. c =L c / L,L c Let L be the distance from the crack to the fixed end of the cantilever beam, and L be the blade length. Transfer matrix T radial Each term in the equation is an expression for the natural frequency of the blade. Combining this with the cantilever beam boundary conditions, the frequency equation for the radial vibration of the blade can be obtained.

[0105] T 22 =0 (2)

[0106] Solving equation (2) yields the natural frequencies ω of the radial vibration of the cracked blade. 1i , where i represents the modal order. Therefore, according to the vibration theory of beams, the mode shape equations for the radial vibration of the blades on both sides of the crack can be expressed as:

[0107]

[0108] In the formula, E and ρ are the elastic modulus and density of the blade material, respectively. 1i ~D 4i It is a constant and can be calculated based on the boundary conditions of the cantilever beam and the compatibility conditions at the crack.

[0109] Reference Figure 3 Calculate the natural frequency and mode shape equation of the bending vibration of the cracked blade in step (2).

[0110] Similarly, the cracked blade can be simplified to a cracked Timoshenko cantilever beam. Figure 3 The left end shown is the fixed end of the cantilever beam, and the right end is the free end. Considering the blade width direction ( Figure 3 The bending vibration (in the z-direction shown) divides the beam into two segments at the crack, and the crack is equivalent to a torsional spring and a tension spring. According to the beam vibration theory and the transfer matrix method, the overall transfer matrix T of the bending vibration of the cracked beam is... bending It can be represented as

[0111]

[0112] Where T1(ξ) represents the transfer matrix of the cantilever beam at ξ, T c1 This represents the transfer matrix at the crack. Based on the boundary conditions of the cantilever beam, the frequency equation for the bending vibration of the cantilever beam can be obtained.

[0113]

[0114] In the formula, det represents the determinant of the matrix. Expanding and solving (5) yields the natural frequencies ω of the cracked blade's bending vibration along the width direction. 3i According to beam vibration theory, the displacement mode equation φ for the bending vibration of the blades on both sides of the crack in the width direction is... 3i l (ξ), φ 3i r (ξ) and the cross-sectional rotation mode equation φ 5i l (ξ), φ 5i r (ξ) can be expressed as

[0115]

[0116]

[0117] In the formula, β 1i β 2i m 1i and m 2i C is a coefficient related to the blade's natural frequency and blade material parameters. 1i ~C 8i It is a constant and can be determined based on the boundary conditions of the cantilever beam and the compatibility conditions at the crack.

[0118] For the blade thickness direction ( Figure 3 The bending vibration (as shown in the y-direction) of the crack can be represented by treating the crack as a torsional spring and a tension spring. By repeating the above calculation process, the natural frequencies ω of the bending vibration of the cracked blade along the thickness direction can be obtained. 2i According to beam vibration theory, the displacement mode equation φ for the bending vibration of the blades on both sides of the crack in the width direction is... 2i l (ξ), φ 2i r (ξ) and the cross-sectional rotation mode equation φ 4i l (ξ), φ 4i r (ξ) can be expressed as

[0119]

[0120]

[0121] In the formula, β 3i β 4i m 3i and m 4i This is a coefficient related to the blade's natural frequency and blade material parameters. B 1i ~B 8i It is a constant and can be determined based on the boundary conditions of the cantilever beam and the compatibility conditions at the crack.

[0122] Reference Figure 4 Calculate the vibration response of the cracked blade in step (3).

[0123] A dynamic model is established using a torsion blade with a through-crack at the trailing edge as the research object. (Refer to...) Figure 4 The origin O of the global coordinate system XYZ is located at the rotor center, and the Z-axis coincides with the rotor's rotation axis. Rotating coordinate system x... r y r z r z r The x-axis coincides with the Z-axis. r The axis is perpendicular to the cross-section of the cracked blade. The rotating coordinate system is rotated by an angle θ = Ωt relative to the global coordinate system, where Ω represents the rotor speed. The local coordinate system xyz at the blade root is located at the center of the blade root cross-section, and the x-axis is perpendicular to the x-axis. r The axes are aligned in the same direction, with the y-axis and z-axis perpendicular to the two sides of the blade root section, respectively. The installation angle of the blade root section is γ0, and the torsional angle of any section on the blade is γ(x), where x represents the distance from any section of the blade to the blade root. Based on the above coordinate relationship, the total kinetic energy of the blade can be easily calculated; considering the radial deformation, bending deformation, shear deformation, and rotational effects of the cracked blade, the total potential energy of the blade can be calculated; considering the aerodynamic load F in the blade thickness direction... v and width direction aerodynamic load F w The work done by external forces on the blade can be calculated. Based on Hamilton's principle, the system of differential equations of motion for the cracked blade can be obtained. Using the Galerkin method, the mode shape equations for the radial and bending vibrations of the cracked blade can be combined. as well as Discretize the system of differential equations of motion for the cracked blade to obtain the dynamic model of the cracked blade.

[0124]

[0125] Where M is the blade mass matrix, G is the blade Coriolis force matrix, C is the blade damping matrix, K is the overall blade stiffness matrix, and F is the external force vector. These represent the generalized displacement, velocity, and acceleration vectors, respectively. The Newmark-β numerical algorithm is used to solve the above dynamic model, yielding the generalized displacement q of the cracked blade.

[0126] q = [Ui (t),V i (t),W i (t),Ψ i (t),Φ i (t)] T ,i=1,2,...,N (11)

[0127] In the formula, N is the modal cutoff number. i (t), V i (t), W i (t), Ψ i (t) and Φ i (t)(i=1,2,…,N) represent the radial displacement, displacement along the blade thickness direction, displacement along the blade width direction, cross-sectional rotation along the blade thickness direction, and cross-sectional rotation along the blade width direction for each mode of the cracked blade, respectively. Based on the modal superposition principle, the radial displacement u(t) of the blade tip surface and the cross-sectional rotation along the blade thickness direction can be obtained. Dynamic response of the blade width section rotation angle φ(t)

[0128]

[0129] Reference Figure 5 Calculate the three-dimensional tip gap dynamic response of the cracked blade in step (4).

[0130] The initial position of the blade tip surface of the aero-engine is A. L After the blade deforms, the tip surface deflects, and its position changes to A1. Taking the tip surface A... L Establish a spatial rectangular coordinate system u with the center as the origin. L v L w L coordinate axis u L Perpendicular to A L face, v L and w L The two sides perpendicular to the surface of the blade tip can be represented as follows:

[0131] v L =[0,1,0] T ,w L =[0,0,1] T (13)

[0132] After the blade tip surface deflects, the coordinate axis v L and w L These are transformed into the v1 and w1 axes within plane A1, respectively. Based on geometric relationships, v1 and w1 can be expressed as...

[0133]

[0134] In the formula, Indicates around w L Rotate the axis clockwise The rotation matrix of degrees; R v (φ(t)) represents the revolution around v L The rotation matrix for a counterclockwise rotation of the axis by φ(t) degrees. Assuming the normal vector to surface A1 is n1, then n1 is perpendicular to v1 and w1, from which we can obtain...

[0135]

[0136] According to equations (13) to (15), the normal vector n1 of the blade tip surface A1 after deflection can be calculated.

[0137]

[0138] The coordinate axes Y and Z are located at A L In the plane, pointing to the circumferential and axial directions of the aero-engine rotor respectively, we can obtain the following based on geometric relationships:

[0139] Y = R u (γ L )v L Z = R u (γ L )w L (17)

[0140] In the formula, γ L R is the tip surface twist angle. u (γ L ) indicates revolving around u L Rotate the axis counterclockwise γ L The rotation matrix of degrees. After the blade tip surface is deflected, the coordinate axes Y and Z become the Y1 axis and Z1 axis in the A1 plane, respectively. According to geometric relationships, Y1 and Z1 can be expressed as...

[0141] Y1=R(f z ,-β(t))Y,Z1=R(f y ,α(t))Z (18)

[0142] In the formula, f y and f z Let represent the unit vectors in the Y and Z directions, respectively; α(t) and β(t) are the dynamic responses of the blade's axial deflection angle and circumferential slip angle to be solved; R(f z ,-β(t)) represents the orbit around f z The rotation matrix for rotating the axis clockwise by β(t) degrees; R(f) y ,α(t)) represents the orbit around f yThe rotation matrix for rotating the axis counterclockwise by α(t) degrees. Since Y1 and Z1 are located in plane A1, the normal vectors n1 of Y1 and Z1 perpendicular to plane A1 can be obtained by combining equations (16) to (18).

[0143]

[0144] Solving equation (19) yields the dynamic response α(t) of the axial deflection angle and the dynamic response β(t) of the circumferential slip angle of the cracked blade.

[0145]

[0146] Based on the dynamic response u(t) of the radial displacement of the blade tip surface and the initial blade tip clearance value r0 of the aero-engine, the dynamic response r(t) of the radial clearance of the cracked blade can be calculated.

[0147] r(t)=r0-u(t) (21)

[0148] Thus, based on equations (20) to (21), the three-dimensional tip gap dynamic response r(t), α(t), and β(t) of the blade with cracks are obtained.

[0149] like Figure 6 As shown, the present invention also provides a three-dimensional blade tip clearance dynamic response simulation analysis system for blades with cracks, comprising:

[0150] The module for calculating natural frequencies and mode shapes is used to simplify cracked blades into cracked Timoshenko beams. For radial vibration of the blade, the crack is equivalent to a radial tension spring, and the module calculates the natural frequencies and mode shapes of the radial vibration. For bending vibration of the blade, the crack is equivalent to a torsional spring and a tension spring in the width / thickness direction of the blade, and the module calculates the natural frequencies of the bending vibration and the mode shapes of the displacement and cross-sectional rotation caused by bending.

[0151] The dynamic model solving module is used to study a uniform cross-section torsion blade with a penetrating crack at the trailing edge. Based on Hamilton's principle and combined with the Galerkin method, a dynamic model of the cracked blade is established, the dynamic model is solved, and the dynamic response of the blade tip surface is obtained.

[0152] The three-dimensional blade tip clearance dynamic response calculation module is used to calculate the three-dimensional blade tip clearance dynamic response of a blade with cracks based on the dynamic response of the blade tip surface and the geometric relationship before and after blade deformation.

[0153] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention and not to limit it. Although the present invention has been described in detail with reference to the above embodiments, those skilled in the art should understand that modifications or equivalent substitutions can still be made to the specific implementation of the present invention. Any modifications or equivalent substitutions that do not depart from the spirit and scope of the present invention should be covered within the scope of protection of the claims of the present invention.

Claims

1. A method for simulating dynamic response of a three-dimensional tip clearance gap with a cracked blade, characterized by, include: The cracked blade is simplified as a cracked Timoshenko beam. For radial vibration of the blade, the crack is equivalent to a radial tension spring of the blade, and the natural frequency and mode shape equation of the radial vibration of the blade are calculated. For bending vibration of the blade, the crack is equivalent to a torsional spring and a tension spring in the width / thickness direction of the blade, and the natural frequency of bending vibration of the blade and the mode shape equation of the displacement and cross-sectional rotation caused by bending are calculated. Taking a torsion blade with a through crack at the trailing edge as the research object, a dynamic model of the blade with cracks is established based on Hamilton's principle and combined with Galerkin's method. The dynamic model is solved and the dynamic response of the blade tip surface with cracks is obtained. The three-dimensional dynamic response of the blade tip gap was calculated based on the dynamic response of the blade tip surface and the geometric relationship before and after blade deformation. The method for addressing blade bending vibration treats the crack as an equivalent to a torsional spring and a tension spring along the blade's width / thickness direction, calculating the natural frequency of the blade bending vibration and the mode shape equations for the displacement and cross-sectional rotation caused by bending; including: The crack is equivalent to a massless torsional spring and a tensile spring according to the blade width direction displacement and section rotation angle caused by the blade bending vibration ω 3i The displacement mode equation of the blade bending vibration along the blade width direction on both sides of the crack is solved according to the transfer matrix method and The section rotation angle mode equation is and ​ In the formula, β 1i , β 2i , m 1i and m 2i These are coefficients related to the blade's natural frequency and blade material parameters; C 1i ~ C 8i It is a constant, determined based on the boundary conditions of the cantilever beam and the compatibility conditions at the crack; For the bending vibration in the thickness direction of the blade, its natural frequency is calculated according to the vibration theory of beams. ω 2i The displacement mode equations of the bending vibrations of the blades on both sides of the crack along the thickness direction are solved using the transfer matrix method. and and the cross-sectional rotation mode equation and In the formula, β 3i , β 4i , m 3i and m 4i These are coefficients related to the blade's natural frequency and blade material parameters; B 1i ~ B 8i It is a constant, determined based on the boundary conditions of the cantilever beam and the compatibility conditions at the crack.

2. The three-dimensional tip clearance gap dynamic response simulation analysis method of a cracked blade according to claim 1, characterized in that, The process involves simplifying the cracked blade into a cracked Timoshenko beam, and for radial vibration of the blade, treating the crack as an equivalent radial tension spring, and calculating the natural frequencies and mode shapes of the blade's radial vibration; including: The cracked blade is simplified into a Timoshenko cantilever beam, and the cantilever beam is divided into two segments at the crack. For the radial vibration of the blade, the crack is equivalent to a massless tension spring connecting two beam segments; the natural frequency of the blade's radial vibration is calculated based on beam vibration theory. ω 1i , i The natural frequency order is represented, and the mode shape equations of the radial vibration of the blades on both sides of the crack are solved using the transfer matrix method. and In the formula, ξ = x / L , x Let be the distance from any section of the cantilever beam to the fixed end. L For the blade length, , E and ρ These are the elastic modulus and density of the blade material, respectively. D 1i ~ D 4i It is a constant, calculated based on the boundary conditions of the cantilever beam and the compatibility conditions at the crack.

3. The three-dimensional tip clearance gap dynamic response simulation method of claim 1, wherein, The study focuses on a torsion blade with a penetrating crack at the trailing edge, including: A dynamic model is established using a torsion blade with a through crack at the trailing edge as the research object, with a global coordinate system. XYZ The origin O Located at the center of the rotor, Z The shaft coincides with the rotor's rotation axis; rotating coordinate system x r y r z r of z r shaft and Z Axis coincidence, x r The axis is perpendicular to the cross-section of the cracked blade, and the angle through which the rotating coordinate system rotates relative to the global coordinate system is . θ = Ωt ,in Ω Represents the rotor speed; blade root local coordinate system xyz Located at the center of the blade root section, x shaft and x r The axes are in the same direction. y shaft and z The axes are perpendicular to both sides of the blade root section; the installation angle of the blade root section is... γ The twist angle of any section on the blade is 0. γ ( x ),in x This represents the distance from any cross section of the blade to the blade root.

4. The three-dimensional tip clearance gap dynamic response simulation analysis method of a cracked blade according to claim 1, characterized in that, The dynamic model of the cracked blade, established based on Hamilton's principle and combined with Galerkin's method, includes: Taking a uniform cross-section torsion blade with a penetrating crack at the trailing edge as the research object, this paper considers the influence of radial deformation, bending deformation, shear deformation, rotational effect, and aerodynamic load on the blade. Based on Hamilton's principle, the differential equations of blade motion are derived. The resulting vibration mode equations of the cracked blade are then used as the basis for further analysis. and The Galerkin method was used to discretize the differential equations of blade motion, resulting in a dynamic model of a cracked aero-engine blade. Where M is the blade mass matrix, G is the blade Coriolis force matrix, C is the blade damping matrix, and K is the blade overall stiffness matrix. These are the generalized displacement vector, generalized velocity vector, and generalized acceleration vector, respectively; F is the external force vector, including the centrifugal load in the radial direction of the blade and the aerodynamic load along the width and thickness directions of the blade.

5. The three-dimensional tip clearance gap dynamic response simulation method of claim 1, wherein, The solution of the dynamic model and the acquisition of the dynamic response of the blade tip surface containing the crack include: Using Newmark- β The numerical algorithm solves the above dynamic model to obtain the generalized displacement q of the cracked blade: In the formula, N Let be the modal cutoff number. U i ( t ), V i ( t ), W i ( t ), Ψ i ( t )and Φ i ( t )( i =1,2,…, N The values ​​(x, y) represent the radial displacement, displacement along the blade thickness direction, displacement along the blade width direction, cross-sectional rotation along the blade thickness direction, and cross-sectional rotation along the blade width direction of the cracked blade. Based on the modal superposition principle, the dynamic response of the radial displacement of the cracked blade tip surface is calculated. u ( t Dynamic response of cross-sectional rotation angle in the thickness direction of the blade φ ( t Dynamic response of blade width section rotation angle ( t ) （12）。 6. The three-dimensional tip clearance gap dynamic response simulation analysis method of a cracked blade according to claim 1, characterized in that, The calculation of the three-dimensional tip clearance dynamic response of the cracked blade based on the dynamic response of the tip surface and the geometric relationship before and after blade deformation includes: Under the action of external forces, the blades of an aero-engine will deform, and the blade tip surface will deflect accordingly; a spatial rectangular coordinate system is established with the center of the blade tip surface before deflection as the origin. u L v L w L , where the coordinate axes u L Perpendicular to the leaf tip surface, v L and w L Located within the blade tip surface; dynamic response based on the cross-sectional rotation angle along the blade thickness direction. φ ( t Dynamic response of blade width section rotation angle ( t By combining the geometric relationship of the blade tip surface before and after deflection, the normal vector n1 of the blade tip surface after deflection is solved. （16） According to the geometric relationship before and after the tip surface deflection, combined with the twist angle of the tip surface γ L , the following equation group is further established （19） in, α ( t )and β ( t f represents the dynamic response of the blade's axial deflection angle and circumferential slip angle to be solved; y and f z R represents the unit vector along the circumferential and axial directions of the aero-engine rotor within the blade tip surface, respectively; u ( γ L ) indicates revolving around u L Rotate the axis counterclockwise γ L The rotation matrix of degrees; R(f) z , β ( t )) indicates revolving around f z Rotate the axis clockwise β ( t The rotation matrix of degree R(f) y , α ( t )) indicates revolving around f y Rotate the axis counterclockwise α ( t The rotation matrix of degree is obtained; the dynamic response of the axial deflection angle of the cracked blade is obtained by solving the problem. α ( t ) and circumferential slip angle dynamic response β ( t ) （20） Based on the obtained dynamic response of radial displacement of the blade tip surface u ( t ), combined with the initial radial clearance r 0, Calculate the dynamic response of radial clearance. r ( t ) Further, a three-dimensional tip clearance dynamic response of a blade with a crack is obtained r ( t ), α ( t ) and β ( t ).

7. A system for simulating and analyzing the dynamic response of a three-dimensional tip clearance gap with a cracked blade, based on the method for simulating and analyzing the dynamic response of a three-dimensional tip clearance gap with a cracked blade according to any one of claims 1 to 6, characterized in that, include: The module for calculating natural frequencies and mode shapes is used to simplify cracked blades into cracked Timoshenko beams. For radial vibration of the blade, the crack is equivalent to a radial tension spring, and the module calculates the natural frequencies and mode shapes of the radial vibration. For bending vibration of the blade, the crack is equivalent to a torsional spring and a tension spring in the width / thickness direction of the blade, and the module calculates the natural frequencies of the bending vibration and the mode shapes of the displacement and cross-sectional rotation caused by bending. The dynamic model solving module is used to study a uniform cross-section torsion blade with a penetrating crack at the trailing edge. Based on Hamilton's principle and combined with the Galerkin method, a dynamic model of the cracked blade is established, the dynamic model is solved, and the dynamic response of the blade tip surface is obtained. The three-dimensional blade tip clearance dynamic response calculation module is used to calculate the three-dimensional blade tip clearance dynamic response of a blade with cracks based on the dynamic response of the blade tip surface and the geometric relationship before and after blade deformation.

Images