A semi-analytical method for calculating vibration response of a cracked variable thickness plate structure

Abstract

The application discloses a kind of crack variable-thickness plate structure vibration response semi-analytical calculation method, belong to engineering structure defect calculation and analysis field, including: crack variable-thickness plate structure is divided into crack region and crack-free region, the vibration of crack region is described using finite element, and crack-free region is described using analytical wave;Crack-free region is dispersed into multiple sections along the thickness variation direction, and the vibration of each section is described using analytical wave;Based on the compatibility relationship of different sections when coupling and the structure boundary condition, the structure dynamic equilibrium equation under wave space is established;Solving structure dynamic equilibrium equation obtains the wave amplitude of each order elastic wave, and obtains the vibration response of each section in crack variable-thickness plate structure based on elastic wave theory.The application solves the problem of low efficiency of existing method.

Description

Technical Field

[0001] This invention belongs to the field of engineering structural defect calculation and analysis, specifically involving a semi-analytical calculation method for the vibration response of a cracked variable thickness plate structure. Background Technology

[0002] Variable thickness plate structures, due to their significant geometric features, are widely used in industrial equipment structures. However, these structures are prone to cracking defects during manufacturing and use, which significantly negatively impacts their mechanical properties. Furthermore, efficient detection of cracks in variable thickness plate structures is crucial for the safe operation of equipment, and identifying cracks based on changes in vibration characteristics is one of the effective non-destructive testing methods. Therefore, it is necessary to conduct relevant analytical research on the impact of cracks on the structural dynamics.

[0003] Currently, efficient calculation methods are severely lacking for the vibration characteristics of variable-thickness plate structures with cracks, especially for plates with diagonal cracks. Using the traditional finite element method to analyze the entire plate results in excessively low computational efficiency. Furthermore, traditional analytical methods cannot handle conditions containing diagonal cracks. Therefore, there is an urgent need to develop more efficient calculation methods. Summary of the Invention

[0004] To address the aforementioned shortcomings in existing technologies, this invention provides a semi-analytical calculation method for the vibration response of cracked variable thickness plate structures, which solves the problem of low computational efficiency in existing methods.

[0005] To achieve the above-mentioned objectives, the technical solution adopted by this invention is as follows: a semi-analytical calculation method for the vibration response of a cracked variable thickness plate structure, comprising the following steps:

[0006] S1: The cracked variable thickness plate structure is divided into cracked region and crack-free region. The vibration of the cracked region is described by finite element method, and the crack-free region is described by analytical wave method.

[0007] S2: The crack-free region is discretized into multiple segments along the thickness variation direction, and the vibration of each segment is described by analytical waves;

[0008] S3: Based on the compatibility relationship and structural boundary conditions of different segments during coupling, establish the structural dynamic equilibrium equations in wave space;

[0009] S4: Solve the structural dynamic equilibrium equations to obtain the amplitude of each order of elastic waves, and obtain the vibration response of each section in the cracked variable thickness plate structure based on elastic wave theory.

[0010] Furthermore, step S3 includes the following sub-steps:

[0011] S31: For adjacent sections in the crack-free region, the amplitude correlation equation of the adjacent sections is established by utilizing the displacement continuity and internal force equilibrium conditions at both ends.

[0012] S32: Introduce a boundary condition indicator matrix to apply boundary conditions to the uncoupled boundary of the crack-free region and obtain the amplitude relationship at the boundary.

[0013] S33: By utilizing the displacement continuity and internal force equilibrium conditions at the coupling interface between the crack-free region and the cracked region, establish the mechanical correlation equations for the two types of regions.

[0014] S34: By combining the finite element stiffness matrix and mass matrix of the crack region, the structural dynamic equilibrium equations in wave space are obtained.

[0015] Furthermore, for adjacent segments in the crack-free region, the amplitude correlation equation for adjacent segments is established using the displacement continuity and internal force equilibrium conditions at both ends, specifically including:

[0016] Considering the section At its right end and the Based on the displacement continuity and internal force equilibrium conditions at the left end of each segment, we obtain:

[0017]

[0018] in, For the first Waveform matrix of each segment For the first Wave propagation matrix of each segment, For the first The amplitude of each segment, For the first Considering the wave propagation matrix of the negative excitation wave in each section, For the first The amplitude of the excitation wave in each segment For the first Waveform matrix of each segment For the first Wave propagation matrix of each segment, The length of each section, For the first The amplitude of each segment, For the first Considering the wave propagation matrix of the positively excited wave in each segment, For the first The amplitude of the excitation wave in each segment;

[0019] For the crack-free plate region on the left, we get:

[0020]

[0021] For the crack-free plate region on the right, we get:

[0022]

[0023] in, The first area without cracks on the left The amplitude of the positive wave in each segment, The first area without cracks on the left The amplitude of the negative wave in each segment The first area without cracks on the left The scattering relationship matrix between the wave amplitudes of each segment The first area without cracks on the left The amplitude of the positive wave in each segment, The first area without cracks on the left The amplitude of the negative wave in each segment, The first area without cracks on the left The relationship vector between the excitation waves of each segment The crack-free area on the right side The amplitude of the positive wave in each segment, The crack-free area on the right side The amplitude of the negative wave in each segment, The crack-free area on the right side The scattering relationship matrix between the segments The crack-free area on the right side The amplitude of the positive wave in each segment, The crack-free area on the right side The amplitude of the negative wave in each segment The crack-free area on the right side The relationship vector between the excitation waves of each segment.

[0024] Furthermore, the introduction of a boundary condition indicator matrix to apply boundary conditions to the uncoupled boundary of the crack-free region to obtain the amplitude relationship at the boundary specifically includes:

[0025] Introducing boundary condition indicator matrix For the crack-free regions on the left and right sides, the boundary conditions on the uncoupled boundaries are expressed as follows:

[0026]

[0027]

[0028] By left multiplication and along 0 to Perform integration and introduce... For the crack-free areas on both the left and right sides, the following applies:

[0029]

[0030] To further simplify:

[0031]

[0032]

[0033] in, This is the indicator matrix for the boundary conditions on the uncoupled boundary of the first segment in the crack-free region on the left. The waveform of the first segment in the crack-free area on the left is shown. This represents the wave propagation matrix of the first segment in the crack-free region on the left. The amplitude of the wave in the first segment of the crack-free region on the left. This is the wave propagation matrix for the first segment of the crack-free region on the left, considering only the negative excitation wave. The amplitude of the direct excitation wave in the first segment of the crack-free region on the left is shown. The crack-free area on the right side Indicator matrix of boundary conditions on the uncoupled boundary of each segment. The crack-free area on the right side The waveform of each segment The crack-free area on the right side Wave propagation matrix of each segment, The crack-free area on the right side The length of each section The crack-free area on the right side The amplitude of each segment, The crack-free area on the right side The wave propagation matrix of each segment only considers the positive excitation wave. The crack-free area on the right side The amplitude of the direct excitation wave in each section, The first segment of the crack-free area on the left or the first segment of the crack-free area on the right. Transpose of the positive wave waveform matrix of each segment It is a 2nd order symplectic matrix. The matrix generated during the operation. for The left half of the matrix, for The right half of the matrix, The first segment of the crack-free area on the left or the first segment of the crack-free area on the right. Waveform matrix of each segment It is the transpose of the positive wave amplitude vector. It is the transpose of the negative wave amplitude vector. A vector related to the amplitude of the excitation wave. The crack-free area on the right side Negative wave amplitude vector of each segment This represents the positive wave amplitude vector of the first segment in the crack-free region on the left. The crack-free area on the right side Wave reflection relationship matrix on the uncoupled boundary of each segment This is the wave reflection relation matrix on the uncoupled boundary of the first segment in the crack-free region on the left. The crack-free area on the right side The positive wave amplitude vector of each segment This represents the negative wave amplitude vector of the first segment in the crack-free region on the left. To the area without cracks on the right Related vectors, To the left side without cracks Related vectors.

[0034] Furthermore, the establishment of mechanical correlation equations for the two types of regions by utilizing the displacement continuity and internal force equilibrium conditions at the coupling interface between the crack-free region and the cracked region specifically includes:

[0035] The displacement and force of the rightmost segment in the crack-free region on its right-hand coupling interface are expressed as follows:

[0036]

[0037]

[0038] The displacement and force of the first segment in the crack-free region on its left-end coupling interface are expressed as follows:

[0039]

[0040]

[0041] in, The first area without cracks on the left Waveform matrix of each segment The first area without cracks on the left Wave propagation matrix of each segment, The first area without cracks on the left The length of each section The first area without cracks on the left Wave reflection relationship matrix of each segment As a unit array, The first area without cracks on the left The amplitude of the negative wave in each segment The first area without cracks on the left Vectors related to the excitation wave of each segment The first area without cracks on the left Each segment only considers the wave propagation matrix of the positive wave. The first area without cracks on the left The amplitude of the excitation wave in each segment, The cracked area and the non-cracked area on the left are shown in section 1. Kinematic variables on the coupling boundary of each segment The first area without cracks on the left The waveform integral correlation matrix of each segment The cracked area and the non-cracked area on the left are shown in section 1. Vectors relating to the integral of dynamic variables on the coupling boundary of each segment. This is the waveform matrix for the first segment of the crack-free region on the right. This represents the wave propagation matrix for the first segment of the crack-free region on the right. This is the wave reflection relation matrix for the first segment of the crack-free region on the right. The amplitude of the positive wave in the first segment of the crack-free region on the right. This is the vector related to the excitation wave of the first segment in the crack-free region on the right. For the first segment in the crack-free region on the right, only the wave propagation matrix of the negative wave is considered. The amplitude of the excitation wave in the first segment of the crack-free region on the right side. Let be the kinematic variables on the coupling boundary of the first segment between the cracked region and the crack-free region on the right. This is the waveform integral correlation matrix for the first segment in the crack-free region on the right. This is a vector relating the integral of the dynamic variables on the first segment of the coupling boundary between the cracked region and the uncracked region on the right.

[0042] Combining the displacement and internal forces at the coupling interfaces on both sides of the crack region, the mechanical correlation equations for the two types of regions are obtained as follows:

[0043]

[0044] in, Let be the kinematic variables at the coupled boundary between the cracked and uncracked regions. The matrix introduced during the calculation process. These are the dynamic variables at the coupled boundary between the cracked and crack-free regions. This refers to the matrix introduced during the calculation process.

[0045] Furthermore, the finite element stiffness matrix and mass matrix of the combined crack region are used to assemble the structural dynamic equilibrium equations in wave space, specifically including:

[0046] Introducing the stiffness and mass matrices from the finite element model of the crack region, the structural dynamic equilibrium equations of the overall structure in wave space are obtained as follows:

[0047]

[0048]

[0049]

[0050] in, These are the dynamic variables at the coupled boundary between the cracked and crack-free regions. As a unit array, The matrix introduced during the calculation process. The matrix introduced during the calculation process. The matrix introduced during the calculation process. These are the matrix elements corresponding to the coupled and uncoupled degrees of freedom in the dynamic stiffness matrix of the crack region. These are the matrix elements corresponding to the uncoupled degrees of freedom in the dynamic stiffness matrix of the crack region. These are the matrix elements corresponding to the uncoupled and coupled degrees of freedom in the dynamic stiffness matrix of the crack region. These are the matrix elements corresponding to the coupled degrees of freedom in the dynamic stiffness matrix of the crack region. Here is the dynamic stiffness matrix of the crack region. The angular frequency of vibration, The stiffness matrix is ​​given in the finite element model of the crack region. The mass matrix is ​​the mass matrix in the finite element model of the crack region.

[0051] Furthermore, in S4, the amplitude of each order of elastic wave is obtained by solving the structural dynamic equilibrium equations, and the vibration response of each segment in the cracked variable thickness plate structure is obtained based on elastic wave theory, specifically including:

[0052] By introducing the relevant vector of external excitation, the vibration states at any position of each segment in the crack-free regions on the left and right sides are obtained as follows:

[0053]

[0054]

[0055] in, The first area without cracks on the left The state vector of each segment The crack-free area on the right side The state vector of each segment For spatial coordinates, The first area without cracks on the left The waveform of each segment The crack-free area on the right side The waveform of each segment The first area without cracks on the left Wave propagation matrix of each segment, The crack-free area on the right side Wave propagation matrix of each segment, The first area without cracks on the left The amplitude of the positive wave in each segment, The first area without cracks on the left The amplitude of the negative wave in each segment The crack-free area on the right side The amplitude of the positive wave in each segment, The crack-free area on the right side The amplitude of the negative wave in each segment This refers to the vector related to external stimuli.

[0056] The beneficial effects of this invention are:

[0057] (1) This invention divides the cracked variable thickness plate structure into different regions, including cracked regions and crack-free regions. The cracked region should be as small as possible while enclosing the crack. The vibration of the crack-free region is described based on analytical waves, and the vibration of the cracked region is described using the finite element method. This avoids the dilemma that the cracked region cannot be described by analytical waves and that the finite element discretization calculation for the entire plate structure is too high.

[0058] (2) In this invention, the crack-free region is discretized into multiple segments along the thickness variation direction, and each segment is approximately a plate of uniform thickness, so that the analytical elastic wave of each segment can be easily obtained.

[0059] (3) This invention utilizes the displacement continuity and internal force balance conditions at the coupling points of each segment to establish the dynamic equilibrium equation of the variable thickness plate structure in wave space. Compared with the finite element modeling and analysis of the whole plate, the dynamic equilibrium equation in wave space has very few degrees of freedom, which can significantly reduce the amount of calculation and reduce the calculation cost.

[0060] (4) Solving this equation yields the amplitude of each order of elastic wave, and thus the vibration response of each segment in a cracked variable thickness plate structure. Since it is based on analytical wave description, parametric analysis can be performed more conveniently. Therefore, based on the calculation method of this invention, the influence of crack location and length on structural vibration characteristics can be studied efficiently. Attached Figure Description

[0061] Figure 1 This is a flowchart of a semi-analytical calculation method for the vibration response of a cracked variable thickness plate structure.

[0062] Figure 2This is a configuration diagram of a variable thickness plate structure containing crack defects.

[0063] Figure 3 This is a schematic diagram of the segment division and elastic wave of a variable thickness plate structure with crack defects.

[0064] Figure 4 This is a schematic diagram showing the vertical positions of different cracks in the cracked area.

[0065] Figure 5 A comparison of frequency response curves for a variable thickness plate structure with cracks and defects.

[0066] Figure 6 This is a schematic diagram to consider the influence of different longitudinal crack locations (along the y-direction of the plate) on the vibration response of a variable thickness plate structure.

[0067] Figure 7 This diagram illustrates the influence of different vertical crack locations (along the x-direction of the plate) on the vibration response of a variable thickness plate structure. Detailed Implementation

[0068] The present invention will be further described below with reference to the accompanying drawings and specific embodiments.

[0069] like Figure 1 As shown, a semi-analytical calculation method for the vibration response of a cracked variable thickness plate structure includes the following steps:

[0070] S1: The cracked variable thickness plate structure is divided into cracked region and crack-free region. The vibration of the cracked region is described by finite element method, and the crack-free region is described by analytical wave method.

[0071] like Figure 2 As shown, the cracked variable thickness plate is divided into cracked and crack-free regions. The cracked region is described using the finite element method (FEM), while the crack-free region is described using analytical wave analysis. The thickness of the plate at the left end. The thickness of the plate at the right end. The crack-free area on the left side Length in direction, The crack-free area on the right side Length in direction, For the edge of the board Length in direction, The length of the crack. The angle of the crack direction, For external loads, The positive direction of the coordinate axis.

[0072] S2: The crack-free region is discretized into multiple segments along the thickness variation direction, and the vibration of each segment is described by analytical waves;

[0073] To describe the crack-free region using analytical waves, the crack-free region is discretized into multiple segments, such as... Figure 3 As shown, in this example, the crack-free region on the left is discretized into... Each section discretizes the crack-free region on the right side into... Each segment is approximately a plate of uniform thickness; therefore, the vibration of each segment can be described using analytical elastic waves. and The first area without cracks on the left The positive wave (propagating to the right) and negative wave (propagating to the left) of each segment, subscript " "This refers to the crack-free area on the left, indicated by the subscript " "Refers to the crack-free area on the right side, subscript" correspond Figure 3 The different segments shown are labeled "+" and "-" to indicate positive and negative waves, respectively. Similarly... and The crack-free area on the right side The positive and negative waves of each segment.

[0074] S3: Based on the compatibility relationship and structural boundary conditions of different segments during coupling, establish the structural dynamic equilibrium equations in wave space;

[0075] For the crack-free areas on the left and right sides, the following derivation is performed, so the subscript no longer indicates "". "and" "Introduction" It is the first The waveform matrix of each segment is introduced. and , It is the crack-free area. The elastic wave amplitude of each segment It is the first The amplitude of the excitation wave in each segment is determined. A wave propagation matrix is ​​introduced. diag{ , , ,…, }, It is the first Step wave propagation parameters, introduced , , , It is a 2nd order symplectic matrix. It refers to the length of each section. The incentive point is Coordinates of direction.

[0076] S3 includes the following steps:

[0077] S31: For adjacent sections in the crack-free region, using the displacement continuity and internal force equilibrium conditions at both ends, establish the wave amplitude correlation equation for adjacent sections, specifically including:

[0078] Considering the section At its right end and the Based on the displacement continuity and internal force equilibrium conditions at the left end of each segment, we obtain:

[0079]

[0080] in, For the first Waveform matrix of each segment For the first Wave propagation matrix of each segment, For the first The amplitude of each segment, For the first Considering the wave propagation matrix of the negative excitation wave in each section, For the first The amplitude of the excitation wave in each segment For the first Waveform matrix of each segment For the first Wave propagation matrix of each segment, The length of each section, For the first The amplitude of each segment, For the first Considering the wave propagation matrix of the positively excited wave in each segment, For the first The amplitude of the excitation wave in each segment;

[0081] Introduction , , , It is an m-order symplectic matrix. It is the edge of the board. The length of the direction, multiplied by the left side of the above formula and along 0 to Integrating, we get:

[0082]

[0083] Introduction , .

[0084] in, , and It is an intermediate quantity introduced to facilitate calculation. , , , for matrix elements, , , , for matrix elements, , for matrix elements, , , , for matrix elements, For the crack-free area The elastic wave amplitude of each segment The first area without cracks on the left The relationship vector between the excitation waves of each segment.

[0085] For the crack-free plate region on the left, we get from the above formula:

[0086]

[0087] For the crack-free plate region on the right, we get:

[0088]

[0089] in, The first area without cracks on the left The amplitude of the positive wave in each segment, The first area without cracks on the left The amplitude of the negative wave in each segment The first area without cracks on the left The scattering relationship matrix between the wave amplitudes of each segment The first area without cracks on the left The amplitude of the positive wave in each segment, The first area without cracks on the left The amplitude of the negative wave in each segment, The first area without cracks on the left The relationship vector between the excitation waves of each segment The crack-free area on the right side The amplitude of the positive wave in each segment, The crack-free area on the right side The amplitude of the negative wave in each segment, The crack-free area on the right side The scattering relationship matrix between the segments The crack-free area on the right side The amplitude of the positive wave in each segment, The crack-free area on the right side The amplitude of the negative wave in each segment The crack-free area on the right side The relationship vector between the excitation waves of each segment.

[0090] S32: Introducing a boundary condition indicator matrix, boundary conditions are applied to the uncoupled boundary of the crack-free region to obtain the amplitude relationship at the boundary, specifically including:

[0091] Introducing boundary condition indicator matrix For the crack-free regions on the left and right sides, the boundary conditions on the uncoupled boundaries are expressed as follows:

[0092]

[0093]

[0094] By left multiplication and along 0 to Perform integration and introduce... For the crack-free areas on both the left and right sides, the following applies:

[0095]

[0096] Introduction , , , , , , The crack-free area on the right side The positive wave waveform of each segment The crack-free area on the right side The waveform of each segment The waveform is the positive wave pattern of the first segment in the crack-free region on the left. Given the waveform of the first segment in the crack-free region on the left, we can obtain the following from the above formula:

[0097]

[0098]

[0099] in, This is the indicator matrix for the boundary conditions on the uncoupled boundary of the first segment in the crack-free region on the left. The waveform of the first segment in the crack-free area on the left is shown. This represents the wave propagation matrix of the first segment in the crack-free region on the left. The amplitude of the wave in the first segment of the crack-free region on the left. This is the wave propagation matrix for the first segment of the crack-free region on the left, considering only the negative excitation wave. The amplitude of the direct excitation wave in the first segment of the crack-free region on the left is shown. The crack-free area on the right side Indicator matrix of boundary conditions on the uncoupled boundary of each segment. The crack-free area on the right side The waveform of each segment The crack-free area on the right side Wave propagation matrix of each segment, The crack-free area on the right side The length of each section The crack-free area on the right side The amplitude of each segment, The crack-free area on the right side The wave propagation matrix of each segment only considers the positive excitation wave. The crack-free area on the right side The amplitude of the direct excitation wave in each section, The first segment of the crack-free area on the left or the first segment of the crack-free area on the right. Transpose of the positive wave waveform matrix of each segment It is a 2nd order symplectic matrix. The matrix generated during the operation. for The left half of the matrix, for The right half of the matrix, The first segment of the crack-free area on the left or the first segment of the crack-free area on the right. Waveform matrix of each segment It is the transpose of the positive wave amplitude vector. It is the transpose of the negative wave amplitude vector. A vector related to the amplitude of the excitation wave. The crack-free area on the right side Negative wave amplitude vector of each segment This represents the positive wave amplitude vector of the first segment in the crack-free region on the left. The crack-free area on the right side Wave reflection relationship matrix on the uncoupled boundary of each segment This is the wave reflection relation matrix on the uncoupled boundary of the first segment in the crack-free region on the left. The crack-free area on the right side The positive wave amplitude vector of each segment This represents the negative wave amplitude vector of the first segment in the crack-free region on the left. To the area without cracks on the right Related vectors, To the left side without cracks Related vectors.

[0100] S33: Utilizing the displacement continuity and internal force equilibrium conditions at the coupling interface between the crack-free and cracked regions, establish the mechanical correlation equations for the two types of regions, specifically including:

[0101] For the crack-free area on the left, consider the relationships between the various sections. And assembled , Introduction , , …, , ,…, , , , ,…, , ,…, , Unit array ,available:

[0102]

[0103] in, For the sections The resulting matrix is ​​assembled. The vector is composed of the amplitudes of the positive and negative waves from the crack-free region on the left. To be a vector reflecting the contributions of the excitation wave and boundary conditions, and The first area without cracks on the left The amplitudes of positive and negative waves in each segment , , and for matrix elements, and for Vector elements.

[0104] Introduction , From the above formula, we can further obtain:

[0105]

[0106] Similarly, for the crack-free area on the right, consider the relationships between the various sections. And assembled , Introduction , , …, , ,…, , , ,…, , ,…, , Then we can get:

[0107]

[0108] in, and The amplitudes of the positive and negative waves in the first segment of the crack-free region on the right are shown. The vector is composed of the amplitudes of the positive and negative waves in the crack-free region on the right.

[0109] Introduction , From the above formula, we can further obtain:

[0110]

[0111] Introduction , , , , It is the force exerted by the cracked region on the coupling interface of the uncracked plate region on the left. , It is the force acting on the coupling interface between the cracked region and the uncracked plate region on the right. [ , , , , , ], , , ,in, ( =1,2, (n) are the coordinates of discrete points on the coupling interface of the crack region. The first area without cracks on the left The waveform integral correlation matrix of each segment The first area without cracks on the left Transpose of the positive wave waveform matrix of each segment This is the internal force indication matrix. This is the waveform integral correlation matrix for the first segment in the crack-free region on the right. This is the transpose of the positive wave waveform matrix of the first segment in the crack-free region on the right. The first area without cracks on the left The waveform matrix of each segment, involving only kinematic variables, takes values ​​at discrete points. The waveform matrix, involving only kinematic variables, is represented at discrete points for the first segment of the crack-free region on the right. It is a matrix composed of the values ​​of the waveform matrix, which involves only kinematic variables, at discrete points. for Elements related to lateral displacement for Elements related to angular displacement , , For discrete point coordinates, Let the internal force vectors be discrete points. , , These are the internal force values ​​at discrete points. and Intermediate quantities are introduced to facilitate calculations.

[0112] The displacement and force of the rightmost segment in the crack-free region on its right-hand coupling interface are expressed as follows:

[0113]

[0114]

[0115] The displacement and force of the first segment in the crack-free region on its left-end coupling interface are expressed as follows:

[0116]

[0117]

[0118] in, The first area without cracks on the left Waveform matrix of each segment The first area without cracks on the left Wave propagation matrix of each segment, The first area without cracks on the left The length of each section The first area without cracks on the left Wave reflection relationship matrix of each segment As a unit array, The first area without cracks on the left The amplitude of the negative wave in each segment The first area without cracks on the left Vectors related to the excitation wave of each segment The first area without cracks on the left Each segment only considers the wave propagation matrix of the positive wave. The first area without cracks on the left The amplitude of the excitation wave in each segment, The cracked area and the non-cracked area on the left are shown in section 1. Kinematic variables on the coupling boundary of each segment The first area without cracks on the left The waveform integral correlation matrix of each segment The cracked area and the non-cracked area on the left are shown in section 1. Vectors relating to the integral of dynamic variables on the coupling boundary of each segment. This is the waveform matrix for the first segment of the crack-free region on the right. This represents the wave propagation matrix for the first segment of the crack-free region on the right. This is the wave reflection relation matrix for the first segment of the crack-free region on the right. The amplitude of the positive wave in the first segment of the crack-free region on the right. This is the vector related to the excitation wave of the first segment in the crack-free region on the right. For the first segment in the crack-free region on the right, only the wave propagation matrix of the negative wave is considered. The amplitude of the excitation wave in the first segment of the crack-free region on the right side. Let be the kinematic variables on the coupling boundary of the first segment between the cracked region and the crack-free region on the right. This is the waveform integral correlation matrix for the first segment in the crack-free region on the right. This is a vector relating to the integral of the dynamic variables on the first segment of the coupled boundary between the cracked region and the uncracked region on the right.

[0119] Introduction , , , , , , , , , , and These represent the displacement and internal forces at the coupling interfaces on the left and right sides of the crack region, respectively. , , and It is an intermediate quantity introduced to make the expression more concise. The first area without cracks on the left The length of each segment.

[0120] Combining the displacements and internal forces at the coupling interfaces on both sides of the crack region, the mechanical correlation equations for the two types of regions are obtained as follows:

[0121]

[0122] in, Let be the kinematic variables at the coupled boundary between the cracked and uncracked regions. The matrix introduced during the calculation process. These are the dynamic variables at the coupled boundary between the cracked and crack-free regions. This refers to the matrix introduced during the calculation process.

[0123] S34: Combining the finite element stiffness matrix and mass matrix of the crack region, the structural dynamic equilibrium equations in wave space are assembled, specifically including:

[0124] like Figure 4 The diagram shows the vertical locations of different cracks in the cracked region. The region shown is the area with cracks, which will be discretized and modeled using the finite element method.

[0125] Introduction and These are the stiffness matrix and mass matrix under the finite element model of the crack region, respectively. The structural dynamic equilibrium equations of the overall structure in wave space are obtained as follows:

[0126]

[0127]

[0128]

[0129]

[0130] in, These are the dynamic variables at the coupled boundary between the cracked and crack-free regions. As a unit array, The matrix introduced during the calculation process. The matrix introduced during the calculation process. The matrix introduced during the calculation process. These are the matrix elements corresponding to the coupled and uncoupled degrees of freedom in the dynamic stiffness matrix of the crack region. These are the matrix elements corresponding to the uncoupled degrees of freedom in the dynamic stiffness matrix of the crack region. These are the matrix elements corresponding to the uncoupled and coupled degrees of freedom in the dynamic stiffness matrix of the crack region. These are the matrix elements corresponding to the coupled degrees of freedom in the dynamic stiffness matrix of the crack region. Here is the dynamic stiffness matrix of the crack region. The angular frequency of vibration, The stiffness matrix is ​​given in the finite element model of the crack region. The mass matrix is ​​given in the finite element model of the crack region. , , , They are respectively The region elements of the matrix, , , , They are respectively The region element.

[0131] S4: Solve the structural dynamic equilibrium equations to obtain the amplitude of elastic waves of each order, and based on elastic wave theory, obtain the vibration response of each segment in a cracked, variable-thickness plate structure, specifically including:

[0132] Introducing external stimulus related vectors for , and , The vibration states at any position in each section of the crack-free region on both the left and right sides were obtained as follows:

[0133]

[0134]

[0135] in, The first area without cracks on the left The state vector of each segment The crack-free area on the right side The state vector of each segment For spatial coordinates, The first area without cracks on the left The waveform of each segment The crack-free area on the right side The waveform of each segment The first area without cracks on the left Wave propagation matrix of each segment, The crack-free area on the right side Wave propagation matrix of each segment, The first area without cracks on the left The amplitude of the positive wave in each segment, The first area without cracks on the left The amplitude of the negative wave in each segment The crack-free area on the right side The amplitude of the positive wave in each segment, The crack-free area on the right side The amplitude of the negative wave in each segment For external stimulus-related vectors, As the incentive point Coordinates of direction.

[0136] In one embodiment of the present invention, Figure 5 This is a comparison of the frequency response curves of a variable thickness plate structure with cracks and defects. The figure compares the frequency response results calculated by the method of this invention and by full-plate finite element modeling simulation. Figure 6 This diagram illustrates the impact of different longitudinal crack locations on the vibration response of a variable-thickness plate structure. The diagram compares a crack-free plate (intact) with plates where cracks are located at different longitudinal positions (along the plate). The frequency response characteristics (direction) show the influence of the crack's location on the structure's frequency response characteristics. Figure 7 This diagram illustrates the impact of different vertical crack locations on the vibration response of a variable-thickness plate structure. The diagram compares a crack-free plate (intact) with plates where cracks are located at different vertical positions (along the plate). The frequency response characteristics (direction) are shown, demonstrating the influence of the crack's location on the structural frequency response characteristics from different vertical positions. Based on the above embodiments, it is evident that the method proposed in this invention is effective.

[0137] Those skilled in the art will recognize that the embodiments described herein are intended to help the reader understand the principles of the invention, and should be understood that the scope of protection of the invention is not limited to such specific statements and embodiments. Those skilled in the art can make various other specific modifications and combinations based on the technical teachings disclosed in this invention without departing from the spirit of the invention, and these modifications and combinations are still within the scope of protection of the invention.

Claims

1. A semi-analytical calculation method for the vibration response of a cracked, variable-thickness plate structure, characterized in that, Includes the following steps: S1: The cracked variable thickness plate structure is divided into cracked region and crack-free region. The vibration of the cracked region is described by finite element method, and the crack-free region is described by analytical wave method. S2: The crack-free region is discretized into multiple segments along the thickness variation direction, and the vibration of each segment is described by analytical waves; S3: Based on the compatibility relationship and structural boundary conditions of different segments during coupling, establish the structural dynamic equilibrium equations in wave space; S4: Solve the structural dynamic equilibrium equations to obtain the amplitude of each order of elastic waves, and obtain the vibration response of each section in the cracked variable thickness plate structure based on elastic wave theory.

2. The semi-analytical calculation method for the vibration response of a cracked variable thickness plate structure according to claim 1, characterized in that, S3 includes the following steps: S31: For adjacent sections in the crack-free region, the amplitude correlation equation of the adjacent sections is established by utilizing the displacement continuity and internal force equilibrium conditions at both ends. S32: Introduce a boundary condition indicator matrix to apply boundary conditions to the uncoupled boundary of the crack-free region and obtain the amplitude relationship at the boundary. S33: By utilizing the displacement continuity and internal force equilibrium conditions at the coupling interface between the crack-free region and the cracked region, establish the mechanical correlation equations for the two types of regions. S34: By combining the finite element stiffness matrix and mass matrix of the crack region, the structural dynamic equilibrium equations in wave space are obtained.

3. The semi-analytical calculation method for the vibration response of a cracked variable thickness plate structure according to claim 2, characterized in that, For adjacent sections in the crack-free region, the amplitude correlation equation of the adjacent sections is established by utilizing the displacement continuity and internal force equilibrium conditions at both ends, specifically including: Considering the section At its right end and the Based on the displacement continuity and internal force equilibrium conditions at the left end of each segment, we obtain: in, For the first Waveform matrix of each segment For the first Wave propagation matrix of each segment, For the first The amplitude of each segment, For the first Considering the wave propagation matrix of the negative excitation wave in each section, For the first The amplitude of the excitation wave in each segment For the first Waveform matrix of each segment For the first Wave propagation matrix of each segment, The length of each section, For the first The amplitude of each segment, For the first Considering the wave propagation matrix of the positively excited wave in each segment, For the first The amplitude of the excitation wave in each segment; For the crack-free plate region on the left, we get: For the crack-free plate region on the right, we get: in, The first area without cracks on the left The amplitude of the positive wave in each segment, The first area without cracks on the left The amplitude of the negative wave in each segment The first area without cracks on the left The scattering relationship matrix between the wave amplitudes of each segment The first area without cracks on the left The amplitude of the positive wave in each segment, The first area without cracks on the left The amplitude of the negative wave in each segment, The first area without cracks on the left The relationship vector between the excitation waves of each segment The crack-free area on the right side The amplitude of the positive wave in each segment, The crack-free area on the right side The amplitude of the negative wave in each segment, The crack-free area on the right side The scattering relationship matrix between the segments The crack-free area on the right side The amplitude of the positive wave in each segment, The crack-free area on the right side The amplitude of the negative wave in each segment The crack-free area on the right side The relationship vector between the excitation waves of each segment.

4. The semi-analytical calculation method for the vibration response of a cracked variable thickness plate structure according to claim 3, characterized in that, The introduction of the boundary condition indicator matrix applies boundary conditions to the uncoupled boundary of the crack-free region, resulting in the amplitude relationship at the boundary, specifically including: Introducing boundary condition indicator matrix For the crack-free regions on the left and right sides, the boundary conditions on the uncoupled boundaries are expressed as follows: By left multiplication and along 0 to Perform integration and introduce... For the crack-free areas on both the left and right sides, the following applies: To further simplify: in, This is the indicator matrix for the boundary conditions on the uncoupled boundary of the first segment in the crack-free region on the left. The waveform of the first segment in the crack-free area on the left is shown. This represents the wave propagation matrix of the first segment in the crack-free region on the left. The amplitude of the wave in the first segment of the crack-free region on the left. This is the wave propagation matrix for the first segment of the crack-free region on the left, considering only the negative excitation wave. The amplitude of the direct excitation wave in the first segment of the crack-free region on the left is shown. The crack-free area on the right side Indicator matrix of boundary conditions on the uncoupled boundary of each segment. The crack-free area on the right side The waveform of each segment The crack-free area on the right side Wave propagation matrix of each segment, The crack-free area on the right side The length of each section The crack-free area on the right side The amplitude of each segment, The crack-free area on the right side The wave propagation matrix of each segment only considers the positive excitation wave. The crack-free area on the right side The amplitude of the direct excitation wave in each section, The first segment of the crack-free area on the left or the first segment of the crack-free area on the right. Transpose of the positive wave waveform matrix of each segment It is a 2nd order symplectic matrix. The matrix generated during the operation. for The left half of the matrix, for The right half of the matrix, The first segment of the crack-free area on the left or the first segment of the crack-free area on the right. Waveform matrix of each segment It is the transpose of the positive wave amplitude vector. It is the transpose of the negative wave amplitude vector. A vector related to the amplitude of the excitation wave. The crack-free area on the right side Negative wave amplitude vector of each segment This represents the positive wave amplitude vector of the first segment in the crack-free region on the left. The crack-free area on the right side Wave reflection relationship matrix on the uncoupled boundary of each segment This is the wave reflection relation matrix on the uncoupled boundary of the first segment in the crack-free region on the left. The crack-free area on the right side The positive wave amplitude vector of each segment This represents the negative wave amplitude vector of the first segment in the crack-free region on the left. To the area without cracks on the right Related vectors, To the left side without cracks Related vectors.

5. The semi-analytical calculation method for the vibration response of a cracked variable thickness plate structure according to claim 4, characterized in that, The method utilizes the displacement continuity and internal force equilibrium conditions at the coupling interface between the crack-free region and the cracked region to establish the mechanical correlation equations for the two types of regions, specifically including: The displacement and force of the rightmost segment in the crack-free region on its right-hand coupling interface are expressed as follows: The displacement and force of the first segment in the crack-free region on its left-end coupling interface are expressed as follows: in, The first area without cracks on the left Waveform matrix of each segment The first area without cracks on the left Wave propagation matrix of each segment, The first area without cracks on the left The length of each section The first area without cracks on the left Wave reflection relationship matrix of each segment As a unit array, The first area without cracks on the left The amplitude of the negative wave in each segment The first area without cracks on the left Vectors related to the excitation wave of each segment The first area without cracks on the left Each segment only considers the wave propagation matrix of the positive wave. The first area without cracks on the left The amplitude of the excitation wave in each segment, The cracked area and the non-cracked area on the left are shown in section 1. Kinematic variables on the coupling boundary of each segment The first area without cracks on the left The waveform integral correlation matrix of each segment The cracked area and the non-cracked area on the left are shown in section 1. Vectors relating to the integral of dynamic variables on the coupling boundary of each segment. This is the waveform matrix for the first segment of the crack-free region on the right. This represents the wave propagation matrix for the first segment of the crack-free region on the right. This is the wave reflection relation matrix for the first segment of the crack-free region on the right. The amplitude of the positive wave in the first segment of the crack-free region on the right. This is the vector related to the excitation wave of the first segment in the crack-free region on the right. For the first segment in the crack-free region on the right, only the wave propagation matrix of the negative wave is considered. The amplitude of the excitation wave in the first segment of the crack-free region on the right side. Let be the kinematic variables on the coupling boundary of the first segment between the cracked region and the crack-free region on the right. This is the waveform integral correlation matrix for the first segment in the crack-free region on the right. This is a vector relating the integral of the dynamic variables on the first segment of the coupling boundary between the cracked region and the uncracked region on the right. Combining the displacement and internal forces at the coupling interfaces on both sides of the crack region, the mechanical correlation equations for the two types of regions are obtained as follows: in, Let be the kinematic variables at the coupled boundary between the cracked and uncracked regions. The matrix introduced during the calculation process. These are the dynamic variables at the coupled boundary between the cracked and crack-free regions. This refers to the matrix introduced during the calculation process.

6. The semi-analytical calculation method for the vibration response of a cracked variable thickness plate structure according to claim 5, characterized in that, The finite element stiffness matrix and mass matrix of the combined crack region are used to assemble the structural dynamic equilibrium equations in wave space, specifically including: Introducing the stiffness and mass matrices from the finite element model of the crack region, the structural dynamic equilibrium equations of the overall structure in wave space are obtained as follows: in, These are the dynamic variables at the coupled boundary between the cracked and crack-free regions. As a unit array, The matrix introduced during the calculation process. The matrix introduced during the calculation process. The matrix introduced during the calculation process. These are the matrix elements corresponding to the coupled and uncoupled degrees of freedom in the dynamic stiffness matrix of the crack region. These are the matrix elements corresponding to the uncoupled degrees of freedom in the dynamic stiffness matrix of the crack region. These are the matrix elements corresponding to the uncoupled and coupled degrees of freedom in the dynamic stiffness matrix of the crack region. These are the matrix elements corresponding to the coupled degrees of freedom in the dynamic stiffness matrix of the crack region. Here is the dynamic stiffness matrix of the crack region. The angular frequency of vibration, The stiffness matrix is ​​given in the finite element model of the crack region. The mass matrix is ​​the mass matrix in the finite element model of the crack region.

7. The semi-analytical calculation method for the vibration response of a cracked variable thickness plate structure according to claim 6, characterized in that, In S4, the amplitudes of elastic waves of each order are obtained by solving the structural dynamic equilibrium equations, and the vibration response of each segment in the cracked variable thickness plate structure is obtained based on elastic wave theory, specifically including: By introducing the relevant vector of external excitation, the vibration states at any position of each segment in the crack-free regions on the left and right sides are obtained as follows: in, The first area without cracks on the left The state vector of each segment The crack-free area on the right side The state vector of each segment For spatial coordinates, The first area without cracks on the left The waveform of each segment The crack-free area on the right side The waveform of each segment The first area without cracks on the left Wave propagation matrix of each segment, The crack-free area on the right side Wave propagation matrix of each segment, The first area without cracks on the left The amplitude of the positive wave in each segment, The first area without cracks on the left The amplitude of the negative wave in each segment The crack-free area on the right side The amplitude of the positive wave in each segment, The crack-free area on the right side The amplitude of the negative wave in each segment This refers to the vector related to external stimuli.

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