A method for calculating the vibration response of a layered composite laminate
By dividing the composite laminate into layered regions and decoupling its dynamics, and combining this with elastic wave theory, vibration state vectors and dynamic equilibrium equations are established. This solves the problem of low computational efficiency in vibration response calculation of composite laminate structures, and enables efficient analysis and detection of layered defects.
Patent Information
- Authority / Receiving Office
- CN · China
- Patent Type
- Applications(China)
- Current Assignee / Owner
- CHINA AIRPLANT STRENGTH RES INST
- Filing Date
- 2026-05-08
- Publication Date
- 2026-06-02
Smart Images
Figure CN122135855A_ABST
Abstract
Description
Technical Field
[0001] This invention belongs to the field of engineering structural defect calculation and analysis, and relates to a method for calculating the vibration response of laminated plates containing layered composite materials. Background Technology
[0002] Composite laminate structures are widely used in industrial equipment structures due to their significant high specific stiffness and damping characteristics. However, these structures are prone to delamination defects during manufacturing and use, especially at the material interfaces of multi-material structures, which can significantly negatively impact the mechanical properties of the structure. Furthermore, efficient detection of delamination defects in composite laminate structures is crucial for the safe use of equipment, and identifying delamination defects 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 delamination defects on the structural dynamics.
[0003] Currently, there is a severe lack of efficient calculation methods for the vibration characteristics of composite laminate structures with delamination defects. Traditional finite element method-based analysis methods are computationally inefficient, necessitating the development of more efficient methods. Summary of the Invention
[0004] To address the aforementioned shortcomings in the existing technology, this invention provides a method for calculating the vibration response of laminated plates containing layered composite materials, which solves the problem of low calculation efficiency in existing methods.
[0005] To achieve the above-mentioned objectives, the technical solution adopted by this invention is as follows: a method for calculating the vibration response of a laminated plate containing layered composite materials, comprising the following steps:
[0006] S1: The composite laminate containing delamination defects is divided into delamination regions and intact regions according to the damage condition. The delamination regions include two states: incomplete separation and complete separation. The interaction between adjacent layers in the delamination region under the incomplete separation state is described by uniformly distributed spring connections. When the spring stiffness is zero, it degenerates into the complete separation state.
[0007] S2: Dynamically decouple the layered regions. Based on the theory of elastic wave propagation, use elastic waves to describe the vibration characteristics of each component in the layered and intact regions respectively, and establish the vibration state vector expression for each region.
[0008] S3: Based on the displacement continuity and internal force equilibrium conditions at the coupling point between the layered region and the intact region, the compatibility relationship is determined, and combined with the boundary conditions of the composite laminate, the structural dynamic equilibrium equation in wave space is established.
[0009] S4: Solve the dynamic equilibrium equation of the structure to obtain the amplitude of each order of elastic waves, and obtain the vibration response of each region of the composite laminate based on elastic wave theory and amplitude calculation.
[0010] Furthermore, the dynamic decoupling of the layered regions in S2 specifically includes:
[0011] The spring layer describing the interlayer action in the layered regions is equivalent to a continuous elastic layer, and the stiffness of the elastic layer between the layered regions is determined as follows:
[0012]
[0013] in, For the stiffness of the elastic layer, For spring stiffness, The number of springs and The length and width of the laminated plate in the layered region;
[0014] Establish the governing equations for bending vibration in the layered region:
[0015]
[0016] in, For the first The bending stiffness of the plate. For operators, , and These represent the displacements of each component. For surface density, For the first The board and the first The stiffness of the elastic layer between the plates For the first The board and the first The stiffness of the elastic layer between the plates As an external force, For time;
[0017] Considering the steady-state vibration problem, , Substituting into the bending vibration control equation, we obtain the vibration equation in matrix form:
[0018]
[0019] Decoupling it yields the generalized vibration equation:
[0020] or
[0021] in, For spatial coordinates, For the first External load on the plate For the first The displacement of the plate Indicates the first Step, For the excitation frequency, Let be the diagonal matrix composed of the bending stiffnesses of components 2 and 3 in the layered region. Let be the displacement vector composed of the lateral displacements of components 2 and 3. Here is the stiffness matrix of the elastic layer. The diagonal matrix is composed of the surface densities of components 2 and 3. The vector composed of the external forces acting on components 2 and 3. This is a vector composed of the ratios of stiffness to mass of each layer of the plate. for The eigenvalue matrix, It is a generalized displacement vector. For generalized load vectors, For the first The ratio of plate stiffness to mass. For the first Generalized displacement of a plate The parameters introduced for the purpose of concise expression. For the first Generalized load.
[0022] Furthermore, in S2, elastic waves are used to describe the vibration characteristics of each component in the layered region and the intact region, respectively, and vibration state vector expressions for each region are established, specifically including:
[0023] For the delamination region of the laminate, the corresponding first State vector of the generalized vibration problem for:
[0024]
[0025] in, For waveform matrix, A matrix composed of wave propagation matrices. It is the amplitude vector composed of positive and negative waves;
[0026] When the layered region is subjected to external excitation, resulting in a direct excitation wave, the formula is:
[0027]
[0028] in, and These are positive and negative excitation waves, respectively. for 1st-fold matrix, It is a 2nd order symplectic matrix. As an external incentive, For the Dirac function, The spatial coordinates of the excitation point;
[0029] Then the first layer of the hierarchical region The expression for the first-order vibration state vector is:
[0030]
[0031] in, For the first The elastic wave amplitude vector of the plate, It is the product of the unit step function and the wave propagation matrix. For excitation waves;
[0032] Vibration state vector expression of the intact region for:
[0033]
[0034] in, For waveform matrix, A matrix composed of wave propagation matrices. For the first The elastic wave amplitude vector of the plate, It is the product of the unit step function and the wave propagation matrix. To excite the wave, These correspond to component 1 and component 4, respectively. Component 1 and component 4 are intact areas located on the left and right sides of the layered region.
[0035] Furthermore, the determination of compatibility relationships in S3 based on displacement continuity and internal force equilibrium conditions at the coupling point between the layered region and the intact region specifically includes:
[0036] The internal force equilibrium conditions and displacement continuity conditions at the coupling points between the left and right ends of components 2 and 3 and the intact region in the layered region are described respectively, resulting in a total of six compatibility conditions:
[0037] The sum of the internal forces of component 2 at its left end and component 3 at its left end is equal to the internal force of component 1 at its right end.
[0038] The sum of the internal forces of component 2 at its right end and component 3 at its right end is equal to the internal force of component 4 at its left end.
[0039] The displacement of component 2 at its left end is equal to the displacement of component 1 at its right end;
[0040] The displacement of component 2 at its right end is equal to the displacement of component 4 at its left end;
[0041] The displacement of component 3 at its left end is equal to the displacement of component 1 at its right end;
[0042] The displacement of component 3 at its right end is equal to the displacement of component 4 at its left end.
[0043] Furthermore, the boundary conditions of the composite material laminate in S3 are as follows:
[0044] The boundary conditions for component 1 at its left end are:
[0045]
[0046] Left multiplied Integrating, we get:
[0047]
[0048] in, The indicator matrix describes the left boundary conditions. This is the waveform matrix for the first board. The wave propagation matrix of the first plate. The amplitude vector of the first plate. This is the negative wave waveform matrix of the first plate. This is the product of the unit step function of the first plate and the wave propagation matrix. Let be the coordinates of the excitation position of the first plate. The negative excitation wave of the first plate, This is the positive waveform matrix of the first board. For matrix transpose, and Let these be the positive and negative wave amplitude vectors of the first plate. and for regional elements, This is the intermediate process matrix introduced. For the relevant excitation vector;
[0049] The boundary conditions for component 4 at its right end are:
[0050]
[0051] Left multiplication Integrating, we get:
[0052]
[0053] in, The indicator matrix describes the right-hand boundary conditions. The waveform matrix of the fourth plate. The wave propagation matrix of the fourth plate. The length of the fourth board, The amplitude vector of the fourth plate. The positive wave waveform matrix of the fourth plate. and These are the positive and negative amplitude vectors of the fourth plate. and for regional elements, This is the intermediate process matrix introduced.
[0054] Furthermore, the structural dynamic equilibrium equations established in wave space in S3 are as follows:
[0055]
[0056]
[0057]
[0058]
[0059] in, , , , This is an intermediate quantity that is introduced. , , , For the region elements of the introduced intermediate process matrix.
[0060] The beneficial effects of this invention are:
[0061] First, the composite laminate structure is divided into different regions based on the damage condition, including delaminated regions and intact regions (i.e., non-delaminated regions). The delaminated regions are connected to the intact regions at their left and right ends. This invention considers two states of the delaminated regions: partial separation and complete separation. Complete separation means that adjacent parts of the delaminated region have no relationship, while partial separation means that there is still a coupling relationship between two adjacent parts, although the coupling stiffness is attenuated to varying degrees compared to the intact state. The method of this invention considers the case of incomplete separation in delaminated damage and can degenerate into the case of complete separation, possessing generality and enabling more efficient and comprehensive delaminated damage analysis.
[0062] Secondly, the dynamics of the layered regions are decoupled, and the vibration of each component in each region is described based on the theory of elastic wave propagation. The dynamic decoupling technique used in this invention can transform the coupled multi-layered dynamic equations into multiple independent generalized single-layered dynamic problems, resulting in higher computational efficiency, especially for layered problems with a large number of layers.
[0063] Then, by utilizing the displacement continuity and internal force equilibrium conditions at the coupling points of the layered and intact regions, as well as the structural boundary conditions, the dynamic equilibrium equations of the structure in wave space are established. The method of this invention solves the vibration response of layered composite laminate structures in wave space, exhibiting higher convergence compared to the traditional modal superposition method. This is because the wave modes used in this invention precisely satisfy the vibration equations of the structure at the analyzed frequencies, while the modal superposition method uses natural modes to superimpose the vibration response of the structure, and the natural modes only satisfy the vibration equations of the structure at the natural frequencies.
[0064] Finally, solving this equation yields the amplitude of each order of elastic wave, and thus the vibration response of each component in the composite laminate structure. Based on the invented calculation method, the influence of delamination location and length on the structural vibration characteristics can be studied efficiently. The method of this invention is based on analytical waves, therefore it can perform parameter influence analysis more efficiently than the traditional finite element method, without the need for repetitive simulation model building. Attached Figure Description
[0065] Figure 1 This is a flowchart of a method for calculating the vibration response of a laminated plate containing layered composite materials.
[0066] Figure 2 This is a schematic diagram showing the configuration and dimensions of a laminated plate structure containing layered composite materials.
[0067] Figure 3 Frequency response curves for composite laminate structures with fully separated delamination.
[0068] Figure 4 Frequency response curves for composite laminate structures with incompletely separated delamination.
[0069] Figure 5 A schematic diagram showing the influence of different layering positions on the vibration response of composite laminate structures.
[0070] Figure 6 A schematic diagram showing the effect of different layer lengths on the vibration response of composite laminate structures. Detailed Implementation
[0071] The present invention will be further described below with reference to the accompanying drawings and specific embodiments.
[0072] like Figure 1 As shown, a method for calculating the vibration response of a laminated plate containing layered composite materials includes the following steps:
[0073] S1: The composite laminate containing delamination defects is divided into delamination regions and intact regions according to the damage condition. The delamination regions include two states: incomplete separation and complete separation. The interaction between adjacent layers in the delamination region under the incomplete separation state is described by uniformly distributed spring connections. When the spring stiffness is zero, it degenerates into the complete separation state.
[0074] This embodiment uses Figure 2 Taking the composite material plate structure shown as an example, consider a rectangular composite laminate plate with a layered region inside. For example... Figure 2 As shown, delamination occurs at the mid-surface of the laminate, naturally dividing it into four parts. Parts 1 and 4 are two intact regions; parts 2 and 3 have the same geometric features and together constitute the delaminated region. The different parts of the laminate exhibit displacement continuity and internal force equilibrium, as shown... Figure 2 As shown, it is expressed as follows:
[0075] , , ,
[0076] , , , (1)
[0077] in, , , , These are lateral displacement, angular displacement, lateral shear force, and bending moment, respectively, indicated by the subscript "". "and" "Refers to "left" and "right" respectively, for example, This means that the right end of the first component, the left end of the second component, and the left end of the third component have the same lateral displacement.
[0078] The case of incomplete separation between the layered components (component 2 and component 3) is considered. A uniformly distributed spring connection is used to describe the interaction between component 2 and component 3, such as... Figure 2 As shown. If the spring stiffness is set to zero, it degenerates into a completely separated case.
[0079] S2: Dynamically decouple the layered regions. Based on the theory of elastic wave propagation, use elastic waves to describe the vibration characteristics of each component in the layered and intact regions respectively, and establish the vibration state vector expression for each region.
[0080] Dynamic decoupling of layered regions specifically includes:
[0081] The spring layer describing the interlayer action in the layered region is equivalent to a continuous elastic layer; therefore, the stiffness of the elastic layer between component 2 and component 3 is... ,in, For the stiffness of the elastic layer, For spring stiffness, The number of springs and The length and width of the laminated plate in the layered region;
[0082] Establish the governing equations for bending vibration in the layered region:
[0083] (2)
[0084] in, , For the first The bending stiffness of the slab, , , , and These are the bending stiffness, lateral displacement, elastic modulus, thickness, and Poisson's ratio of each component. For operators, and These represent the displacements of each component. For surface density, , It is volume density. For the first Shelves and the first The stiffness of the elastic layer between the layers, For the first Shelves and the first The stiffness of the elastic layer between the layers, As an external force, Corresponding to components 2 and 3 respectively, For time;
[0085] Considering the steady-state vibration problem, , Substituting into the bending vibration control equation, we obtain the vibration equation in matrix form:
[0086] (3)
[0087] Decoupling it yields the generalized vibration equation:
[0088] or (4)
[0089] in, For spatial coordinates, For the first The external load on each plate For the first The displacement of the plate Indicates the first Step, For the excitation frequency, Let be the diagonal matrix composed of the bending stiffnesses of components 2 and 3 in the layered region. Let be the displacement vector composed of the lateral displacements of components 2 and 3. Here is the stiffness matrix of the elastic layer. The diagonal matrix is composed of the surface densities of components 2 and 3. The vector composed of the external forces acting on components 2 and 3. Let be a vector composed of the ratios of stiffness to mass of each plate. for The eigenvalue matrix, , It is a generalized displacement vector. It is the eigenvector matrix. For generalized load vectors, For the first The ratio of plate stiffness to mass For the first The bending stiffness of the plate. For the first The quality of each board , For the first The generalized displacement of the plate, Parameters introduced for the sake of simplicity. yes The eigenvalues of order, For the first Generalized load.
[0090] Starting from equation (4), new kinematic and dynamic variables can be introduced, namely... , , , , , , , Furthermore, from equation (4), the following partial differential equation can be obtained:
[0091] (5)
[0092] in, It is a state vector composed of new kinematic and dynamic variables. It is a partial differential matrix operator, expressed as:
[0093]
[0094] in, For the first The Poisson's ratio of each plate, equations (4) and (5) are in fact generalized vibration equations. By solving equation (5), the first plate can be obtained. Wave propagation parameters of generalized vibration problems and waveform .
[0095] Elastic waves are used to describe the vibration characteristics of each component in the layered region and the intact region, respectively, and vibration state vector expressions for each region are established, specifically including:
[0096] For the delamination region of the laminate, the corresponding first State vector of the generalized vibration problem for:
[0097] (6)
[0098] in, For waveform matrix, For the first The positive and negative waveform matrices of each board. For the first Before the board Step waveform, The number of positively traveling waves. ( A matrix composed of wave propagation matrices. For the wave propagation matrix, It is the amplitude vector composed of positive and negative waves. For the first The positive wave amplitude on the left side and the negative wave amplitude on the right side of the board, with the superscript "+" and "-" indicating positive and negative movement respectively;
[0099] When the layered region is subjected to external excitation, resulting in a direct excitation wave, the formula is:
[0100] (7)
[0101] in, and These are positive and negative excitation waves, respectively. , , , They are 2nd order and An identity matrix of order 1 As an external incentive, For the Dirac function, The spatial coordinates of the excitation point;
[0102] At this point, the generalized vibration response must consider the direct excitation wave. The contribution of the hierarchical region is the first The expression for the first-order vibration state vector is:
[0103] (8)
[0104] in, For the first The amplitude vector of each plate, unit step function With wave propagation matrix The product of , ; , ;
[0105] Vibration state vector expression for intact regions (components 1 and 4) for:
[0106] (9)
[0107] in, For waveform matrix, A matrix composed of wave propagation matrices. For the first The amplitude vector of each plate, It is the product of the unit step function and the wave propagation matrix. To excite the wave, These correspond to component 1 and component 4, respectively. Component 1 and component 4 are intact areas located on the left and right sides of the layered region.
[0108] S3: Based on the displacement continuity and internal force equilibrium conditions at the coupling point between the layered region and the intact region, the compatibility relationship is determined, and combined with the boundary conditions of the composite laminate, the structural dynamic equilibrium equation in wave space is established.
[0109] The compatibility relationship is determined based on the displacement continuity and internal force equilibrium conditions at the coupling point between the layered and intact regions, specifically including:
[0110] Introduction This represents the contribution of the first generalized vibration response to component 1. This represents the contribution of the second generalized vibration response to component 1. This represents the contribution of the first generalized vibration response to component 2. This represents the contribution of the second generalized vibration response to component 2. This indicates that only internal force elements are extracted. This indicates that only the shifted elements are extracted, where, , , , This represents the contribution coefficient to the kinematic variable. , , , This represents the contribution coefficient to the dynamic variable. Represents a unit array.
[0111] The internal force equilibrium conditions and displacement continuity conditions at the coupling points between the left and right ends of components 2 and 3 and the intact region in the layered region are described respectively, resulting in a total of six compatibility conditions:
[0112] The sum of the internal forces at the left end of component 2 and the left end of component 3 is equal to the internal force at the right end of component 1.
[0113] (10)
[0114] in, , , This represents the waveform matrix for the 1st, 2nd, and 3rd plates. , , The wave propagation matrices for the 1st, 2nd, and 3rd plates are shown. , , The amplitude vectors of the 1st, 2nd, and 3rd plates are... The length of the first board, The positive wave waveform matrix of the first plate. Let these be the coordinates of the location of the first board excitation. This is the positive excitation wave for the first plate;
[0115] The sum of the internal forces at the right end of component 2 and the right end of component 3 is equal to the internal force at the left end of component 4.
[0116] (11)
[0117] in, The waveform matrix of the fourth plate. The wave propagation matrix of the fourth plate. The amplitude vector of the fourth plate. The length of the second plate;
[0118] The displacement of component 2 at its left end is equal to the displacement of component 1 at its right end:
[0119] (12)
[0120] The displacement of component 2 at its right end is equal to the displacement of component 4 at its left end:
[0121] (13)
[0122] The displacement of component 3 at its left end is equal to the displacement of component 1 at its right end:
[0123] (14)
[0124] The displacement of component 3 at its right end is equal to the displacement of component 4 at its left end:
[0125] (15)
[0126] Multiply the left side of equation (10) by Integrate and introduce intermediate quantities , , , Then we get:
[0127] (16)
[0128] Multiply the left side of equation (11) by Integrate and introduce intermediate quantities , , Then we get:
[0129] (17)
[0130] Multiply the left side of equation (12) by Integrate and introduce intermediate quantities , , , Then we get:
[0131] (18)
[0132] Multiply the left side of equation (13) by Integrate and introduce intermediate quantities Then we get:
[0133] (19)
[0134] Multiply the left side of equation (14) by Integrate and introduce intermediate quantities , , , Then we get:
[0135] (20)
[0136] Multiply the left side of equation (15) by Integrate and introduce intermediate quantities Then we get:
[0137] (twenty one)
[0138] The boundary conditions for the composite laminate are:
[0139] Introduction and These refer to the left and right boundary conditions, respectively. For example, for a fixed boundary, there is... Considering the boundary conditions of the laminate at the left end, the boundary conditions of component 1 at its left end are as follows:
[0140] (twenty two)
[0141] Considering the boundary conditions of the laminate at the right end, i.e., the boundary conditions of component 4 at its right end, are as follows:
[0142] (twenty three)
[0143] Multiply equations (22) and (23) on the left by respectively , And integrate, to obtain
[0144] (twenty four)
[0145] (25)
[0146] in, The waveform matrix of the first board. Here is the wave propagation matrix for the first plate. Let the amplitude vector of the first plate be... The negative wave waveform matrix of the first plate. This is the product of the wave propagation matrix of the first plate and the unit step function. Let be the coordinates of the excitation position of the first plate. The negative excitation wave of the first plate, The positive wave waveform matrix of the first plate. For matrix transpose, and Let these be the positive and negative amplitude vectors of the first plate. and for regional elements, The matrix introduced for the intermediate process. The matrix introduced for the intermediate process. The matrix introduced for the intermediate process. A matrix introduced for the intermediate process; The waveform matrix of the fourth plate. The wave propagation matrix of the fourth plate. The length of the fourth board, The amplitude vector of the fourth plate. This is the positive waveform matrix of the fourth plate. and The positive and negative wave amplitude vectors of the fourth plate. and for regional elements, The matrix introduced for the intermediate process. The matrix introduced for the intermediate process, superscript " "" represents finding the inverse of a matrix.
[0147] The structural dynamic equilibrium equations in wave space are established as follows:
[0148] From equations (18) and (19), we can obtain:
[0149] (26)
[0150] in, , , , These are all intermediate values introduced for convenience.
[0151] From equations (20) and (21), we can obtain:
[0152] (27)
[0153] in, , , , This intermediate quantity was also introduced for convenience.
[0154] From equations (26) and (27), we can further obtain:
[0155] (28)
[0156] (29)
[0157] From equations (28) and (29), we can obtain:
[0158] (30)
[0159] For convenience, an intermediate quantity is introduced here, namely... , , , .
[0160] In fact, It is always a zero matrix, therefore and Since it is a zero matrix, we can obtain the following from equations (16) and (17):
[0161] (31)
[0162] From equations (30) and (31), we can obtain:
[0163] (32)
[0164] For convenience, an intermediate quantity is introduced, namely... , , , , , , For the region elements of the introduced intermediate process matrix;
[0165] Will Write the component region matrix, that is ,Will and Write the component region vector, that is , Then equation (32) can be written as:
[0166] (33)
[0167] Combining equations (24), (25) and (33), we can obtain:
[0168] (34)
[0169] For convenience, an intermediate quantity is introduced, namely... , , .
[0170] S4: Solve the dynamic equilibrium equation of the structure to obtain the amplitude of each order of elastic waves, and obtain the vibration response of each region of the composite laminate based on elastic wave theory and amplitude calculation.
[0171] The amplitude can be obtained from equation (34). and Then, from equations (24) and (25), we can obtain... and That is, we obtained and Then, from equation (30), we obtain Then, from equation (29), we obtain .Depend on and Combining equation (9) yields the vibration responses of components 1 and 4. and Combining equation (8) yields the state vector of the generalized vibration problem in the layered region, and then through... , , , Then the vibration response of the layered region can be obtained.
[0172] In one embodiment of the present invention, Figure 3 The frequency response curve of a composite laminate structure with completely separated delamination is shown. The vibration response at the excitation point in the completely separated state of components 2 and 3 was calculated using both the finite element method and the method of this invention, namely the analytical wave method. It can be seen that the calculation results of the two methods are consistent, verifying the effectiveness of the method of this invention in solving the vibration response of composite laminates with completely separated delamination.
[0173] Figure 4 The frequency response curve is shown for a composite laminate structure with incomplete delamination. The vibration response at the excitation point in the incompletely delamination state of components 2 and 3 was calculated using both the finite element method and the method of this invention (analytical wave method). It can be seen that the calculation results of the two methods are consistent, verifying the effectiveness of the method of this invention in solving the vibration response of composite laminates with incomplete delamination.
[0174] Figure 5 This diagram illustrates the impact of different delamination locations on the vibration response of composite laminate structures. The vibration response of composite laminate structures with delamination damage occurring at different locations was calculated using the method of this invention. It can be seen that the delamination location significantly affects the vibration response of the structure, providing a basis for detecting the location of delamination damage in composite laminates based on vibration response.
[0175] Figure 6 This diagram illustrates the influence of different delamination lengths on the vibration response of composite laminate structures. The vibration response of composite laminate structures with different delamination damage lengths was calculated using the method of this invention. It can be seen that the length of the delamination region significantly affects the vibration response of the structure, providing a basis for detecting the size of delamination damage regions in composite laminates based on vibration response.
[0176] 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 method for calculating the vibration response of a laminated plate containing layered composite materials, characterized in that, Includes the following steps: S1: The composite laminate containing delamination defects is divided into delamination regions and intact regions according to the damage condition. The delamination regions include two states: incomplete separation and complete separation. The interaction between adjacent layers in the delamination region under the incomplete separation state is described by uniformly distributed spring connections. When the spring stiffness is zero, it degenerates into the complete separation state. S2: Dynamically decouple the layered regions. Based on the theory of elastic wave propagation, use elastic waves to describe the vibration characteristics of each component in the layered and intact regions respectively, and establish the vibration state vector expression for each region. S3: Based on the displacement continuity and internal force equilibrium conditions at the coupling point between the layered region and the intact region, the compatibility relationship is determined, and combined with the boundary conditions of the composite laminate, the structural dynamic equilibrium equation in wave space is established. S4: Solve the dynamic equilibrium equation of the structure to obtain the amplitude of each order of elastic waves, and obtain the vibration response of each region of the composite laminate based on elastic wave theory and amplitude calculation.
2. The method for calculating the vibration response of laminated plates containing layered composite materials according to claim 1, characterized in that, The dynamic decoupling of the layered regions in S2 specifically includes: The spring layer describing the interlayer action in the layered regions is equivalent to a continuous elastic layer, and the stiffness of the elastic layer between the layered regions is determined as follows: ； in, For the stiffness of the elastic layer, For spring stiffness, The number of springs and The length and width of the laminated plate in the layered region; Establish the governing equations for bending vibration in the layered region: ； in, For the first Bending stiffness of the slab, For operators, , and These represent the displacements of each component. For surface density, For the first Shelves and the first The stiffness of the elastic layer between the layers, For the first Shelves and the first The stiffness of the elastic layer between the layers, As an external force, For time; Considering the steady-state vibration problem, , Substituting into the bending vibration control equation, we obtain the vibration equation in matrix form: ； Decoupling it yields the generalized vibration equation: or ; in, For spatial coordinates, For the first The external load on each plate For the first Displacement of the shelf, Indicates the first Step, For the excitation frequency, Let be the diagonal matrix composed of the bending stiffnesses of components 2 and 3 in the layered region. Let be the displacement vector composed of the lateral displacements of components 2 and 3. Here is the stiffness matrix of the elastic layer. The diagonal matrix is composed of the surface densities of components 2 and 3. The vector composed of the external forces acting on components 2 and 3. This is a vector composed of the ratios of stiffness to mass of each layer of the plate. for The eigenvalue matrix, It is a generalized displacement vector. For generalized load vectors, For the first The ratio of plate stiffness to mass. For the first Generalized displacement of a plate The parameters introduced for the purpose of concise expression. For the first Generalized load.
3. The method for calculating the vibration response of laminated plates containing layered composite materials according to claim 2, characterized in that, In S2, elastic waves are used to describe the vibration characteristics of each component in the layered region and the intact region, respectively, and vibration state vector expressions for each region are established, specifically including: For the delamination region of the laminate, the corresponding first State vector of the generalized vibration problem for: ； in, For waveform matrix, A matrix composed of wave propagation matrices. It is the amplitude vector composed of positive and negative waves; When the layered region is subjected to external excitation, resulting in a direct excitation wave, the formula is: ； in, and These are positive and negative excitation waves, respectively. for 1st-fold matrix, It is a 2nd order symplectic matrix. As an external incentive, For the Dirac function, The spatial coordinates of the excitation point; Then the first layer of the hierarchical region The expression for the first-order vibration state vector is: ； in, For the first The elastic wave amplitude vector of the plate, It is the product of the unit step function and the wave propagation matrix. For excitation waves; Vibration state vector expression of the intact region for: ； in, For waveform matrix, A matrix composed of wave propagation matrices. For the first The elastic wave amplitude vector of the plate, It is the product of the unit step function and the wave propagation matrix. To excite the wave, These correspond to component 1 and component 4, respectively. Component 1 and component 4 are intact areas located on the left and right sides of the layered region.
4. The method for calculating the vibration response of laminated plates containing layered composite materials according to claim 3, characterized in that, The compatibility relationship determined in S3 based on the displacement continuity and internal force equilibrium conditions at the coupling point between the layered region and the intact region specifically includes: The internal force equilibrium conditions and displacement continuity conditions at the coupling points between the left and right ends of components 2 and 3 and the intact region in the layered region are described respectively, resulting in a total of six compatibility conditions: The sum of the internal forces of component 2 at its left end and component 3 at its left end is equal to the internal force of component 1 at its right end. The sum of the internal forces of component 2 at its right end and component 3 at its right end is equal to the internal force of component 4 at its left end. The displacement of component 2 at its left end is equal to the displacement of component 1 at its right end; The displacement of component 2 at its right end is equal to the displacement of component 4 at its left end; The displacement of component 3 at its left end is equal to the displacement of component 1 at its right end; The displacement of component 3 at its right end is equal to the displacement of component 4 at its left end.
5. The method for calculating the vibration response of laminated plates containing layered composite materials according to claim 4, characterized in that, The boundary conditions for the composite laminate in S3 are as follows: The boundary conditions for component 1 at its left end are: ； Left multiplied Integrating, we get: ； in, The indicator matrix describes the left boundary conditions. This is the waveform matrix for the first board. The wave propagation matrix of the first plate. The amplitude vector of the first plate. This is the negative wave waveform matrix of the first plate. This is the product of the unit step function of the first plate and the wave propagation matrix. Let be the coordinates of the excitation position of the first plate. The negative excitation wave of the first plate, This is the positive waveform matrix of the first board. For matrix transpose, and Let these be the positive and negative wave amplitude vectors of the first plate. and for regional elements, This is the intermediate process matrix introduced. For the relevant excitation vector; The boundary conditions for component 4 at its right end are: ； Left multiplication Integrating, we get: ； in, The indicator matrix describes the right-hand boundary conditions. The waveform matrix of the fourth plate. The wave propagation matrix of the fourth plate. The length of the fourth board. The amplitude vector of the fourth plate. The positive wave waveform matrix of the fourth plate. and These are the positive and negative amplitude vectors of the fourth plate. and for regional elements, This is the intermediate process matrix introduced.
6. The method for calculating the vibration response of laminated plates containing layered composite materials according to claim 5, characterized in that, The structural dynamic equilibrium equations established in wave space in S3 are as follows: ； ； ； ； in, , , , This is an intermediate quantity that is introduced. , , , For the region elements of the introduced intermediate process matrix.