Method for identifying elastic modulus and loss factor of functionally graded coating in thermal environment based on vibration test

Abstract

The functionally graded coating elastic modulus and loss factor identification method based on vibration test in thermal environment belongs to the technical field of structural vibration and material parameter identification, and comprises the following steps: a metal plate specimen coated with a functionally graded coating is prepared, and structural geometric parameters, component material parameters and thermal vibration environment parameters are obtained; vibration excitation is applied to the metal plate specimen under thermal environment conditions, and the test inherent frequency and test damping ratio of the target order are obtained; an equivalent material parameter model of the functionally graded coating is established, the equivalent Young's modulus, shear modulus, Poisson's ratio and density of the coating material are obtained, and the transformation stiffness coefficient is calculated; based on the first-order shear deformation theory and the thermal plasticity theory, an analytical dynamics model of the metal plate specimen under thermal environment is established, and the theoretical inherent frequency and theoretical damping ratio of the corresponding order are obtained; an error function with inherent frequency and resonance response as constraints is constructed, and the equivalent elastic modulus and loss factor of the functionally graded coating are obtained through iterative optimization based on the particle swarm algorithm.

Description

Technical Field

[0001] This invention belongs to the field of structural vibration and material parameter identification technology, and relates to a method for identifying the elastic modulus and loss factor of functionally graded coatings under thermal conditions based on vibration testing. Background Technology

[0002] Functionally graded coatings (FGCs), as advanced surface engineering materials with continuously or graded variations in composition and properties along their thickness, have been widely used in aerospace, energy, and high-end equipment to enhance the high-temperature resistance, oxidation resistance, wear resistance, corrosion resistance, and vibration and noise reduction performance of substrate structures. Existing research on the mechanical properties of coating materials largely focuses on material preparation processes, gradient structure design, interfacial bonding quality, microhardness, and protective performance evaluation. However, research on the changes in the mechanical properties of coatings under actual service conditions (such as thermal vibration environments), especially the identification and performance changes of key dynamic mechanical parameters such as elastic modulus, shear modulus, Poisson's ratio, and loss factor, remains limited.

[0003] In the limited existing research involving the identification of coating material parameters, most methods focus on room temperature conditions, homogeneous coatings, or the static mechanical properties of coatings. These methods typically employ indentation, bending, ultrasonic, or room temperature modal testing to obtain parameters such as the equivalent elastic modulus and damping ratio of the coating. While these methods can standardize the dynamic mechanical parameters of the coating to some extent, they generally do not consider the impact of thermal environment changes on the coating material properties, particularly the influence of thermal environment changes on the elastic modulus and loss factor of functionally graded coatings, which exhibit anisotropic mechanical characteristics. For example, Chinese patent CN201610872414.2 proposes a method and system for identifying the mechanical property parameters of hard coating materials. This patent establishes a finite element model of a hard-coated composite thin-shell structure and matches the experimentally measured natural frequencies and resonance responses with theoretical calculations to inversely deduce the storage modulus and loss factor of the hard coating material. However, this patent primarily focuses on the identification of parameters for homogeneous hard coatings under room temperature conditions and does not address the impact of material parameter changes under thermal environment on the structural dynamics. Chinese patent CN202010443618.0 proposes a method and device for identifying the mechanical properties of vibration-damping coatings. This method targets vibration-damping coatings applied to the inner spherical surface of the outer ring of a spherical bearing. It uses finite element analysis to obtain the simulated loading curve exponent and maximum indentation depth, and then combines this with nanoindentation test results to identify the coating's mechanical property parameters. However, this patent is essentially a localized identification method combining nanoindentation and finite element analysis, and it does not consider the influence of material parameter changes under thermal conditions on the identification results. Chinese patent CN201610799244.X discloses a method for measuring the elastic modulus of a coating. This method obtains parameters such as the elastic modulus of the coating composite sample, the elastic modulus of the substrate, and the thickness of the substrate and coating. This patent has some practical value for the rapid acquisition of the coating's elastic modulus, but it focuses on measuring a single elastic modulus parameter and lacks a comprehensive consideration of the influence of temperature effects on the coating's mechanical parameters. Chinese patent CN201610289698.2 proposes a method for ultrasonic non-destructive measurement of the thickness and elastic modulus of wear-resistant coatings. This method, based on pulse-echo technology, extracts longitudinal and transverse wave signals and performs spectral analysis and parameter inversion to achieve simultaneous measurement of the coating thickness and elastic modulus. However, this patent does not consider the impact of changes in coating material parameters caused by thermal environment variations. Chinese patent CN201410667957.1 proposes a method and apparatus for identifying material damping and elastic modulus. This method uses a benchmarking approach between experimental and simulated transfer functions, obtaining the sample's frequency response function through a hammer impact method, and continuously adjusting the structural damping and elastic modulus parameters in the simulation model to ensure consistency between simulation and experimental results, thereby obtaining the material's damping and elastic modulus. However, this method does not consider the impact of temperature changes under thermal environment conditions on the identification results of the elastic modulus and damping parameters. Summary of the Invention

[0004] To address the aforementioned problems in identification, given that the material composition in functionally graded coatings changes continuously along the thickness direction, and existing identification methods do not consider the changes in material mechanical parameters and structural dynamics under temperature, it is difficult to accurately invert and identify relevant parameters under thermal conditions. This invention proposes a method for identifying the elastic modulus and loss factor of functionally graded coatings under thermal conditions based on vibration testing, so as to achieve accurate identification of key dynamic parameters of functionally graded coatings under thermal vibration conditions.

[0005] This invention provides a method for identifying the elastic modulus and loss factor of functionally graded coatings under thermal conditions based on vibration testing, comprising:

[0006] Step 1: Prepare metal plate specimens coated with functionally graded coatings, and obtain the structural geometric parameters, composition material parameters, and thermal shock environment parameters of the metal plate specimens;

[0007] Step 2: Apply basic vibration excitation to the metal plate specimen under thermal environment conditions, obtain the test natural frequency of the target order by frequency sweep test, obtain the test damping ratio of the metal plate specimen by fixed frequency resonance test, and use the test natural frequency and test damping ratio as the constraint quantities for material parameter identification in Step 5.

[0008] Step 3: Establish the equivalent material parameter model of the functionally graded coating based on the volume fraction distribution law, obtain the equivalent Young's modulus, shear modulus, Poisson's ratio and density of the coating material according to the Voigt mixing rule, and characterize the damping using the complex modulus method. Write the elastic modulus and shear modulus in complex form, and obtain the transformation stiffness coefficients of the functionally graded coating and the substrate plate.

[0009] Step 4: Based on the first-order shear deformation theory, thermoplasticity theory and Rayleigh-Ritz method, establish an analytical dynamic model of the metal plate specimen under thermal conditions, give the displacement field, strain field, constitutive relation, internal force and moment expressions and energy expressions, and obtain the theoretical natural frequencies and theoretical damping ratios of the corresponding orders.

[0010] Step 5: Construct an error function constrained by the natural frequency and resonant response. Based on the particle swarm optimization algorithm, iteratively update the equivalent elastic modulus and loss factor material parameters of the functionally graded coating within a preset parameter range and step size, and output the identification results according to the convergence criterion.

[0011] This invention discloses a method for identifying the elastic modulus and loss factor of functionally graded coatings under thermal conditions based on vibration testing. It combines frequency sweeping to obtain the natural frequency with fixed-frequency resonance to obtain the amplitude response, constructing a joint "frequency-amplitude" error function. Based on first-order shear deformation theory, the Ritz analytical model, and complex modulus damping characterization, a particle swarm optimization algorithm is used for iterative solution, achieving traceable identification of the coating's equivalent elastic modulus and loss factor. This method has low computational cost, a clear mechanism, and controllable convergence, making it more efficient than finite element inversion and more suitable for functionally graded coatings than homogeneous coating methods. Attached Figure Description

[0012] Figure 1 This is a flowchart of a method for identifying the elastic modulus and loss factor of a functionally graded coating under thermal conditions based on vibration testing, according to the present invention. Detailed Implementation

[0013] like Figure 1 As shown, the method for identifying the elastic modulus and loss factor of functionally graded coatings under thermal conditions based on vibration testing according to the present invention includes:

[0014] Step 1: Prepare metal plate specimens coated with functionally graded coatings, and obtain the structural geometric parameters, compositional material parameters, and thermal shock environment parameters of the metal plate specimens, specifically:

[0015] Step 1.1: Prepare the mixed slurry required for each layer of the functional gradient coating according to the preset gradient distribution law. Add component material A and component material B to multiple containers in different proportions and add curing agent. After stirring, the spray slurry for each layer is obtained.

[0016] Step 1.2: After roughening the surface of the substrate plate by grinding, the slurry is deposited layer by layer on both sides of the substrate plate by atomization spraying. After each layer is sprayed, a curing process is performed. The spraying-curing process is repeated until a functional gradient coating with a preset number of layers is formed, and a metal plate specimen coated with a functional gradient coating is produced.

[0017] Step 1.3: Define a global coordinate system o-xyz on the metal plate specimen coated with functionally graded coating, located on the mid-surface of the metal plate specimen.

[0018] Step 1.4: Obtain the structural geometric parameters of the metal plate specimen, including the length, width and thickness of the metal plate specimen, the total thickness of the functional gradient coating applied to one side of the substrate plate and the total number of layers.

[0019] Step 1.5: Obtain the component material parameters of the metal plate specimen, including the Young's modulus, shear modulus, Poisson's ratio, density, and loss factor of functionally graded coating component material A, component material B, and the substrate plate.

[0020] Step 1.6: Obtain the initial parameters for iteration, including the functionally graded coating component material A, component material B, and the equivalent Young's modulus and loss factor of each coating layer. Based on this, set the initial values, allowable value ranges, and iteration step size of each parameter to be identified, forming the initial parameter vector used for iterative identification.

[0021] Step 2: Apply basic vibration excitation to the metal plate specimen under thermal conditions. Obtain the target order test natural frequency using frequency sweep testing, and obtain the test damping ratio of the metal plate specimen using constant frequency resonance testing. Use the test natural frequency and test damping ratio as constraint quantities for material parameter identification in Step 5, specifically:

[0022] Step 2.1: Specimen installation and boundary condition realization: Install the metal plate specimen on the special fixture of the vibration table, and install the fixture on the vibration table to form a single-sided fixed support boundary condition; then extend the metal plate specimen into the hot box.

[0023] Step 2.2: Basic excitation loading method and control parameter setting: A thermal environment is established using a heating chamber, temperature controller, and thermocouple sensor. The metal plate specimen is heated to several preset temperatures and then kept stable. Subsequently, a simple harmonic base motion is applied using a vibration table as the basic excitation method. The base displacement is expressed as:

[0024]

[0025] Where Y is the basic displacement amplitude. f is the excitation angular frequency; f is the frequency, and the test frequency band is set. Frequency sweep resolution Scan frequency And set the excitation amplitude to remain constant.

[0026] Step 2.3: Obtain the natural frequency of the test by sweeping frequency test. Perform a sweeping frequency test on the metal plate specimen within the set test frequency band, collect the input signal and output response signal, obtain the vibration frequency response curve of the target order, and then obtain the natural frequency of the corresponding target order.

[0027] Step 2.4: Obtaining damping parameters through constant-frequency resonance test. Using each natural frequency obtained in Step 2.3 as the target frequency, constant-frequency excitation is performed under the same excitation amplitude. After the system reaches steady-state response, the output response signal is collected to obtain the corresponding vibration frequency response curve. Subsequently, the half-power bandwidth method is used to analyze the vibration frequency response curve to obtain the test damping ratio of the metal plate specimen.

[0028] Step 3: Based on the volume fraction distribution law, establish the equivalent material parameter model of the functionally graded coating. According to the Voigt mixing rule, obtain the equivalent Young's modulus, shear modulus, Poisson's ratio, and density of the coating material. Characterize the damping using the complex modulus method. Express the elastic modulus and shear modulus in complex form, and calculate the transformation stiffness coefficients of the functionally graded coating and the substrate plate. Specifically:

[0029] Step 3.1: Propose a volume fraction distribution model for the functionally graded coating along the thickness direction. Assuming that the material properties of the mixed functionally graded coating vary with thickness and follow a distribution law, the volume fraction is expressed as:

[0030]

[0031]

[0032] in, z represents the coordinates of any point on the coating along the z-axis, z0 represents the coordinates of the first coating layer along the z-axis, z k d represents the coordinate of the outer coating layer in the z-axis direction; d represents the volume gradient index of the functionally graded coating, which is used to determine the variation trend of component material B to component material A along the thickness direction of the functionally graded coating. and These are the volume fractions of component materials A and B, respectively.

[0033] Step 3.2: Establish equivalent material parameters based on Voigt mixing rules:

[0034]

[0035] in, It is the equivalent Young's modulus of the functionally graded coating. and These represent the Young's moduli of component materials A and B, respectively. It is the equivalent Poisson's ratio of the functionally graded coating. and These represent the Poisson's ratios of components A and B, respectively. It is the equivalent density of the functionally graded coating. and These represent the densities of component materials A and B, respectively. It is the equivalent shear modulus of the functionally graded coating.

[0036] Step 3.3: Characterize the damping and obtain the complex modulus using the complex modulus method;

[0037]

[0038]

[0039]

[0040] in, This represents the complex Young's modulus of a functionally graded coating material; This represents the complex shear modulus of a functionally graded coating material; and These are the loss factors of component material A and component material B, respectively. This is the equivalent loss factor of the mixed functional gradient coating.

[0041] Step 3.4: Considering the effect of thermal environment temperature, establish an expression for the relationship between the equivalent material parameters of the functionally graded coating and temperature:

[0042]

[0043]

[0044]

[0045] Where ΔT represents the temperature change and t is the degradation time. This represents the equivalent dynamic Young's modulus of a functionally graded coating that takes into account the effects of temperature changes and degradation time in the thermal environment. This represents the equivalent dynamic loss factor of a functionally graded coating that takes into account the effects of thermal environment temperature changes and degradation time. This represents a series of correlation fitting coefficients that need to be determined, taking into account the effect of temperature on the coating. This represents the equivalent dynamic shear modulus of a functionally graded coating that takes into account the effects of temperature changes and degradation time in the thermal environment.

[0046] Step 3.5: Establish the expression for the equivalent thermal expansion coefficient of the functionally graded coating based on the thermal expansion coefficients of the component materials, and obtain the transformed stiffness coefficient of the functionally graded coating under thermal conditions. :

[0047]

[0048]

[0049]

[0050]

[0051] The transformation stiffness coefficients of the remaining functionally graded coatings are all 0.

[0052] Step 3.6: Derive the stiffness coefficient matrix of the base plate. Since the base plate is considered to be unaffected by temperature, the stiffness coefficient matrix of the base plate is... Represented as:

[0053]

[0054] In the formula, This represents the Young's modulus of a thin sheet of metal substrate. Poisson's ratio for a thin sheet of metal matrix is the transformation stiffness coefficient of the base plate.

[0055] Step 4: Based on the first-order shear deformation theory, thermoplasticity theory, and Rayleigh-Ritz method, establish an analytical dynamic model of the metal plate specimen under thermal conditions, giving the displacement field, strain field, constitutive relation, expressions for internal forces and moments, and energy expressions, and obtaining the theoretical natural frequencies and theoretical damping ratios of the corresponding orders, specifically:

[0056] Step 4.1: Define the displacement field of the structure using the first-order shear deformation theory. To account for the influence of shear deformation, based on the first-order shear deformation theory, the displacement expression for any point on the metal plate specimen is obtained as follows:

[0057]

[0058] Where u, v, w represent the displacement of any point on the metal plate specimen; u0, v0, w0 are the displacements of a point in the mid-plane in the x, y, z directions; φ, ψ are the rotation angles about the y and x axes; t is time; and z is the thickness coordinate.

[0059] Step 4.2: Strain-Displacement Relationship and Shear Correction. The strain expression is derived from the displacement field, and a thermal strain term is introduced into the constitutive relation. Due to the symmetry of the metal plate specimen, the bending and tensile motions are not coupled, and u0 and v0 are neglected. Therefore, when the displacement expression of the metal plate specimen is known, the structural strain expression is as follows:

[0060]

[0061] in, and Let x and y represent the normal strain at any point on the metal plate specimen at room temperature, respectively, in the x and y directions. , and These represent the shear strain at any point on the metal plate specimen at room temperature in the xy, xz, and yz planes, respectively. , , , , , , , .

[0062] Step 4.3: Establish the thermoelastic constitutive relation of the metal plate specimen according to the generalized Hooke's law:

[0063]

[0064] In the formula, Let be the normal stress along the x-direction of the metal plate specimen. Let be the normal stress along the y-direction of the metal plate specimen. Let be the shear stress of the metal plate specimen in the xy plane. Let be the shear stress of the metal plate specimen in the yz plane. The shear stress of the metal plate specimen in the xz plane; This is the stiffness coefficient. These represent the substrate material and the functionally graded coating material, respectively.

[0065] The strain of a metal plate specimen in a thermal environment is expressed according to thermoplastic theory:

[0066]

[0067]

[0068]

[0069] In the formula, and This represents the equivalent coefficient of thermal expansion of the functionally graded coating and the substrate.

[0070] Step 4.4: Calculate the internal forces, bending moments, shear forces, thermal internal forces, and thermal moments at any location:

[0071]

[0072]

[0073]

[0074]

[0075] In the formula, Let x be the components of the internal force at any point on the metal plate specimen in the x, y, z directions. Let x be the components of the bending moment at any point on the metal plate specimen in the x, y, z directions. The shear force at any point on the metal plate specimen is expressed in the x and y directions. Let x be the components of the thermal internal force at any point on the metal plate specimen in the x, y, z directions. Let x be the components of the thermal torque at any point on the metal plate specimen in the x, y, z directions. This is the shearing correction factor, with a value of 5 / 6; and The tensile, tensile-bending coupling, and bending stiffness coefficients of the metal plate specimen are represented by the following expressions:

[0076]

[0077] Among them, h p h is the thickness of the base plate. c The total thickness of the functional gradient coating applied to one side of the substrate plate.

[0078] Step 4.5: Establish the expressions for the kinetic energy, strain energy, and thermal potential energy of the upper coating, lower coating, and substrate plate, and establish the expression for the work done by the external force in conjunction with the basic harmonic excitation.

[0079] Kinetic energy of upper and lower coatings and substrate and Assume:

[0080]

[0081]

[0082]

[0083] in, is the density of the substrate; h is the thickness of the metal plate specimen; A represents the area of ​​the mid-surface region of the metal plate specimen.

[0084] Strain energy of upper and lower coatings and substrate , and Represented as:

[0085]

[0086]

[0087]

[0088] In the formula, Let x be the components of the internal force at any point in the functionally graded coating in the x, y, z directions. Let x be the components of the bending moment at any point in the functionally graded coating in the x, y, z directions. The components of the shear force at any point in the functionally graded coating in the x and y directions. Let x be the components of the internal force at any point on the base plate in the x, y, z directions. Let x be the components of the bending moment at any point on the base plate in the x, y, z directions. Let x be the component of the shear force at any point on the substrate plate in the x and y directions.

[0089] Considering the temperature field, the thermal potential energy generated by the metal plate specimen can be expressed as:

[0090] The work done by the fundamental harmonic excitation on each part is expressed as follows:

[0091]

[0092] Step 4.6: Using orthogonal polynomials satisfying the boundary conditions as Ritz trial functions, the Rayleigh-Ritz method is used to obtain the equations of motion for the metal plate specimen:

[0093] The displacement of the metal plate specimen under lateral vibration is expressed as:

[0094]

[0095]

[0096]

[0097]

[0098]

[0099] In the formula, The excitation frequency is used, and the angular frequency is used when solving for the eigenvalues. and Assume unknown eigenvectors; and These are orthogonal polynomial terms, obtained by orthogonalizing polynomial functions that satisfy the boundary conditions:

[0100]

[0101]

[0102]

[0103]

[0104] in, , and These are all coefficient functions, and their expressions are:

[0105]

[0106]

[0107] in, It is a weight function, usually taken as... Since the structure is under cantilever boundary conditions, the polynomial and The expression is:

[0108]

[0109]

[0110]

[0111]

[0112] Where a and b are the length and width of the metal plate specimen, respectively.

[0113] The Lagrangian energy function of the coated plate structure is expressed as follows:

[0114]

[0115] According to the Rayleigh-Ritz method, by taking the partial derivatives of the Ritz coefficients in the Lagrange energy function constructed above, we obtain the following:

[0116]

[0117] The equation of motion for the metal plate specimen under foundation harmonic excitation is obtained as follows:

[0118]

[0119] in, M and F represent the complex stiffness, complex mass matrix, and excitation force vector of the metal plate specimen, respectively, while K and C are the stiffness and damping matrices. Let q represent the natural frequency, and q be the response vector.

[0120] Step 4.7: Obtain the theoretical natural frequency and theoretical damping ratio of the metal plate specimen by solving the equations of motion. To solve for the fundamental frequency of the metal plate specimen, set... Using this condition, we obtain the following formula:

[0121]

[0122] By solving the above equation, the structural eigenvectors and natural frequencies can be obtained. To solve for the damping parameters of the metal plate specimen, the strain energy method is used, specifically the i-th order damping ratio of the metal plate specimen. It is represented as:

[0123]

[0124] In the formula, This represents the vector of the i-th mode shape.

[0125] Step 5: Construct an error function constrained by the natural frequency and resonant response. Based on a particle swarm optimization algorithm, iteratively update the equivalent elastic modulus and loss factor material parameters of the functionally graded coating within a preset parameter range and step size. Output the identification results according to the convergence criterion. Specifically:

[0126] Step 5.1: Use the particle swarm optimization algorithm to iteratively calculate the Young's modulus of the functionally graded coating material.

[0127] First, based on the theoretical elastic modulus of component material A and component material B provided by the manufacturer... and Centered on this, by introducing a value coefficient and Limit the range of Young's modulus values ​​for component materials A and B in the initial population:

[0128]

[0129] Then, within the above value range, multiple sets of Young's moduli for component material A and component material B are randomly generated, and particle position vectors are constructed:

[0130]

[0131] in, Indicates the first Each particle position vector.

[0132] No. The velocity of each particle is also a 2-dimensional vector, denoted as:

[0133]

[0134] Step 5.2: After setting the thermal environment temperature and degradation time, construct a fitness function based on the deviation between the theoretically calculated natural frequencies and the thermal vibration test results. The performance of each particle depends on the error objective function. Determined fitness value:

[0135]

[0136] In the formula, The number of mode shapes contained in the frequency band. The i-th natural frequency is obtained from theoretical calculations. The i-th natural frequency is obtained from experimental testing.

[0137] Step 5.3: Update particle velocity and position based on the individual particle's optimal position and the global optimal position:

[0138]

[0139] In the formula, ts is the current iteration number. and These are self-learning and group learning factors, respectively. and These are random numbers that follow a uniform distribution within the interval [0,1]. For the first The best position of an individual particle found through iterative search so far. This is the globally optimal position found through iterative searching of the entire particle swarm so far.

[0140] Step 5.4: Using the solution formula derived in Step 4, iteratively calculate the theoretical natural frequency from the updated position vector. When the iterative error function of the natural frequency obtained by theoretical calculation When the set value is reached, the particle swarm iteration calculation stops, and the Young's modulus of component material A and component material B that meet the iteration accuracy requirements is output.

[0141] Step 5.5: Iteratively calculate the loss factor of the functionally graded coating material using the particle swarm optimization algorithm; the particle swarm initialization, particle velocity and position update process is the same as the process of identifying the elastic modulus.

[0142] A fitness function is constructed based on the deviation between the theoretically calculated damping parameters and the results of thermal vibration tests.

[0143]

[0144] In the formula, Represented as the first Each order The maximum number of iterations to participate in the calculation. This represents the damping ratio obtained from experimental testing; This represents the calculated theoretical damping ratio; the damping ratio of the coating was tested and calculated under the same excitation amplitude and measuring point location, and the error function was used to determine the damping ratio. When the iteration termination condition is met, output the loss factors of component material A and component material B that meet the iteration accuracy requirements. .

[0145] Step 5.6: After resetting the thermal environment temperature and degradation time, repeat steps 5.1-5.5 until the elastic modulus and loss factor at all temperatures and degradation times are identified.

[0146] Step 5.7: Substitute all the identified elastic moduli, loss factors, thermal environment temperatures, and degradation times into the formula in Step 3.4, and use the method of undetermined coefficients to calculate the elastic modulus and loss factor of the functionally graded coating under the thermal environment of the vibration test.

[0147] The above description is only a preferred embodiment of the present invention and is not intended to limit the ideas of the present invention. Any modifications, equivalent substitutions, improvements, etc., made within the spirit and principles of the present invention should be included within the protection scope of the present invention.

Claims

1. A method for identifying the elastic modulus and loss factor of functionally graded coatings under thermal conditions based on vibration testing, characterized in that, include: Step 1: Prepare metal plate specimens coated with functionally graded coatings, and obtain the structural geometric parameters, composition material parameters, and thermal shock environment parameters of the metal plate specimens; Step 2: Apply basic vibration excitation to the metal plate specimen under thermal environment conditions, obtain the test natural frequency of the target order by frequency sweep test, obtain the test damping ratio of the metal plate specimen by fixed frequency resonance test, and use the test natural frequency and test damping ratio as the constraint quantities for material parameter identification in Step 5. Step 3: Establish the equivalent material parameter model of the functionally graded coating based on the volume fraction distribution law, obtain the equivalent Young's modulus, shear modulus, Poisson's ratio and density of the coating material according to the Voigt mixing rule, and characterize the damping using the complex modulus method. Write the elastic modulus and shear modulus in complex form, and obtain the transformation stiffness coefficients of the functionally graded coating and the substrate plate. Step 4: Based on the first-order shear deformation theory, thermoplastic theory and Rayleigh-Ritz method, establish an analytical dynamic model of the metal plate specimen under thermal conditions, give the displacement field, strain field, constitutive relation, internal force and moment expressions and energy expressions, and obtain the theoretical natural frequencies and theoretical damping ratios of the corresponding orders. Step 5: Construct an error function constrained by the natural frequency and resonant response. Based on the particle swarm optimization algorithm, iteratively update the equivalent elastic modulus and loss factor material parameters of the functionally graded coating within a preset parameter range and step size, and output the identification results according to the convergence criterion.

2. The method for identifying the elastic modulus and loss factor of functionally graded coatings under thermal conditions based on vibration testing according to claim 1, characterized in that, Step 1 specifically involves: Step 1.1: Prepare the mixed slurry required for each layer of the functional gradient coating according to the preset gradient distribution law. Add component material A and component material B to multiple containers in different proportions and add curing agent. After stirring, the spray slurry for each layer is obtained. Step 1.2: After roughening the surface of the substrate plate by grinding, the slurry is deposited layer by layer on both sides of the substrate plate by atomization spraying. After each layer is sprayed, a curing process is performed. The spraying-curing process is repeated until a functional gradient coating with a preset number of layers is formed, and a metal plate specimen coated with a functional gradient coating is made. Step 1.3: Define a global coordinate system o-xyz on the metal plate specimen coated with functionally graded coating, located on the mid-surface of the metal plate specimen; Step 1.4: Obtain the structural geometric parameters of the metal plate specimen, including the length, width and thickness of the metal plate specimen, the total thickness and total number of functional gradient coatings applied to one side of the substrate plate; Step 1.5: Obtain the component material parameters of the metal plate specimen, including the Young's modulus, shear modulus, Poisson's ratio, density and loss factor of functionally graded coating component material A, component material B and substrate plate; Step 1.6: Obtain the initial parameters for iteration, including the functionally graded coating component material A, component material B, and the equivalent Young's modulus and loss factor of each coating layer. Based on this, set the initial values, allowable value ranges, and iteration step size of each parameter to be identified, forming the initial parameter vector used for iterative identification.

3. The method for identifying the elastic modulus and loss factor of functionally graded coatings under thermal conditions based on vibration testing according to claim 1, characterized in that, Step 2 specifically involves: Step 2.1: Install the metal plate specimen onto the special fixture of the vibration table, and install the fixture onto the vibration table to form a single-sided fixed support boundary condition; then insert the metal plate specimen into the hot box; Step 2.2: Establish a thermal environment using a heating chamber, temperature controller, and thermocouple sensors. Heat the metal plate specimen to several preset temperatures and maintain stability. Then, apply simple harmonic base motion using a vibration table excitation method. The base displacement is expressed as: Where Y is the basic displacement amplitude. f is the excitation angular frequency; f is the frequency, and the test frequency band is set. Frequency sweep resolution Scan frequency And set the excitation amplitude to remain constant; Step 2.3: Obtain the natural frequency of the test using a frequency sweep test: Within the set test frequency band, the metal plate specimen is subjected to a frequency sweep test. The input signal and output response signal are collected to obtain the vibration frequency response curve of the target order, and then the natural frequency of the corresponding target order is obtained. Step 2.4: Obtain damping parameters through constant-frequency resonance test: Using the natural frequency of each test obtained in step 2.3 as the target frequency, a fixed-frequency excitation is performed under the same excitation amplitude. After the system reaches a steady-state response, the output response signal is collected to obtain the corresponding vibration frequency response curve. Then, the half-power bandwidth method is used to analyze the vibration frequency response curve to obtain the test damping ratio of the metal plate specimen.

4. The method for identifying the elastic modulus and loss factor of functionally graded coatings under thermal conditions based on vibration testing according to claim 1, characterized in that, Step 3 specifically involves: Step 3.1: Assuming that the material properties of the mixed functionally graded coating vary with thickness and follow a distribution law, the volume fraction is expressed as: in, z represents the coordinates of any point on the coating along the z-axis, z0 represents the coordinates of the first coating layer along the z-axis, z k d represents the coordinate of the outer coating layer in the z-axis direction; d represents the volume gradient index of the functionally graded coating, which is used to determine the variation trend of component material B to component material A along the thickness direction of the functionally graded coating. and These are the volume fractions of components A and B, respectively. Step 3.2: Establish equivalent material parameters based on Voigt mixing rules: in, It is the equivalent Young's modulus of a functionally graded coating. and These represent the Young's moduli of component materials A and B, respectively. It is the equivalent Poisson's ratio of the functionally graded coating. and These represent the Poisson's ratios of component materials A and B, respectively. It is the equivalent density of the functionally graded coating. and These represent the densities of component materials A and B, respectively. It is the equivalent shear modulus of the functionally graded coating; Step 3.3: Characterize the damping and obtain the complex modulus using the complex modulus method; in, This represents the complex Young's modulus of a functionally graded coating material; This represents the complex shear modulus of a functionally graded coating material; and These are the loss factors of component material A and component material B, respectively. The equivalent loss factor of the hybrid functionally graded coating; Step 3.4: Considering the effect of thermal environment temperature, establish an expression for the relationship between the equivalent material parameters of the functionally graded coating and temperature: Where ΔT represents the temperature change and t is the degradation time. This represents the equivalent dynamic Young's modulus of a functionally graded coating that takes into account the effects of temperature changes and degradation time in the thermal environment. This represents the equivalent dynamic loss factor of a functionally graded coating that takes into account the effects of thermal environment temperature changes and degradation time. This represents a series of correlation fitting coefficients that need to be determined, taking into account the effect of temperature on the coating. This represents the equivalent dynamic shear modulus of a functionally graded coating that takes into account the effects of temperature changes and degradation time in the thermal environment. Step 3.5: Establish the expression for the equivalent thermal expansion coefficient of the functionally graded coating based on the thermal expansion coefficients of the component materials, and obtain the transformed stiffness coefficient of the functionally graded coating under thermal conditions. : The transformation stiffness coefficients of the remaining functionally graded coatings are all 0; Step 3.6: Derive the stiffness coefficient matrix of the base plate. Since the base plate is considered to be unaffected by temperature, the stiffness coefficient matrix of the base plate is... Represented as: In the formula, This represents the Young's modulus of a thin sheet of metal substrate. Poisson's ratio for a thin sheet of metal matrix is the transformation stiffness coefficient of the base plate.

5. The method for identifying the elastic modulus and loss factor of functionally graded coatings under thermal conditions based on vibration testing according to claim 4, characterized in that, Step 4 specifically involves: Step 4.1: To account for the influence of shear deformation, based on the first-order shear deformation theory, the displacement expression for any point on the metal plate specimen is obtained as follows: Where u, v, w represent the displacement of any point on the metal plate specimen; u0, v0, w0 are the displacements of a point in the mid-plane in the x, y, z directions; φ, ψ are the rotation angles about the y and x axes; t is time; and z is the thickness coordinate. Step 4.2: When the displacement expression of the metal plate specimen is known, the structural strain expression is as follows: in, and Let x and y represent the normal strain at any point on the metal plate specimen at room temperature, respectively, in the x and y directions. , and These represent the shear strain at any point on the metal plate specimen at room temperature in the xy, xz, and yz planes, respectively. , , , , , , , ; Step 4.3: Establish the thermoelastic constitutive relation of the metal plate specimen according to the generalized Hooke's law: In the formula, Let be the normal stress along the x-direction of the metal plate specimen. Let be the normal stress along the y-direction of the metal plate specimen. Let be the shear stress of the metal plate specimen in the xy plane. Let be the shear stress of the metal plate specimen in the yz plane. The shear stress of the metal plate specimen in the xz plane; This is the stiffness coefficient. These represent the substrate material and the functionally graded coating material, respectively. The strain of a metal plate specimen in a thermal environment is expressed according to thermoplastic theory: In the formula, and Indicates the equivalent coefficient of thermal expansion of the functionally graded coating and the substrate; Step 4.4: Calculate the internal forces, bending moments, shear forces, thermal internal forces, and thermal moments at any location: In the formula, Let x be the components of the internal force at any point on the metal plate specimen in the x, y, z directions. Let x be the components of the bending moment at any point on the metal plate specimen in the x, y, z directions. The shear force at any point on the metal plate specimen is expressed in the x and y directions. Let x be the components of the thermal internal force at any point on the metal plate specimen in the x, y, z directions. Let x be the components of the thermal torque at any point on the metal plate specimen in the x, y, z directions. This is the shearing correction factor, with a value of 5 / 6; and The tensile, tensile-bending coupling, and bending stiffness coefficients of the metal plate specimen are represented by the following expressions: Among them, h p h is the thickness of the base plate. c The total thickness of the functional gradient coating applied to one side of the substrate plate; Step 4.5: Establish the expressions for the kinetic energy, strain energy, and thermal potential energy of the upper coating, lower coating, and substrate, and establish the expression for the work done by the external force in conjunction with the basic harmonic excitation; Kinetic energy of upper and lower coatings and substrate and Assume: in, is the density of the substrate; h is the thickness of the metal plate specimen; A represents the area of ​​the mid-surface region of the metal plate specimen; Strain energy of upper and lower coatings and substrate , and Represented as: In the formula, Let x be the components of the internal force at any point in the functionally graded coating in the x, y, z directions. Let x be the components of the bending moment at any point in the functionally graded coating in the x, y, z directions. The components of the shear force at any point in the functionally graded coating in the x and y directions. Let x be the components of the internal force at any point on the base plate in the x, y, z directions. Let x be the components of the bending moment at any point on the base plate in the x, y, z directions. The components of the shear force at any point on the base plate in the x and y directions; Considering the temperature field, the thermal potential energy generated by the metal plate specimen can be expressed as: The work done by the fundamental harmonic excitation on each part is expressed as follows: Step 4.6: Using orthogonal polynomials satisfying the boundary conditions as Ritz trial functions, the Rayleigh-Ritz method is used to obtain the equations of motion for the metal plate specimen: The displacement of the metal plate specimen under lateral vibration is expressed as: In the formula, The excitation frequency is used, and the angular frequency is used when solving for the eigenvalues. and Assume unknown eigenvectors; and These are orthogonal polynomial terms, obtained by orthogonalizing polynomial functions that satisfy the boundary conditions: in, , and These are all coefficient functions, and their expressions are: in, It is a weight function, take Since the structure is under cantilever boundary conditions, the polynomial and The expression is: Where a and b are the length and width of the metal plate specimen, respectively; The Lagrangian energy function of the coated plate structure is expressed as follows: According to the Rayleigh-Ritz method, by taking the partial derivatives of the Ritz coefficients in the Lagrange energy function constructed above, we obtain the following: The equation of motion for the metal plate specimen under foundation harmonic excitation is obtained as follows: in, M and F represent the complex stiffness, complex mass matrix, and excitation force vector of the metal plate specimen, respectively, while K and C are the stiffness and damping matrices. The natural frequency is represented by q, and the response vector is q. Step 4.7: Obtain the theoretical natural frequency and theoretical damping ratio of the metal plate specimen by solving the equations of motion: To determine the fundamental frequency of the metal plate specimen, we set... Using this condition, we obtain the following formula: By solving the above equation, the structural eigenvectors and natural frequencies can be obtained. To solve for the damping parameters of the metal plate specimen, the strain energy method is used, specifically the i-th order damping ratio of the metal plate specimen. It is represented as: In the formula, This represents the vector of the i-th mode shape.

6. The method for identifying the elastic modulus and loss factor of functionally graded coatings under thermal conditions based on vibration testing according to claim 4, characterized in that, Step 5 specifically involves: Step 5.1: Iteratively calculate the Young's modulus of the functionally graded coating material using the particle swarm optimization algorithm; First, based on the theoretical elastic modulus of component material A and component material B provided by the manufacturer... and Centered on this, by introducing a value coefficient and Limit the range of Young's modulus values ​​for component materials A and B in the initial population: Then, within the above value range, multiple sets of Young's moduli for component material A and component material B are randomly generated, and particle position vectors are constructed: in, Indicates the first The position vectors of the particles; No. The velocity of each particle is also a 2-dimensional vector, denoted as: Step 5.2: After setting the thermal environment temperature and degradation time, construct a fitness function based on the deviation between the theoretically calculated natural frequency and the thermal vibration test results: The performance of each particle depends on the error objective function. Determined fitness value: In the formula, The number of mode shapes contained in the frequency band. The i-th natural frequency is obtained from theoretical calculations. The i-th order natural frequency obtained from experimental testing; Step 5.3: Update particle velocity and position based on the individual particle's optimal position and the global optimal position: In the formula, ts is the current iteration number. and These are self-learning and group learning factors, respectively. and These are random numbers that follow a uniform distribution within the interval [0,1]. For the first The best position of an individual particle found through iterative search so far. This is the globally optimal position found through iterative searching of the entire particle swarm so far. Step 5.4: Using the solution formula derived in Step 4, iteratively calculate the theoretical natural frequency from the updated position vector. When the iterative error function of the natural frequency obtained by theoretical calculation When the set value is reached, the particle swarm iteration calculation stops, and the Young's modulus of component material A and component material B that meet the iteration accuracy requirements is output. Step 5.5: Iteratively calculate the loss factor of the functionally graded coating material using the particle swarm optimization algorithm; the particle swarm initialization, particle velocity and position update process is the same as the process of identifying the elastic modulus; A fitness function is constructed based on the deviation between the theoretically calculated damping parameters and the results of thermal vibration tests. In the formula, Represented as the first Each order The maximum number of iterations to participate in the calculation. This represents the damping ratio obtained from experimental testing; This represents the calculated theoretical damping ratio; the damping ratio of the coating was tested and calculated under the same excitation amplitude and measuring point location, and the error function was used to determine the damping ratio. When the iteration termination condition is met, output the loss factors of component material A and component material B that meet the iteration accuracy requirements. ; Step 5.6: After resetting the thermal environment temperature and degradation time, repeat steps 5.1-5.5 until the elastic modulus and loss factor at all temperatures and degradation times are identified; Step 5.7: Substitute all the identified elastic moduli, loss factors, thermal environment temperatures, and degradation times into the formula in Step 3.4, and use the method of undetermined coefficients to calculate the elastic modulus and loss factor of the functionally graded coating under the thermal environment of the vibration test.

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