A method for fault characterization and dynamics analysis of a rotating airfoil variable cross-section crack blade

Abstract

The application discloses a rotating wing type variable cross-section crack blade fault characterization and dynamics analysis method, belongs to the field of rotating wing type variable cross-section crack blade fault characterization and dynamics analysis, and comprises the following steps: according to the finite element theory and the plate shell theory, based on centrifugal rigidification, rotation softening and the coriolis force effect, a motion differential equation of the wing type variable cross-section blade is established; according to the motion differential equation of the wing type variable cross-section blade, based on the crack existing in the blade, the breathing effect between crack surfaces is equivalent by adopting a spring unit, a rotating wing type variable cross-section crack blade dynamics model containing the breathing effect is established, vibration energy theory is adopted to convert the displacement response into energy response, based on the rotating crack blade crack diagnosis index with the breathing effect, the crack characteristics of the blade are obtained, and the application is more effective and accurate in analyzing the rotating crack blade fault, and can provide theoretical guidance for detection and diagnosis of the rotating crack blade.

Description

A method for fault characterization and dynamic analysis of cracked airfoil blades Technical Field

[0001] This invention belongs to the field of fault characterization and dynamic analysis of cracked blades with variable cross-section of rotating airfoil, and relates to a method for fault characterization and dynamic analysis of cracked blades with variable cross-section of rotating airfoil. Background Technology

[0002] Blades, as one of the most important components of rotating machinery, are widely used in modern mechanical equipment such as aero engines and gas turbines. Rotating blades are constantly operating in extreme environments such as high temperature, high pressure, and high speed. In addition, they are subject to alternating loads, aerodynamic loads, and foreign object damage, making them highly susceptible to high-cycle fatigue, which affects their service life, leading to cracks or even breakage, thus impacting the performance of the rotating machinery. Blade breakage can cause engine failure, hindering mission completion, or even result in serious accidents.

[0003] For the dynamic analysis of cracked blades, many researchers have simplified them to cantilever beam models. Compared to cantilever beam models, flat plate models are closer to the actual blade structure. However, due to the complex and thin-walled structure of aero-engine blades, simplification to cantilever beam and flat plate models is flawed. Based on this, some scholars have also conducted extensive research on the vibration characteristics of actual blades using three-dimensional finite element models in ANSYS software. Although blade modeling has been widely studied, finite element modeling of aero-engine blades with complex cross-sections is still necessary to simultaneously consider computational efficiency and solution accuracy.

[0004] Fault diagnosis and detection of cracked blades is a crucial component of rotating machinery research. Timely detection of fault signals and appropriate measures can prevent serious accidents. Vibration response-based crack fault diagnosis has been extensively studied, primarily focusing on the nonlinear characteristics of rotating cracked blades based on displacement response, velocity response, and frequency domain analysis. However, when small cracks exist in the structure, the system stiffness remains almost unchanged, making it difficult to collect or even detect fault signals based on time or frequency domain analysis. Vibration is essentially the conversion and transfer of energy, and research on the diagnosis and detection of rotating cracked blades based on vibration energy characteristics is scarce. To address these issues, it is essential to research and design a novel dynamic simulation model for rotating airfoil variable cross-section cracked blades incorporating a breathing effect. This approach aims to resolve the problems existing in current rotating airfoil variable cross-section cracked blade modeling methods and dynamic analysis techniques. Summary of the Invention

[0005] To address the shortcomings of existing dynamic analysis techniques for rotating airfoil cracked blades, such as the lack of research on the dynamic characteristics of airfoil-shaped variable cross-section cracked blades with breathing effects and the lack of energy indicators for diagnosing cracked blade faults, this invention provides a technical solution: a method for fault characterization and dynamic analysis of rotating airfoil-shaped variable cross-section cracked blades, comprising the following steps:

[0006] Based on finite element theory and plate and shell theory, and taking into account centrifugal stiffening, rotational softening and Coriolis force effects, the motion differential equations of the airfoil variable cross section blade are established.

[0007] Based on the differential equation of motion of the airfoil variable cross section blade, and considering the existing cracks in the blade, a spring element is used to represent the breathing effect between the crack surfaces, and a dynamic model of the rotating airfoil variable cross section cracked blade with breathing effect is established.

[0008] By employing vibration energy theory, the displacement response is converted into an energy response, and the structural characteristics of the blade crack are obtained based on the crack diagnosis index of the rotating crack blade with breathing effect.

[0009] Furthermore, this method is applicable to the dynamic analysis of a single crack.

[0010] Furthermore, based on finite element theory and plate and shell theory, and considering the effects of centrifugal stiffening, rotational softening, and Coriolis force, the differential equations of motion for the airfoil variable cross-section blade are established as follows:

[0011] Using the isoparametric element concept, the coordinates (x, y, z) of any point within the blade shell element can be represented as follows:

[0012]

[0013] In the formula, ξ and η are the node coordinates in the local coordinate system, and N i (ξ, η) are the nodal shape functions in the local coordinate system, expressed as:

[0014]

[0015] In equation (1), x i m y i m With z i m Let V be the coordinates of node i on the midface in the global coordinate system. 3i Let be the vector from the bottom to the top of node i, that is, the normal to the face of node i, and its expression is:

[0016]

[0017] In the formula, x i t y i t and z i t x represents the coordinates of the top of the shell element corresponding to the mid-surface node i; i b y i b and z i b Let i be the coordinates of the bottom of the shell element corresponding to the mid-surface node i;

[0018] S1.2 To obtain the unit normal vectors of each node on the mid-surface of the blade, based on the mid-surface node coordinates obtained in step S1.1, a fifth-order polynomial is used to fit the mid-surface of the blade. Its expression is:

[0019]

[0020] In the formula, P ij (i, j = 0, ..., 5) are the coefficients of the polynomial;

[0021] Based on the unit normal vector of each node and the thickness of each node obtained in S1.2, the displacement of any point in the shell element can be expressed as:

[0022]

[0023] In the formula, t i Let u be the thickness of node i. i v i w i Let α be the displacement of node i along the x, y, and z axes; i β i γ i The normals v of the mid-surface node i are respectively 3i Around two mutually perpendicular orthogonal vectors v 2i v 1i The corner and v 3i Angle of rotation in the global coordinate system; l 3i m 3i n 3i For V 3i unit vector v 3i The direction cosine;

[0024] Based on the large deformation nonlinear plate and shell theory, and considering centrifugal stiffening, rotational softening, and Coriolis force effects, the differential equations of motion for the airfoil variable cross-section blade can be obtained in S1.4. The strain components of the element are:

[0025]

[0026]

[0027] In the formula, ε x ε y ε z For linear strain; γ xy γ yz γ zx For shear strain;

[0028] The relationship between strain ε and stress σ is:

[0029] σ=[σ x σ y σ z τ xy τ yz τ zx ] T =Dε (8)

[0030] In the formula, σ x σ y σ z For normal stress, τ xy τ yz τ zx D is the shear stress; D is the elastic matrix;

[0031] Considering the centrifugal stiffening, rotational softening, and Coriolis force effects of the blade, the differential equation of motion for the airfoil variable cross-section blade can be obtained as follows:

[0032]

[0033] In the formula, M is the mass matrix, G is the Coriolis force matrix, C is the Rayleigh damping matrix, and K is the mass matrix. e K is the structural stiffness matrix. c K is the centrifugal stiffening matrix. s Let F be the rotation softening matrix. c F is the centrifugal force vector. p Let q be the aerodynamic vector. These are displacement, velocity, and acceleration vectors, respectively.

[0034] Furthermore, the process of establishing a dynamic model of a rotating airfoil with a variable cross-section cracked blade, based on the differential equation of motion of the airfoil blade and the presence of cracks in the blade, using spring elements to represent the breathing effect between the crack surfaces, is as follows:

[0035] To simulate the periodic opening and closing of the cracked surface over time, a breathing model of a rotating blade with a penetrating crack is constructed using spring elements. The crack-free surface is constructed with shared nodes, while the nodes of the cracked surface are connected via spring elements. A contact state function for the node pairs is established based on the node displacements.

[0036]

[0037] Among them, K breathing It is a breathing matrix, K spring It is the spring stiffness, z g It is the normal displacement of node g, z j It is the normal displacement of node j;

[0038] Based on the breathing effect of cracked blades, the dynamic model of a rotating airfoil with variable cross-section cracked blade is as follows:

[0039]

[0040] In the formula, M is the mass matrix, G is the Coriolis force matrix, C is the Rayleigh damping matrix, and K is the mass matrix. e K is the structural stiffness matrix. c K is the centrifugal stiffening matrix. s K is the rotation softening matrix. breathing It is the breathing matrix, F c F is the centrifugal force vector. p Let q be the aerodynamic vector. These are displacement, velocity, and acceleration vectors, respectively.

[0041] Furthermore, the process for obtaining the diagnostic index for the rotating cracked blade with breathing effect is as follows:

[0042] S4.1 converts the displacement response into an energy response, the expression of which is:

[0043]

[0044] Where V represents strain energy, q represents nodal displacement at the tip of the cracked blade, and K represents the overall stiffness matrix.

[0045] S4.2 Based on the energy response, first determine the breathing state of the blade, whether it is an open or closed crack; then determine the extreme points of all energy responses under the open crack state, and then determine the maximum value P2 among these extreme points; similarly, the maximum value P1 of all energy responses under the closed crack state can be obtained. Therefore, the difference between the extreme point P1 of the open crack and the extreme point P2 of the closed crack energy response within one cycle is used to characterize the crack fault, thereby constructing a new fault diagnosis index δ for rotating cracked blades with breathing effect, the expression of which is:

[0046] δ=P1-P2(13)

[0047] Where P1 is the maximum response value corresponding to the closed crack state, and P2 is the maximum response value corresponding to the open crack state. P1 and P2 are both extreme points and maximum / minimum points.

[0048] A device for fault characterization and dynamic analysis of a rotating airfoil variable cross-section cracked blade includes:

[0049] Module I: This module is used to establish the differential equations of motion for airfoil-shaped variable cross-section blades based on finite element theory and plate and shell theory, and on the effects of centrifugal stiffening, rotational softening, and Coriolis force.

[0050] Module II: This module is used to establish a dynamic model of a rotating airfoil with a variable cross section crack, based on the differential equation of motion of the airfoil blade and the presence of cracks in the blade, by using spring elements to represent the breathing effect between the crack surfaces.

[0051] The module is used to convert displacement response into energy response using vibration energy theory, and to obtain the structural characteristics of blade cracks based on the crack diagnosis index of rotating cracked blades with breathing effect.

[0052] A computer-readable storage medium storing a computer program, wherein when the computer program is executed, it performs a learning resource recommendation method for a method of fault characterization and dynamic analysis of a rotating airfoil variable cross-section cracked blade.

[0053] This invention provides a method for fault characterization and dynamic analysis of a rotating airfoil with variable cross-section cracks. It proposes a dynamic analysis method for rotating airfoil with variable cross-section cracks, incorporating the breathing effect. This method considers the breathing effect between crack surfaces in the airfoil with variable cross-section cracks and establishes a dynamic model of the rotating blade considering breathing penetration cracks using shell element theory. Then, the displacement response is converted into an energy response using the strain energy formula, and further, crack fault diagnosis indicators are proposed. This method has high solution efficiency, fills the gap in dynamic modeling and analysis of rotating cracked blades considering the breathing effect, and provides support for subsequent dynamic characteristic analysis of rotating cracked blades considering the breathing effect.

[0054] This invention presents a novel dynamic response analysis model for a rotating airfoil with a variable cross-section cracked blade exhibiting a breathing effect, based on finite shell element theory and the vibration energy method. First, the differential equations of motion for the rotating airfoil with a variable cross-section cracked blade are established using shell element theory. Then, considering the breathing effect between the crack surfaces of the cracked blade, the breathing effect is equivalently represented by a linear spring, thus establishing a dynamic model for the rotating airfoil with a breathing effect. Next, the vibration energy method is used to convert the displacement response into an energy response, and based on the energy response, a new index for diagnosing cracked blade faults is proposed. This invention is more effective and accurate in analyzing rotating cracked blade faults and can provide theoretical guidance for the detection and diagnosis of rotating cracked blades. Attached Figure Description

[0055] To more clearly illustrate the technical solutions in the embodiments of the present invention or the prior art, the drawings used in the description of the embodiments or the prior art will be briefly introduced below. Obviously, the drawings described below are some embodiments of the present invention. For those skilled in the art, other drawings can be obtained based on these drawings without creative effort.

[0056] Figure 1 is a schematic diagram of an aero-engine blade;

[0057] Figure 2(a) is a schematic diagram of the geometry of an 8-node shell element, and (b) is a schematic diagram of the surface coordinate system of the shell element.

[0058] Figure 3(a) is a schematic diagram of the breathing model of the cracked blade; (b) is a partial schematic diagram of the breathing model of the cracked blade.

[0059] Figure 4(a) is a schematic diagram of the stiffness assembly of the cracked blade structure, and (b) is a schematic diagram of the stiffness assembly of the spring unit of the cracked blade.

[0060] Figure 5(a) is the time-domain waveform of the displacement response in the superharmonic resonance state of the blade tip node; (b) is the spectrum of the displacement response in the superharmonic resonance state of the blade tip node; (c) is the time-domain waveform of the displacement response in the first-order resonance state of the blade tip node; and (d) is the spectrum of the displacement response in the first-order resonance state of the blade tip node.

[0061] Figure 6 is a flowchart of the calculation of the index δ;

[0062] Figure 7(a1) is the time-domain waveform of the displacement of the γ=0 cracked blade, (a2) is the energy response of the γ=0 cracked blade, and (a3) ​​is the energy-displacement curve of the γ=0 cracked blade.

[0063] (b1) is the time-domain waveform diagram of the displacement of the γ=0.1 cracked blade, (b2) is the energy response diagram of the γ=0.1 cracked blade, and (b3) is the energy-displacement curve diagram of the γ=0.1 cracked blade.

[0064] (c1) is the time-domain waveform diagram of the displacement of the γ=0.2 cracked blade, (c2) is the energy response diagram of the γ=0.2 cracked blade, and (c3) is the energy-displacement curve diagram of the γ=0.2 cracked blade.

[0065] (d1) is the time-domain waveform diagram of the displacement of the γ=0.3 cracked blade, (d2) is the energy response diagram of the γ=0.3 cracked blade, and (d3) is the energy-displacement curve diagram of the γ=0.3 cracked blade.

[0066] (e1) is the time-domain waveform diagram of the displacement of the γ=0.4 cracked blade, (e2) is the energy response diagram of the γ=0.4 cracked blade, and (e3) is the energy-displacement curve diagram of the γ=0.4 cracked blade.

[0067] (f1) is The time-domain waveform of the cracked blade displacement, (f2) is The energy response diagram of the cracked blade, (f3) is Energy-displacement curve of cracked blade;

[0068] Figure 8 shows the trend of the index δ as the crack depth ratio changes;

[0069] Figure 9(a) shows the energy response spectrum of the superharmonic resonance state of the cracked blade at different crack depths; (b) shows the energy response spectrum of the first-order resonance state of the cracked blade at different crack depths.

[0070] Figure 10(a) is a comparison of the amplitude of the 0fr frequency under different crack depth ratios; (b) is a comparison of the amplitude of the 1fr frequency under different crack depth ratios; and (c) is a comparison of the amplitude of the 2fr frequency under different crack depth ratios. Detailed Implementation

[0071] It should be noted that, unless otherwise specified, the embodiments and features in the embodiments of the present invention can be combined with each other. The present invention will be described in detail below with reference to the accompanying drawings and embodiments.

[0072] To make the objectives, technical solutions, and advantages of the embodiments of the present invention clearer, 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. The following description of at least one exemplary embodiment is merely illustrative and is in no way intended to limit the present invention or its application or use. 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.

[0073] It should be noted that the terminology used herein is for the purpose of describing particular embodiments only and is not intended to limit the scope of exemplary embodiments according to the invention. As used herein, the singular form is intended to include the plural form as well, unless the context clearly indicates otherwise. Furthermore, it should be understood that when the terms "comprising" and / or "including" are used in this specification, they indicate the presence of features, steps, operations, devices, components, and / or combinations thereof.

[0074] Unless otherwise specifically stated, the relative arrangement, numerical expressions, and values ​​of the components and steps described in these embodiments do not limit the scope of the invention. It should also be understood that, for ease of description, the dimensions of the various parts shown in the drawings are not drawn to actual scale. Techniques, methods, and devices known to those skilled in the art may not be discussed in detail, but where appropriate, such techniques, methods, and devices should be considered part of the specification. In all examples shown and discussed herein, any specific values ​​should be interpreted as merely exemplary and not as limitations. Therefore, other examples of exemplary embodiments may have different values. It should be noted that similar reference numerals and letters in the following figures denote similar items; therefore, once an item is defined in one figure, it need not be further discussed in subsequent figures.

[0075] In the description of this invention, it should be understood that the orientation or positional relationship indicated by directional terms such as "front, back, up, down, left, right", "horizontal, vertical, horizontal" and "top, bottom" is generally based on the orientation or positional relationship shown in the accompanying drawings, and is only for the convenience of describing this invention and simplifying the description. Unless otherwise stated, these directional terms do not indicate or imply that the device or element referred to must have a specific orientation or be constructed and operated in a specific orientation, and therefore should not be construed as a limitation on the scope of protection of this invention. The directional terms "inner" and "outer" refer to the inner and outer contours relative to the outline of each component itself.

[0076] For ease of description, spatial relative terms such as "above," "over," "on the upper surface of," "above," etc., are used herein to describe the spatial positional relationship of a device or feature as shown in the figures to other devices or features. It should be understood that spatial relative terms are intended to encompass different orientations in use or operation besides the orientation of the device as described in the figures. For example, if the device in the figures is inverted, a device described as "above" or "above" other devices or structures would subsequently be positioned as "below" or "under" other devices or structures. Thus, the exemplary term "above" can include both "above" and "below." The device may also be positioned in other different ways (rotated 90 degrees or in other orientations), and the spatial relative descriptions used herein will be interpreted accordingly.

[0077] Furthermore, it should be noted that the use of terms such as "first" and "second" to define components is merely for the purpose of distinguishing the corresponding components. Unless otherwise stated, the above terms have no special meaning and therefore should not be construed as limiting the scope of protection of this invention.

[0078] A method for fault characterization and dynamic analysis of a rotating airfoil variable cross-section cracked blade, applicable to the dynamic analysis of a single crack, comprising the following steps:

[0079] S1: Based on the finite element theory and plate and shell theory, and based on the effects of centrifugal stiffening, rotational softening and Coriolis force, the motion differential equation of the airfoil variable cross section blade is established.

[0080] S2: Based on the motion differential equation of the airfoil variable cross section blade, and based on the existing cracks in the blade, a dynamic model of a rotating airfoil variable cross section cracked blade with breathing effect is established by using spring elements to represent the breathing effect between the crack surfaces.

[0081] S3: Using vibration energy theory, the displacement response is converted into an energy response. Based on the crack diagnosis index of rotating cracked blades with breathing effect, the structural characteristics of the blade crack are obtained. The structural characteristics of the blade crack include the crack depth and the crack width.

[0082] Steps S1, S2, and S3 are executed sequentially.

[0083] Figure 2(a) is a schematic diagram of the geometry of an 8-node shell element, and (b) is a schematic diagram of the surface coordinate system of the shell element.

[0084] Based on finite element theory and plate and shell theory, and considering the effects of centrifugal stiffening, rotational softening, and Coriolis force, the following differential equations of motion for the airfoil variable cross-section blade are established:

[0085] Using the concept of isoparametric elements, the coordinates (x, y, z) of any point within a shell element can be represented as follows:

[0086]

[0087] In the formula, ξ and η are the node coordinates in the local coordinate system, and N i (ξ, η) are the nodal shape functions in the local coordinate system, expressed as:

[0088]

[0089] In equation (1), x i m y i m With z i m Let V be the coordinates of node i on the midface in the global coordinate system. 3i Let be the vector from the bottom to the top of node i, that is, the normal to the face of node i, and its expression is:

[0090]

[0091] In the formula, x i t y i t and z i t x represents the coordinates of the top of the shell element corresponding to the mid-surface node i; i b y i b and z i b Let i be the coordinates of the bottom of the shell element corresponding to the mid-surface node i;

[0092] S1.2 To accurately obtain the unit normal vectors of each node on the mid-surface of the blade, a fifth-order polynomial is used to fit the mid-surface of the blade based on the mid-surface node coordinates obtained in step S1.1. The expression is as follows:

[0093]

[0094] In the formula, P ij (i, k = 0, ..., 5) are the coefficients of the polynomial;

[0095] Table 1. Coefficients of surface equations

[0096]

[0097]

[0098] Based on the unit normal vector of each node and the thickness of each node obtained in S1.2, the displacement of any point in the shell element can be expressed as:

[0099]

[0100] In the formula, t i Let u be the thickness of node i. i V i W i Let α be the displacement of node i along the x, y, and z axes; i β i γ i The normals v of the mid-surface node i are respectively 3i Around two mutually perpendicular orthogonal vectors v 2i v 1i The corner and v 3i Angle of rotation in the global coordinate system; l 3i m 3i n 3i For V 3i unit vector v 3i The direction cosine.

[0101] Based on the large deformation nonlinear plate and shell theory, and considering centrifugal stiffening, rotational softening, and Coriolis force effects, the differential equations of motion for the airfoil variable cross-section blade can be obtained in S1.4. The strain components of the element are:

[0102]

[0103]

[0104] In the formula, ε x ε y ε z For linear strain; γ xy γ yz γ zx For shear strain;

[0105] The relationship between strain ε and stress σ is:

[0106] σ=[σ x σ y σ z τ xy τ yz τ zx ] T =Dε (8)

[0107] In the formula, σ x σ y σ z For normal stress, τ xy τ yz τ zx denoted as shear stress; D is the elastic matrix.

[0108] Considering the centrifugal stiffening, rotational softening, and Coriolis force effects of the blade, the differential equation of motion for the airfoil variable cross-section blade can be obtained as follows:

[0109]

[0110] In the formula, M is the mass matrix, G is the Coriolis force matrix, C is the Rayleigh damping matrix, and K is the mass matrix. e K is the structural stiffness matrix. c K is the centrifugal stiffening matrix. s Let F be the rotation softening matrix. e F is the centrifugal force vector. p Let q be the aerodynamic vector. These are displacement, velocity, and acceleration vectors, respectively.

[0111] Furthermore, the process of establishing a dynamic model of a rotating airfoil with a variable cross-section cracked blade, based on the differential equation of motion of the airfoil blade and the presence of cracks in the blade, by using spring elements to represent the breathing effect between the crack surfaces, is as follows:

[0112] Figure 3(a) is a schematic diagram of the breathing model of the cracked blade; (b) is a partial schematic diagram of the breathing model of the cracked blade. In order to simulate the periodic opening and closing of the cracked surface over time, a breathing model of a rotating blade with a penetrating crack is established using linear springs. In this model, the crack-free surface is constructed as a common node connection, and the node pairs of the cracked surface are connected by springs. The contact state function of the node pairs is established based on the node displacement.

[0113]

[0114] Among them, K breathing It is a breathing matrix, K spring It is the spring stiffness, z g It is the normal displacement of node g, z j It is the normal displacement of node j;

[0115] Figure 4(a) is a schematic diagram of the stiffness assembly of the cracked blade structure, and (b) is a schematic diagram of the stiffness assembly of the spring element of the cracked blade. By assembling the stiffness matrix of the cracked blade structure and considering the breathing effect of the cracked blade, the dynamic model of the rotating airfoil variable cross-section cracked blade is as follows:

[0116]

[0117] In the formula, M is the mass matrix, G is the Coriolis force matrix, C is the Rayleigh damping matrix, and K is the mass matrix. e K is the structural stiffness matrix. c K is the centrifugal stiffening matrix. s K is the rotation softening matrix. breathing It is the breathing matrix, F c F is the centrifugal force vector. p Let q be the aerodynamic vector. These are displacement, velocity, and acceleration vectors, respectively.

[0118] picture (a) is the time-domain waveform of the displacement response in the superharmonic resonance state of the blade tip node; (b) is the spectrum of the displacement response in the superharmonic resonance state of the blade tip node; (c) is the time-domain waveform of the displacement response in the first-order resonance state of the blade tip node; and (d) is the spectrum of the displacement response in the first-order resonance state of the blade tip node.

[0119] From the figure The results show the displacement response at the blade tip in both the superharmonic and first-order resonance states of the dynamic analysis model and the ANSYS finite element model of the rotating airfoil variable cross-section cracked blade of this application. The dynamic model and the finite element model of the rotating airfoil variable cross-section cracked blade of this application agree well, indicating that simulating the breathing crack using a linear spring is effective. Based on the time-domain waveform and spectrum, the following phenomena can also be observed:

[0120] (1) Under the action of centrifugal force, a constant component appears in the spectrum;

[0121] (2) Due to the nonlinear effect of the cracked blade, there are multiple frequency components (nf) in the figure. r (n = 0, 1, 2...);

[0122] (3) 2f under superharmonic resonance state r The amplitude is closer to 1f than in the first-order resonance state. r The amplitude of the 2f wave is caused by the superharmonic resonance state. r The value is closer to the first natural frequency of the cracked blade.

[0123] The process for obtaining the diagnostic index for the rotating cracked blade with breathing effect is as follows:

[0124] S3.1 The displacement response q of the cracked blade considering the breathing effect, obtained by solving according to equation (11), is converted into an energy response, the expression of which is:

[0125]

[0126] Where V represents strain energy, q represents nodal displacement at the tip of the cracked blade, and K represents the overall stiffness matrix.

[0127] S3.2 Due to the breathing effect of cracked blades, the system stiffness is approximately a rectangular wave within a load cycle. Therefore, the energy response of the system naturally has one or more extreme points within a load cycle, and the energy difference between the extreme points of the open crack state and the closed crack state is different. Based on this phenomenon, an index can be constructed using the energy response to characterize the severity of the blade crack. The specific method is as follows: Based on the energy response, first determine the breathing state of the blade, and then determine whether it is an open or closed crack; then determine the extreme points of all energy responses under the open crack state, and then determine the maximum value P2 among these extreme points; similarly, the maximum value P1 of the extreme points of all energy responses under the closed crack state can be obtained. Therefore, the difference between the extreme points P1 and P2 of the energy response of the open and closed cracks within a cycle is used to characterize the crack fault, thereby constructing a new fault diagnosis index δ for rotating cracked blades with a breathing effect, the expression of which is:

[0128] δ=P1-P2(13)

[0129] Where P1 is the maximum response value corresponding to the closed crack state, and P2 is the maximum response value corresponding to the open crack state. Figure The diagram shows the specific calculation process for the crack fault diagnosis index δ.

[0130] A device for fault characterization and dynamic analysis of a rotating airfoil variable cross-section cracked blade includes:

[0131] Module I: This module is used to establish the differential equations of motion for airfoil-shaped variable cross-section blades based on finite element theory and plate and shell theory, and on the effects of centrifugal stiffening, rotational softening, and Coriolis force.

[0132] Module II: This module is used to establish a dynamic model of a rotating airfoil with a variable cross section crack, based on the differential equation of motion of the airfoil blade and the presence of cracks in the blade, by using spring elements to represent the breathing effect between the crack surfaces.

[0133] The module is used to convert displacement response into energy response using vibration energy theory, and to obtain the structural characteristics of blade cracks based on the crack diagnosis index of rotating cracked blades with breathing effect.

[0134] A computer-readable storage medium storing a computer program, wherein when the computer program is executed, it performs a learning resource recommendation method for a method of fault characterization and dynamic analysis of a rotating airfoil variable cross-section cracked blade.

[0135] Figure 7(a1) is the time-domain waveform of the displacement of the γ=0 cracked blade, (a2) is the energy response of the γ=0 cracked blade, and (a3) ​​is the energy-displacement curve of the γ=0 cracked blade.

[0136] (b1) is the time-domain waveform diagram of the displacement of the γ=0.1 cracked blade, (b2) is the energy response diagram of the γ=0.1 cracked blade, and (b3) is the energy-displacement curve diagram of the γ=0.1 cracked blade.

[0137] (c1) is the time-domain waveform diagram of the displacement of the γ=0.2 cracked blade, (c2) is the energy response diagram of the γ=0.2 cracked blade, and (c3) is the energy-displacement curve diagram of the γ=0.2 cracked blade.

[0138] (d1) is the time-domain waveform diagram of the displacement of the γ=0.3 cracked blade, (d2) is the energy response diagram of the γ=0.3 cracked blade, and (d3) is the energy-displacement curve diagram of the γ=0.3 cracked blade.

[0139] (e1) is the time-domain waveform diagram of the displacement of the γ=0.4 cracked blade, (e2) is the energy response diagram of the γ=0.4 cracked blade, and (e3) is the energy-displacement curve diagram of the γ=0.4 cracked blade.

[0140] (f1) is The time-domain waveform of the cracked blade displacement, (f2) is The energy response diagram of the cracked blade, (f3) is Energy-displacement curve of cracked blade;

[0141] Figure 7 shows the vibration response of the cracked blade under different crack depth ratios. From the figure, we can see that:

[0142] (1) As the crack depth ratio increases, the vibration displacement amplitude of the rotating cracked blade with breathing effect also increases. The response curve exhibits a sinusoidal function curve. For the energy response, not only does the amplitude increase, but the response curve also becomes less smooth, even showing multiple local maxima within a single period. The energy response curve no longer presents a simple single harmonic vibration. When the crack depth ratio reaches 0.4 and... At this time, the energy response curve exhibits multiple local maxima within one cycle. When the crack depth ratio is 0 (i.e., a healthy blade), the difference between the two maximum extremes within one cycle is zero; when cracks appear on the blade, as the crack depth ratio increases, the difference between the maximum extremes under the open and closed crack states within one cycle gradually increases.

[0143] (2) Figures 7(a3) to (f3) show the relationship between tip displacement and energy response. The crack-free blade exhibits a closed, smooth parabola with the opening facing upwards. As the crack depth ratio increases, the closed curve gradually transforms into an irregular parabola with multiple intersection points, and the curve deformation becomes increasingly severe. Simultaneously, the area of ​​the closed curve also increases. Therefore, the energy-displacement curve can intuitively reflect the change in crack depth.

[0144] (3) In Figures 7(a3) to (f3), the difference between the two maximum energy values ​​corresponding to positive and negative displacements has the same meaning as the index δ. The reason for this phenomenon is that when the displacement is positive, the blade crack is closed, and the system stiffness value corresponding to the closed crack is large, so the energy value of the cracked blade is large.

[0145] Figure 8 shows the trend of the index δ as the crack depth ratio changes; it can be found that when the blade is crack-free, the index δ is zero, and when the crack depth reaches half the blade thickness (i.e., ... When the crack depth increases, the index δ is 0.021 J. It can also be observed that the change in index δ becomes more pronounced as the crack depth gradually increases (i.e., the slope of the curve in the figure becomes increasingly steep). Therefore, index δ is a good quantitative indicator of crack depth variation.

[0146] Figure 9(a) shows the energy response spectrum of the cracked blade in the superharmonic resonance state at different crack depths; (b) shows the energy response spectrum of the cracked blade in the first-order resonance state at different crack depths; Figure 9 shows the nonlinear characteristics of the cracked blade in the superharmonic and first-order resonance states for each crack depth ratio. In the superharmonic resonance state, 0f r 1f r 2f r 3f r The amplitudes are relatively close; for the first-order resonance state, 2f r and 0f r The amplitude is greater than 1f r It can be seen that as the crack depth increases, 3f r and 4f r The increase in amplitude is more pronounced. To visually demonstrate the variation process of multi-frequency amplitude, different crack depths are compared with 0f. r 1f r 2f r The comparison of amplitudes is shown in Figure 10. Regardless of the resonance state, 0f r 1f r With 2f r The amplitude gradually increases with the increase of crack depth.

[0147] When the crack depth ratio reaches 0.3, a more obvious phenomenon appears. Figure 10(a) shows a comparison of the amplitude of the 0fr frequency under different crack depth ratios; (b) shows a comparison of the amplitude of the 1fr frequency under different crack depth ratios; and (c) shows a comparison of the amplitude of the 2fr frequency under different crack depth ratios. Therefore, based on the variation process of the multi-frequency amplitude in the energy spectrum, when the crack depth ratio is 0.1 and 0.2, the amplitude of the nonlinear component is relatively small, which can be considered as a small crack. As the crack depth ratio increases, the severity of the crack also increases accordingly.

[0148] 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 them; although the present invention has been described in detail with reference to the foregoing embodiments, those skilled in the art should understand that modifications can still be made to the technical solutions described in the foregoing embodiments, or equivalent substitutions can be made to some or all of the technical features; and these modifications or substitutions do not cause the essence of the corresponding technical solutions to deviate from the scope of the technical solutions of the embodiments of the present invention.

Claims

1. A method for fault characterization and dynamic analysis of a rotating airfoil variable cross-section cracked blade, characterized in that, The process includes the following steps: Based on finite element theory and plate and shell theory, and considering the effects of centrifugal stiffening, rotational softening, and Coriolis force, the differential equation of motion for the airfoil variable cross-section blade is established; based on the differential equation of motion for the airfoil variable cross-section blade, and considering the existing cracks in the blade, spring elements are used to represent the equivalent breathing effect between the crack surfaces, thus establishing a dynamic model of a rotating airfoil variable cross-section cracked blade with a breathing effect; using vibration energy theory, the displacement response is converted into an energy response, and based on the crack diagnosis index of the rotating cracked blade with a breathing effect, the structural characteristics of the blade crack are obtained; the process of establishing a dynamic model of a rotating airfoil variable cross-section cracked blade with a breathing effect by using spring elements to represent the equivalent breathing effect between the crack surfaces based on the differential equation of motion for the airfoil variable cross-section blade is as follows: To simulate the periodic opening and closing of the crack surface over time, a breathing model of a rotating blade with a penetrating crack is established using spring elements, wherein the crack-free surface is constructed as a common node connection, and the node pairs of the crack surface are connected through spring elements, and the node pair contact state function is established based on the node displacement: (10) Wherein, K breathing It is a breathing matrix, K spring It is the spring stiffness, z g It is the normal displacement of node g, z j It is the normal displacement of node j; based on the breathing effect of the cracked blade, the dynamic model of the rotating airfoil variable cross-section cracked blade is as follows: (11) In the formula, M is the mass matrix, G is the Coriolis force matrix, C is the Rayleigh damping matrix, and K is the mass matrix. e K is the structural stiffness matrix. c K is the centrifugal stiffening matrix. s K is the rotation softening matrix. breathing It is the breathing matrix, F c F is the centrifugal force vector. p Let q be the aerodynamic vector. 、 These are displacement, velocity, and acceleration vectors, respectively. The process for obtaining the crack diagnosis index of the rotating cracked blade with breathing effect is as follows: S4.1 The displacement response is converted into an energy response, the expression of which is: (12) Where V represents strain energy, q represents nodal displacement at the tip of the cracked blade, and K represents the overall stiffness matrix; S4.2 Based on the energy response, first determine the breathing state of the blade, whether it is an open crack or a closed crack; then determine the extreme points of all energy responses under the open crack state, and then determine the maximum value P2 among these extreme points; similarly, the maximum value P1 of the extreme points of all energy responses under the closed crack state can be obtained. Therefore, the difference between the extreme point P1 of the open crack and the extreme point P2 of the energy response of the closed crack within one cycle is used to characterize the crack fault, thereby constructing a new fault diagnosis index δ of rotating cracked blade with breathing effect, the expression of which is: (13) Where P1 is the maximum response value corresponding to the closed crack state, and P2 is the maximum response value corresponding to the open crack state. P1 and P2 are both extreme points and maximum / minimum points.

2. The method for fault characterization and dynamic analysis of a rotating airfoil variable cross-section cracked blade according to claim 1, characterized in that, This method is applicable to the dynamic analysis of a single crack.

3. The method for fault characterization and dynamic analysis of a rotating airfoil variable cross-section cracked blade according to claim 1, characterized in that, Based on finite element theory and plate and shell theory, and considering the effects of centrifugal stiffening, rotational softening, and Coriolis force, the differential equations of motion for the airfoil variable cross-section blade are established as follows: S1.1 Using the isoparametric element concept, the coordinates (x, y, z) of any point within the blade shell element can be represented as: (1) In the formula, ξ and η are the node coordinates in the local coordinate system, N i (ξ,η) is the nodal shape function in the local coordinate system, expressed as: (2) In equation (1), xim, yim, and zim are the coordinates of the mid-surface node i in the global coordinate system, V 3i Let be the vector from the bottom to the top of node i, that is, the normal to the face of node i, and its expression is: (3) In the formula, xit, yit, and zit are the coordinates of the top of the shell element corresponding to the mid-surface node i; xib, yib, and zib are the coordinates of the bottom of the shell element corresponding to the mid-surface node i; S1.2 In order to obtain the unit normal vector of each node of the blade mid-surface, the blade mid-surface is fitted with a fifth-order polynomial based on the mid-surface node coordinates obtained in step S1.1, and its expression is: (4) In the formula, P ij Let i,j be the coefficients of the polynomial, i,j=0, …, 5; S1.3 Based on the unit normal vector of each node and the thickness of each node obtained in S1.2, the displacement of any point in the shell element is expressed as: (5) In the formula, t i Let be the thickness of node i, ui, vi, and wi be the displacements of mid-surface node i along the x, y, and z axes, respectively; αi, βi, and γi are the normals of mid-surface node i, respectively. Orthogonal vectors perpendicular to it 、 The corner and Angle of rotation in the global coordinate system; 、 、 For V 3i unit vector The direction cosine; S1.4 Based on the large deformation nonlinear plate and shell theory, considering centrifugal stiffening, rotational softening and Coriolis force effects, the motion differential equation of the airfoil variable cross section blade is obtained, and the strain components of the element are: (6) In equation (7), 、 、 For linear strain; 、 、 For shear strain; the relationship between strain ε and stress σ is: (8) In the formula, σ x σ y σ z For normal stress, τ xy τ yz τ zx Let D be the shear stress; D be the elastic matrix; considering the centrifugal stiffening, rotational softening, and Coriolis force effects of the blade, the differential equation of motion for the airfoil variable cross-section blade can be obtained as follows: (9) In the formula, M is the mass matrix, G is the Coriolis force matrix, C is the Rayleigh damping matrix, and K is the mass matrix. e K is the structural stiffness matrix. c K is the centrifugal stiffening matrix. s Let F be the rotation softening matrix. c F is the centrifugal force vector. p Let q be the aerodynamic vector. 、 These are displacement, velocity, and acceleration vectors, respectively.

4. A device for fault characterization and dynamic analysis of a rotating airfoil variable cross-section cracked blade, characterized in that, include: Module I is used to establish the differential equations of motion for the airfoil variable cross-section blade based on finite element theory and plate and shell theory, and on the effects of centrifugal stiffening, rotational softening, and Coriolis force. Module II is used to establish a dynamic model of a rotating airfoil variable cross-section cracked blade with a breathing effect, based on the differential equations of motion for the airfoil variable cross-section blade and on the presence of cracks in the blade, using spring elements to represent the breathing effect between crack surfaces. The process of establishing a dynamic model of a rotating airfoil variable cross-section cracked blade with a breathing effect based on the differential equations of motion for the airfoil variable cross-section blade and on the presence of cracks in the blade, using spring elements to represent the breathing effect between crack surfaces, is as follows: To simulate the periodic opening and closing of the crack surface over time, a breathing model of a rotating blade with penetrating cracks is established using spring elements. The crack-free surface is constructed as a common-node connection, and the node pairs of the cracked surface are connected through spring elements. The contact state function of the node pairs is established based on the node displacement. (10) Wherein, K breathing It is a breathing matrix, K spring It is the spring stiffness, z g It is the normal displacement of node g, z j It is the normal displacement of node j; based on the breathing effect of the cracked blade, the dynamic model of the rotating airfoil variable cross-section cracked blade is as follows: (11) In the formula, M is the mass matrix, G is the Coriolis force matrix, C is the Rayleigh damping matrix, and K is the mass matrix. e K is the structural stiffness matrix. c K is the centrifugal stiffening matrix. s K is the rotation softening matrix. breathing It is the breathing matrix, F c F is the centrifugal force vector. p Let q be the aerodynamic vector. 、 These are displacement, velocity, and acceleration vectors, respectively. The module is used to convert the displacement response into an energy response using vibration energy theory, and based on the crack diagnosis index for rotating cracked blades with a breathing effect, to obtain the structural characteristics of the blade crack. The process of obtaining the crack diagnosis index for rotating cracked blades with a breathing effect is as follows: S4.1 Convert the displacement response into an energy response, the expression of which is: (12) Where V represents strain energy, q represents nodal displacement at the tip of the cracked blade, and K represents the overall stiffness matrix; S4.2 Based on the energy response, first determine the breathing state of the blade, whether it is an open crack or a closed crack; then determine the extreme points of all energy responses under the open crack state, and then determine the maximum value P2 among these extreme points; similarly, the maximum value P1 of the extreme points of all energy responses under the closed crack state can be obtained. Therefore, the difference between the extreme point P1 of the open crack and the extreme point P2 of the energy response of the closed crack within one cycle is used to characterize the crack fault, thereby constructing a new fault diagnosis index δ of rotating cracked blade with breathing effect, the expression of which is: (13) Where P1 is the maximum response value corresponding to the closed crack state, and P2 is the maximum response value corresponding to the open crack state. P1 and P2 are both extreme points and maximum / minimum points.

5. A computer-readable storage medium storing a computer program, wherein, When the computer program is executed, it performs the learning resource recommendation method of the method for fault characterization and dynamic analysis of a rotating airfoil variable cross-section cracked blade as described in any one of claims 1-3.

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