Functional gradient coating cone-column shell high-temperature vibration dynamics modeling and analysis method

Abstract

The invention discloses a functional gradient coating cone-cylindrical shell high-temperature vibration dynamics modeling and analysis method. The method comprises the following steps: establishing a linear vibration analysis model and a nonlinear vibration model considering temperature and amplitude dependence; identifying material parameters of the functional gradient coating-composite material combined shell and determining a corresponding fitting coefficient; preparing a functional gradient coating, coating the conical-cylindrical shell matrix structure with the functional gradient coating, and measuring geometric and material parameters of the matrix and the coating; building a high-temperature vibration test system, and measuring the dynamic response characteristics of the test piece under normal-temperature and multi-temperature working conditions; the theoretical model is verified based on an experimental test result, and a mechanism of influence of key parameters on the dynamic performance of the functionally gradient coating cone-cylindrical shell structure in a high-temperature vibration environment is further researched. The method has the advantages that calculation is easy and convenient, precision is high, a reliable theoretical tool is provided for thermal shock resistance design and life prediction of the structure in the extreme environment, and service safety and reliability are remarkably improved.

Description

Technical Field

[0001] The present invention belongs to the field of material science and technology, specifically a composite material structure vibration and performance analysis technology, specifically relates to a high-temperature vibration dynamics modeling and analysis method for functional gradient coatings. Background Art

[0002] Cone-cylindrical shell structures, as a typical engineering structure, are widely used in aerospace propulsion systems, nuclear reactor pressure vessels, and other fields. However, due to the unique geometry of their shape, the dynamic behavior of these structures exhibits strong nonlinear characteristics under high-temperature vibration environments, placing higher demands on their service performance. To address this issue, high-temperature-resistant coatings are widely used in these structures to improve their heat resistance and vibration resistance. However, the sudden change in material properties between traditional uniform coatings and substrates can easily lead to interfacial stress concentration, which can cause coating delamination or substrate damage, seriously compromising the overall performance of the structure. Functionally graded coatings, through continuous gradient changes in material composition, can effectively alleviate this interfacial stress concentration, thereby improving the overall dynamic performance and service life of the structure. Despite this, the dynamic behavior of functionally graded coating-substrate structures under high-temperature vibration environments is extremely complex. Currently, existing research focuses on the vibration analysis of cone-cylindrical shell structures made of uniform materials or traditional composite materials, while relatively little research has been conducted on the dynamic behavior of functionally graded coating cone-cylindrical shell structures under high-temperature vibration environments. Therefore, establishing a dynamic model of functionally gradient coated conical-cylindrical shell structures under high-temperature vibration environments and developing corresponding analysis methods are of great significance for improving the service reliability and safety of such structures in extreme environments.

[0003] There are certain limitations in the existing experimental and theoretical studies on the dynamic performance analysis of conical-cylindrical shell structures under thermo-vibration environments. Experimental studies mainly focus on testing devices and methods. In particular, experimental studies on functionally graded coating conical-cylindrical shell structures have not been carried out yet. Most of the theoretical studies are based on the assumptions of homogeneous materials or traditional composite materials, and fail to fully consider the complex influence of the gradient distribution characteristics of functionally graded coating materials on the dynamic behavior of structures. For example, Patent CN201711057508.5 proposes a method and system for testing the nonlinear dynamic parameters of fiber composite plates under thermo-vibration environments, which can accurately obtain the nonlinear dynamic parameters of structures under thermo-vibration environments, including nonlinear natural frequencies, damping, and modal vibration modes. Patent CN202210373242.X discloses a thermo-modal test system and method for large deformations after buckling of thin-walled structures in high-temperature environments, realizing continuous vibration excitation of thin-walled structures in high-temperature environments. Patent CN202410384219.X provides a thermo-vibration composite fatigue test device and method applicable to thin-walled structures, which can be used for thermo-vibration composite fatigue test and measurement of the hot-end thin-walled structures of aero-engines. However, the above methods are all limited to the scope of experimental testing, lacking in-depth analysis of test results from the perspective of theoretical mechanisms, and it is difficult to reveal the internal mechanism of the change in the dynamic performance of structures under thermo-vibration environments. Patent CN201910446358.X provides a method for analyzing the degradation of the high-temperature dynamic performance of fiber-reinforced composite plates. Shen Huishen et al. (Comp Mater Sci, 2011, 50(8): 2319-2330), Li Hui et al. (Compos Part B-Eng, 2019,162:206-218; Mech Syst Signal Pr, 2021, 156:107665), Yu Kaiping et al. (Thin WallStruct, 2024, 205: 112454) have carried out theoretical modeling studies on the vibration characteristics of composite thin plates, cylindrical shells, and conical shells under thermo-vibration environments, and explored the influence of temperature on the dynamic response characteristics of structures, but none of them considered the role of functionally graded coatings. The existing research in the coating field mainly focuses on the development of preparation methods and equipment, represented by Patent CN202211416873.1, CN202210348132.8, and CN202411516713.3. Although these studies have achieved technological innovations in the preparation of thermal protection, oxidation / ablation and other protective coating materials, they lack in-depth analysis and evaluation of the structural performance of coated coatings, especially the research on functionally graded coatings is more insufficient.

[0004] In summary, although existing studies have conducted varying degrees of research on the dynamic performance of composite structures under thermo-vibration environments through experiments and theoretical analyses, existing theoretical models generally only consider the direct influence of temperature on the dynamic performance of structures and lack in-depth analysis of the influence of coatings. In particular, in the research on the high-temperature vibration dynamics behavior of functionally graded coating-conical cylindrical shell structures, relevant reports are relatively scarce, and there is a lack of systematic modeling and analysis methods for the gradient distribution characteristics of functionally graded coating materials. Therefore, providing a method for modeling and analyzing the high-temperature vibration dynamics of functionally graded coating conical-cylindrical shells is of great theoretical significance and engineering application value for deeply revealing the variation law of the structural dynamic performance under thermo-vibration, optimizing the anti-thermo-vibration design of structures, and improving the accuracy of life prediction. Summary of the Invention

[0005] The present invention provides a method for modeling and analyzing the high-temperature vibration dynamics of functionally graded coating conical-cylindrical shells. After establishing a dynamic analysis model of functionally graded coating conical-cylindrical shell structures considering the influence of temperature and amplitude dependence, the particle swarm optimization algorithm is used to identify the material parameters of the functionally graded coating-composite shell, and then based on a self-built high-temperature vibration test system, the effectiveness of the theoretical model is verified. Further parametric analysis reveals the influence mechanism of the key parameters of the functionally graded coating on the high-temperature dynamic performance of the structure, with simple calculation and high accuracy.

[0006] The technical solution of the present invention is as follows:

[0007] A method for modeling and analyzing the high-temperature vibration dynamics of functionally graded coating conical-cylindrical shells includes the following steps:

[0008] Step 1: Establish a linear vibration analysis model of a composite conical-cylindrical shell structure considering the thermodynamic characteristics of the functionally graded coating under high-temperature vibration environments;

[0009] Step 2: Solve the nonlinear vibration characteristics of the functionally graded coating conical-cylindrical shell structure under thermal environments;

[0010] Step 3: Identify the material parameters of the functionally graded coating-composite shell and determine the corresponding fitting coefficients;

[0011] Step 4: Prepare the functionally graded coating and coat it on the conical-cylindrical shell matrix structure, and measure the geometric and material parameters of the matrix and the coating;

[0012] Step 5: Build a high-temperature vibration test system and measure the dynamic response characteristics of the specimen under normal temperature and multi-temperature conditions;

[0013] Step 6: Verification of the theoretical model based on experimental tests;

[0014] Step 7: Study the dynamic performance mechanism of the functionally graded coating conical-cylindrical shell structure under high-temperature vibration environments.

[0015] Step 1 includes:

[0016] Step 1.1: Establish a coordinate system for the functionally graded coating conical-cylindrical shell structure;

[0017] Step 1.2: Define structure- and material-related parameters;

[0018] Step 1.3: Determine the equivalent material parameters of the composite matrix and the functionally graded coating under a thermal environment;

[0019] Step 1.4: Solve the linear vibration characteristics of the functionally graded coating conical-cylindrical shell structure under a thermal environment.

[0020] Step 2 modifies the material parameters discussed in Step 1.4 based on the Jones-Nelson nonlinear theory, and introduces the amplitude variable into the theoretical model, so that the structure has the characteristics of temperature and amplitude dependence at the same time.

[0021] Step 3 includes:

[0022] Step 3.1 Identify material parameters;

[0023] Use the natural frequency values of the uncoated specimens and the coated specimens obtained by experimental tests under different excitation amplitudes and temperatures as the target values, and construct a natural frequency error function using the particle swarm algorithm, so as to identify the elastic modulus of the composite shell layer and the Young's modulus of the functionally graded coating under different excitation amplitudes and temperature conditions;

[0024] Step 3.2 Determine the relevant fitting coefficients;

[0025] Use the nonlinear material parameters obtained in Section 3.1, combined with the Curve Fitting toolbox in MATLAB software, to solve the fitting coefficients of the material parameters related to temperature and amplitude in the theoretical model of the functionally graded coating-composite shell.

[0026] Step 4 includes:

[0027] Step 4.1: Prepare the matrix sample: A conical-cylindrical shell specimen was prepared using continuous 3D printing technology, and its matrix material is composed of carbon fiber and polylactic acid resin;

[0028] Step 4.2: Spray a functionally graded coating on the surface of the specimen using the atomization deposition technology: This coating is a high-temperature resistant ceramic material, which can form a film with excellent flexibility, impact resistance and chemical stability at room temperature and high temperature environments. The coating is evenly sprayed on the surface of the specimen through an atomizer to ensure uniform coating thickness distribution;

[0029] Step 4.3: The material parameters of the cone-shell structure matrix and the functionally graded coating are determined by the inverse identification method.

[0030] The system in Step 5 mainly consists of the following equipment: a vibration table and its multi-functional controller, a power amplifier, a Polytec PDV-100 Doppler laser vibrometer, a heating chamber and a temperature controller, a thermocouple, a special fixture for the specimen, an LMS data acquisition instrument, and a computer workstation.

[0031] Step 6 includes:

[0032] Step 6.1: Theoretical and experimental comparison verification of the dynamic characteristics of the functionally graded coating cone-shell structure under normal temperature conditions;

[0033] Step 6.2: Theoretical and experimental comparison verification of the dynamic characteristics of the functionally graded coating cone-shell structure under high temperature environment.

[0034] Based on the verified theoretical model, Step 7 conducts a parametric system analysis of the functionally graded coating-composite shell under high temperature vibration environment. The key research includes: the existence effect of the functionally graded coating (comparison with and without coating), the influence law of the coating thickness parameter, the regulation effect of the coating gradient index, and the influence law of key parameters on the structural vibration characteristics.

[0035] Through the systematic parameter sensitivity analysis in Step 7, the action mechanism of the functionally graded coating on the dynamic performance of the composite shell under high temperature vibration conditions is revealed.

[0036] The beneficial effects of the present invention are as follows: A nonlinear dynamic model of the functionally graded coating-composite shell considering the amplitude-temperature dependence is established. Based on the Jones-Nelson theory, a material nonlinear constitutive relation is constructed, and the material parameters are determined by the inverse identification technology. By using the first-order shear deformation theory combined with the Rayleigh-Ritz method and the mode superposition method, the efficient and accurate solution of the structural dynamic response is realized. This method is simple in calculation and high in accuracy, providing a reliable theoretical tool for the anti-thermal vibration design and life prediction of structures in extreme environments, and significantly improving the service safety and reliability. Description of the Drawings

[0037] Figure 1 It is a schematic diagram of the dynamic theory model of the functionally graded coating cone-shell under thermal vibration environment;

[0038] Figure 2 It is a physical diagram of the preparation process flow of the functionally graded coating cone-shell specimen;

[0039] Figure 3 It is a physical diagram of the dynamic performance test system of the functionally graded coating cone-shell under high temperature vibration environment;

[0040] Figure 4 Comparison diagrams of the theoretical solutions and experimental results of the first four natural frequencies of specimens at different temperature conditions (a) 50 °C and (b) 100 °C

[0041] Figure 5 Comparison diagrams of the theoretical solutions and experimental results of the first four resonance responses of specimens at different temperature conditions (a) 50 °C and (b) 100 °C

[0042] Figure 6 Influence diagrams of the functional gradient index on the first four natural frequencies of the structure under conditions of (a) 50 °C and (b) 100 °C

[0043] Figure 7 Influence diagrams of the functional gradient index on the first four resonance responses of the structure under conditions of (a) 50 °C and (b) 100 °C

[0044] Figure 8 Influence diagrams of the coating thickness ratio on the first four natural frequencies of the structure under conditions of (a) 50 °C and (b) 100 °C

[0045] Figure 9 Influence diagrams of the coating thickness ratio on the first four resonance responses of the structure under conditions of (a) 50 °C and (b) 100 °C

[0046] Figure 10 Influence diagrams of the functional gradient index on the nonlinear dynamic characteristics of the structure under the 150 °C condition, where (a) is the curve diagram of the change in the nonlinear natural frequency and (b) is the curve diagram of the amplitude characteristics of the nonlinear vibration response

[0047] Figure 11 Influence diagrams of the coating thickness ratio on the nonlinear dynamic behavior of the structure under the 150 °C condition, where (a) is the trend diagram of the change in the nonlinear natural frequency with the thickness ratio and (b) is the change rule diagram of the vibration response with the thickness ratio Specific implementation manners

[0048] A method for modeling and analyzing the high-temperature vibration dynamics of a conical-cylindrical shell with a functionally graded coating includes the following steps:

[0049] Step 1: Establish a linear vibration analysis model of a composite conical-cylindrical shell structure considering the thermodynamic characteristics of the functionally graded coating under a high-temperature vibration environment

[0050] Step 1.1: Establish a coordinate system for the conical-cylindrical shell structure with a functionally graded coating

[0051] Establish a global coordinate system and , which are respectively located on the middle planes of the left sides of the conical shell and the cylindrical shell structures 、 and (i = 1, 2) represent the axial, circumferential, and radial directions of the structure respectively. A local coordinate system O-123 is established, where "1", "2", and "3" represent the parallel fiber direction, the perpendicular fiber direction, and the direction perpendicular to the fiber plane respectively; represents the direction angle between the "1" direction and the axis direction. respectively represent the displacements at any point on the mid-surface of the structure along , and directions.

[0052] Step 1.2: Define the structure and material-related parameters;

[0053] and respectively represent the lengths of the structure along x , , i , , ,

[0058] , i , ,

[0053] , , , ,

[0054] , , ,

[0055] , i , , , i , , , , , , ,

[0057] , ,

[0056] , θ i , z i directions, the radius of the small end of the conical shell, and the radius of the cylindrical shell. The radius at any point along the length direction of the conical shell can be expressed as , where is the half apex angle of the conical shell. and respectively represent the thicknesses of the shell and the coating, and h is the distance from the mid-plane to the inner surface. The conical and cylindrical shell matrices are composed of N layers of fibers and polymer materials, and are the thickness coordinates of the kth layer of fibers in the functionally graded coating-composite shell.

[0054] In addition, two groups of virtual springs with stiffness values of k i (i = 1, 2, …, 10) are used to simulate the boundary conditions of the structure. At the same time, the continuity conditions between the conical shell and the cylindrical shell are assumed through virtual springs with stiffness values of , , , and .

[0055] Step 1.3: Determine the equivalent material parameters of the composite matrix and the functionally graded coating under the thermal environment;

[0056] The fiber-reinforced composite conical-cylindrical shell is made of a mixture of matrix material polylactic acid resin and carbon fiber, assuming that the two materials are evenly distributed. Based on the micromechanics mixing law and considering the influence of temperature on the properties of the composite material, the effective material parameters of the composite shell structure can be expressed in the following form:

[0057] (1a)

[0058] ​​​ (1b)

[0059] (1c)

[0060] in, , It represents the Young's modulus of the composite shell-matrix structure along the thickness direction considering the influence of temperature; , and is the shear modulus, , , , and is the corresponding loss factor; is the temperature change; represents a series of temperature-dependent fitting coefficients to be determined.

[0061] In addition, based on the complex modulus method, the equivalent complex Young's modulus of the composite shell matrix structure is 、 and complex shear modulus 、 、 It can be expressed as:

[0062] (2)

[0063] in, It is a plural unit.

[0064] The axis stiffness coefficient matrix of the kth layer of the matrix It is necessary to pass the corresponding stiffness coefficient matrix The coordinate transformation is obtained, and the transformation process is as follows:

[0065] (3)

[0066] in, The transformation matrix between the local coordinate system of the k-th fiber layer of the matrix and the global coordinate system is expressed as follows:

[0067] (4)

[0068] Among them, the stiffness coefficient matrix in formula (3) is The expression is:

[0069] (5)

[0070] Where, and is the Poisson's ratio of the fiber in the corresponding direction.

[0071] The functionally graded coating is made by mixing component A and component B in a certain proportion. Based on the Voigt mixing rule, the effective material parameters of the layer (such as Young's modulus, shear modulus, Poisson's ratio, and density, etc.) can be expressed in the following form:

[0072] (6)

[0073] where, and are the volume fractions of component A and component B respectively, and the two satisfy .

[0074] Assume that the material properties of the functionally graded coating vary continuously along the thickness direction and follow the power-law distribution law, and its volume fraction has the following form.

[0075] (7)

[0076] where, d is the volume gradient index of the functionally graded coating, and its value determines the change trend of component B to component A along the thickness direction. In addition, considering the influence of temperature, the Young's modulus and loss factor of component A and component B in the functionally graded coating are assumed as follows:

[0077] (8)

[0078] where, and represent the Young's modulus of component A and component B respectively, and represent their loss factors respectively, , , and represent a series of fitting coefficients related to temperature to be determined.

[0079] Based on the complex modulus method, the equivalent complex Young's modulus , and the equivalent complex shear modulus , of component A and component B in the functionally graded coating can be expressed as:

[0080] (9)

[0081] In the formula, and are the Poisson's ratios of the materials of component A and B respectively.

[0082] Considering the influence of temperature, the equivalent complex Young's modulus , the thermal expansion coefficient , and the Poisson's ratio and density It can be expressed as follows:

[0083] (10)

[0084] Where, , , , are the thermal expansion coefficient and density of component A and B materials respectively.

[0085] Stiffness coefficient matrix of functionally graded coatings It can be expressed as:

[0086] (11)

[0087] Step 1.4: Determine the linear vibration characteristics of the functionally graded coating conical-cylindrical shell structure under thermal conditions.

[0088] It should be emphasized that the following assumptions were made during the modeling process: (1) each fiber layer has the same thickness and volume fraction; (2) the functionally gradient coating and the matrix material are tightly bonded and there is no relative sliding phenomenon; and (3) the nonlinear characteristics of the fiber-reinforced composite material are ignored.

[0089] According to the first-order shear deformation theory, the displacement components of the functionally gradient coating-composite composite shell along the x, θ and z directions are , and It can be expressed as:

[0090] (12)

[0091] in, represents the conical shell and cylindrical shell parts, , and represents the mid-plane displacement components along the x, θ, and z directions, and are the components of the mid-plane rotation displacement relative to its x and θ directions, respectively, and t is the time.

[0092] Based on the above assumptions, the strain-displacement relationship at any point on the midplane of the structure can be expressed as follows:

[0093] (13)

[0094] in, , and are the membrane strain of the structure; , and respectively represent the curvature change of the structure; and are the shear strains along different directions; and respectively represent the Lame coefficient and the radius of curvature. In this structure, their values are in the following forms:

[0095] Truncated conical shell: (14)

[0096] Cylindrical shell: (15)

[0097] In addition, according to the generalized Hooke's law and considering the influence of the temperature field, the stress-strain relationship of the functionally graded coating-composite shell structure can be expressed in the following form.

[0098] (16)

[0099] where and are the normal stresses of the structure along the x and θ directions, , and respectively represent the shear stresses along the x, θ, and z directions, represents the transformed stiffness coefficient, and its expressions are shown in Eqs. (3) and (11). In addition, , , , and represent the thermal strains of the fiber-reinforced composite matrix layer, , , , and represent the thermal strains of the functionally graded coating. According to the thermoelastic theory, the above thermal strains can be expressed as:

[0100] (17)

[0101] where represents the temperature increment, and its expression is , and respectively represent the current temperature and the reference temperature; represents the thermal expansion coefficient of the functionally graded coating, , and respectively represent the thermal expansion coefficients of the fiber-reinforced composite matrix layer along their respective directions, and its expression is:

[0102] (18)

[0103] wherein, and represent the coefficient of thermal expansion in the direction of the fiber-reinforced composite layer.

[0104] Therefore, the elastic internal forces and moments of the structure are expressed as follows:

[0105] (19)

[0106] wherein, represents the shear correction coefficient in the first-order shear deformation theory, taken as 5 / 6. Additionally, 、 and represent the in-plane elastic internal forces, 、 and represent the elastic moments, and represent the transverse shear forces, 、 and represent the tensile stiffness, tensile-bending coupling stiffness, and bending stiffness of the structure, respectively, and their expressions are as follows:

[0107] (20)

[0108] The thermal internal forces 、 and [[ID=5l]] , and the thermal moments 、 and can be expressed as follows:

[0109] (21)

[0110] Based on the above analysis, the energy expression of the functionally graded coating-composite shell structure can be constructed, and its Lagrangian energy equation can be expressed in the following form:

[0111] (22)

[0112] wherein, and represent the kinetic energy and strain energy of the functionally graded coating-composite shell, respectively, represents the thermal potential energy generated by the temperature field, represents the elastic potential energy stored at both ends of the structure by the virtual springs, represents the elastic potential energy stored at the connection between the conical shell and the cylindrical shell by the virtual springs. The kinetic energy expression of the coating-composite shell can be expressed as:

[0113] (23)

[0114] Thus, it can be further obtained that:

[0115] (24)

[0116] Among them, , and represent the inertial terms of the conical shell, cylindrical shell, and functionally graded coating respectively, and their expressions are as follows:

[0117] (25)

[0118] Among them, and represent the density of the functionally graded coating and the composite matrix material.

[0119] To simplify the expression of strain energy, the strain energy is decomposed into tensile strain energy , tensile-bending coupling strain energy and bending strain energy , and their specific expressions are as follows:

[0120] (26)

[0121] (27)

[0122] (28)

[0123] (29)

[0124] Considering the influence of thermal effects on the vibration characteristics of the functionally graded coating-composite shell structure, its thermal potential energy is:

[0125] (30)

[0126] Using virtual springs to simulate arbitrary boundary conditions, for the conical-cylindrical shell coupling structure, the springs are evenly distributed along the edge. Therefore, the boundary potential energy can be expressed as:

[0127] (31)

[0128] Among them, k i (i = 1, 2, …, 10) represent the translational boundary springs and rotational boundary springs respectively.

[0129] Using the virtual spring technology to simulate the displacement and force continuity conditions between the composite cylindrical shell and the conical shell, the control equation of its coupling conditions can be expressed in the following form:

[0130] (32)

[0131] Therefore, the coupling potential energy stored by the virtual spring at the coupling connection of the conical shell and the cylindrical shell can be defined as:

[0132] (33)

[0133] Assume that the structure is subjected to a base excitation load acting, , and are the base excitation amplitude components along the axial, circumferential, and radial directions, respectively, and t is the time variable, and its expression is as follows:

[0134] (34)

[0135] where, , and represent the excitation amplitudes in the axial, circumferential, and radial directions, respectively, is the excitation frequency, and n is the circumferential wave number. In addition, since the structure is only subjected to the base excitation along the w direction, so .

[0136] Therefore, the work done by the base excitation on the functionally graded coating-composite shell can be expressed in the following form: can be expressed as:

[0137] (35)

[0138] Assume that the displacement of the structure at any point can be assumed to be:

[0139] (36)

[0140] where, are the Ritz coefficients, are the Jacobi polynomial coefficients, and their specific expressions are as follows:

[0141] (37)

[0142] where, .

[0143] Using the Rayleigh-Ritz method, taking the partial derivative of the Lagrangian energy functional constructed in Equation (22) with respect to the Ritz coefficients, the following equations can be obtained:

[0144] (38)

[0145] Substituting Equation (22) into the above equations, the motion equations of the structure under study can be expressed in the following form:

[0146] (39)

[0147] Among them, M is the mass matrix of the structure, and K * is the complex stiffness matrix based on the complex modulus method, q is the eigenvector, and F is the exciting force vector. The K of the complex stiffness matrix * has the following expression:

[0148] K * =K + iC (40)

[0149] Among them, C is the damping matrix. In order to solve the natural frequency of the structure, let C = 0 and F = 0, and the following characteristic equation can be obtained:

[0150] (41)

[0151] Natural circular frequency of a conical-cylindrical shell structure with a functionally graded coating under a thermal environment can be obtained by solving the above equation. Substituting the eigenvector q into Equation (36) can obtain the corresponding modal vibration mode.

[0152] In addition, by transforming Equation (39), the r-th order modal vibration mode vector q r in the modal coordinate system can be obtained, and its expression is:

[0153] (42)

[0154] Among them, F r represents the r-th order exciting force vector, is the r-th order natural circular frequency, K r , M r and C r respectively represent the r-th order stiffness matrix, mass matrix, and damping matrix that satisfy the normalization condition in the modal coordinate system, and their expressions are as follows:

[0155] (43)

[0156] In addition, the r-th order modal vibration mode vector q r can be further expressed in the following form:

[0157] (44)

[0158] Finally, based on the modal superposition method, the r-th order resonant displacement response can be expressed in the following form:

[0159] (45)

[0160] Among them, N is the maximum order of the mode.

[0161] Step 2: Solve the nonlinear vibration characteristics of the conical-cylindrical shell structure with functionally graded coatings under thermal environment;

[0162] In order to study the nonlinear vibration characteristics of the functionally graded coating-composite combined shell considering temperature and amplitude dependence, it is necessary to correct the material parameters discussed in Step 1.4 based on the Jones-Nelson nonlinear theory, introduce the amplitude variable into the theoretical model, so that the structure has the characteristics of both temperature and amplitude dependence.

[0163] On the premise of only considering amplitude dependence, based on the Jones-Nelson nonlinear theory, the Young's modulus, shear modulus of the fiber-reinforced composite matrix and the elastic modulus of the functionally graded coating can be expressed in the following forms:

[0164] (46)

[0165] Zwhere and represent the fitting coefficients related to the amplitude of the fiber-reinforced composite matrix, A a , B a and A b , B b represent the amplitude fitting coefficients of component A and component B of the functionally graded coating respectively; is the corresponding dimensionless strain energy density.

[0166] Thus, the elastic modulus of the fiber-reinforced composite matrix considering both amplitude and temperature dependence and the elastic moduli of component A and component B of the functionally graded coating can be obtained, and their expressions are as follows:

[0167] (47)

[0168] The solution idea in this section is similar to that in Step 1.4, and the energy expression is the same as it, so it will not be repeated here. Different from Step 1.4, in this section, the Jacobi polynomial is used to assume the mid-plane displacement components, and its expression is as follows:

[0169] (48)

[0170] where represents the mid-plane displacement component, represents the unknown coefficient in the mid-plane displacement function, is a sub-expression of the Jacobi polynomial, and its expression is shown in formula (37), is the corresponding Fourier expression.

[0171] ​​​​​​​​​Next, using the Lagrangian energy functional L constructed in Step 1.4 and taking the partial derivatives of the unknown coefficients by the Rayleigh-Ritz method, the following matrix equation can be obtained:

[0172] (49)

[0173] where, and represent the complex stiffness matrix, stiffness matrix, and mass matrix respectively, represents the damping matrix, and its expression is as follows:

[0174] (50)

[0175] where c and d are Rayleigh damping coefficients, and their expressions are:

[0176] (51)

[0177] where, and are the two natural frequencies of the structure, and are the corresponding two modal damping ratios.

[0178] Neglecting the damping matrix C in Equation (49), the following characteristic equation can be obtained:

[0179] (52)

[0180] The natural frequencies and modal vibration modes of the structure can be obtained through the above equation.

[0181] Assume that the functionally graded coating-composite shell is subjected to a base harmonic excitation Z b in the z direction, and its expression is:

[0182] (53)

[0183] where, is the acceleration amplitude of the harmonic excitation, is the excitation circular frequency, and t is time.

[0184] Adopting the modal strain energy method, the modal damping ratio and loss factor of the structure can be expressed in the following form:

[0185] (54)

[0186] where, respectively represent the modal vibration mode vector, modal damping ratio, and loss factor related to the r-th mode of the structure.

[0187] To facilitate the solution of the forced vibration response of the composite shell structure, the mass matrix M, stiffness matrix K, and damping matrix C are regularized, and the results are as follows:

[0188] (55)

[0189] where M, K, and C are the regularized mass matrix, stiffness matrix, and damping matrix, respectively. diag represents the diagonal matrix, and are the circular frequency and modal mass of the r-th order mode of the structure, respectively, and their expressions are as follows:

[0190] (56)

[0191] Therefore, the complex frequency response function of the structure can be expressed as:

[0192] (57)

[0193] where N a represents the maximum order of the mode.

[0194] Using the mode superposition method, the steady-state response amplitude function of the composite conical-cylindrical shell structure with a functionally graded coating at the response point can be expressed in the following form:

[0195] (58)

[0196] In the above expression, is the equivalent force load, and represent the r-th order displacement functions related to the direction of the base excitation and the response point , is the r-th order modal stiffness, , , and are expressed as follows:

[0197] (59)

[0198] where Sub(u,v) means substituting the value of variable v into the expression u for calculation.

[0199] Finally, substituting the frequency ω and excitation amplitude Z b of the base excitation into Eq. (58), the acceleration response of the structure under the base harmonic excitation can be calculated, and the corresponding resonant displacement response can be obtained through double integration.

[0200] Step 3: Identify the material parameters of the functionally graded coating-composite shell and determine the corresponding fitting coefficients;

[0201] 3.1 Identify material parameters;

[0202] Using the natural frequency values of the uncoated specimens and the coated specimens obtained from experimental tests under different excitation amplitudes and temperatures as the target values, a natural frequency error function is constructed using the particle swarm algorithm to identify the elastic modulus of the composite shell layer under different excitation amplitudes and temperature conditions 、 、 、 、 and the Young's modulus of the functionally graded coating 、 . The natural frequency error function can be expressed as:

[0203] (60)

[0204] where is the experimental target value of the r-th order natural frequency, and is the corresponding natural frequency obtained from the test.

[0205] Since there are a total of 7 material parameters to be identified, a search space with a dimension of 7 is defined, and the particle swarm Y is initialized in the following form:

[0206] (61)

[0207] where i is the size of the population, and is the position vector of the i-th particle in the population, and its expression is:

[0208] (62)

[0209] During the iteration process, the flight speed V of each particle, the historical optimal individual solution P of each particle in the iterative search, and the overall best historical solution S reached by all particles are defined as:

[0210] (63)

[0211] During the iteration process, the update methods of the position X and speed V of each particle are as follows:

[0212] (64)

[0213] where is the inertia weight, and c1 and c2 are learning factors, which respectively represent the acceleration weight coefficients of the particles when searching for the individual optimal solution and the group optimal solution; and are random numbers uniformly distributed in the interval [0, 1].

[0214] In addition, during the iteration process, since the elements of each particle are randomly initialized, it is necessary to limit their value ranges. Taking the material parameters of the fiber layer of the composite shell without considering material nonlinearity at room temperature , , , , and the Young's modulus of the functionally graded coating , as the intermediate values, and considering the influence of the excitation amplitude and temperature on the material parameters, the variation ranges are set as . Therefore, the search ranges of each particle can be expressed in the following form:

[0215] (65)

[0216] Through the above iterative method, when the iteration termination condition (usually ) is satisfied, the global optimal solution can be output. Repeating the above steps can obtain the dynamic material parameters of the fiber-reinforced composite layer at different temperatures and amplitudes. In addition, after determining the elastic modulus of the fiber-reinforced composite layer, continuing to use the above iterative algorithm can further determine the material parameters of the functionally graded coating.

[0217] The loss factors of the fiber-reinforced composite material matrix and the functionally graded coating are also identified through the above process. It should be noted that during the loss factor identification process, the particle is expressed in the following form:

[0218] (66)

[0219] where respectively represent the dynamic loss factors of the fiber-reinforced composite layer, which can be expressed in the following form:

[0220] (67)

[0221] Taking the loss factors of the fiber layer of the composite shell without considering material nonlinearity at room temperature , , , and as the intermediate values, and considering the influence of temperature and amplitude on the material loss factors, their value ranges can be expressed in the following form:

[0222] (68)

[0223] In addition, the error function of damping can be expressed in the following form:

[0224] (69)

[0225] where is the experimental target value of the r-th order damping, is the corresponding damping value obtained from the test.

[0226] In the process of damping identification, when the iteration termination condition (usually ) is satisfied, the iteration is stopped and the optimal solution is output. By repeating the above operations, the dynamic loss factors at different temperatures and amplitudes can be obtained. In addition, after determining the loss factor of the fiber-reinforced composite, the loss factor of the functionally graded coating can be further determined.

[0227] 3.2 Determine the relevant fitting coefficients;

[0228] Using the nonlinear material parameters obtained in Section 3.1, combined with the Curve Fitting toolbox in MATLAB software, the fitting coefficients of the material parameters related to temperature and amplitude in the theoretical model of the functionally graded coating-composite shell are solved. First, the nonlinear elastic modulus measured for the composite shell under different temperature and amplitude conditions is taken as "YData", and the corresponding dimensionless strain energy density at each excitation amplitude is taken as "X Data". Subsequently, formula (46) is input into the "Custom Equation" of the Curve Fitting toolbox in the form of a custom function, and MATLAB will automatically fit according to the input data and output the required fitting coefficients and .

[0229] Step 4: Prepare the functionally graded coating and coat it on the conical-cylindrical shell substrate structure, and measure the geometric and material parameters of the substrate and the coating;

[0230] Step 4.1: Prepare the substrate sample. A conical-cylindrical shell specimen was prepared using continuous 3D printing technology, and its substrate material consists of carbon fiber and polylactic acid resin;

[0231] Step 4.2: Spray the functionally graded coating on the surface of the specimen using the atomization deposition technique. This coating is a high-temperature resistant ceramic material that can form a film with excellent flexibility, impact resistance, and chemical stability at room temperature and high temperature environments. The coating is evenly sprayed on the surface of the specimen through an atomizer to ensure a uniform coating thickness distribution;

[0232] Step 4.3: The material parameters of the conical-cylindrical shell structure substrate and the functionally graded coating are determined by the inverse identification method.

[0233] Step 5: Build a high-temperature vibration test system to measure the dynamic response characteristics of the specimen under normal temperature and multiple temperature conditions;

[0234] This system mainly consists of the following equipment: a vibration table and its multi-functional controller, a power amplifier, a Polytec PDV-100 Doppler laser vibrometer, a heating chamber and a temperature controller, thermocouples, a specimen fixture, an LMS data acquisition instrument, and two computer workstations.

[0235] Before the experiment, fix the specimen fixture on the vibration table, and use a torque wrench to fasten one end of the conical-cylindrical shell to the fixture to simulate the boundary conditions of one end fixed and one end free, ensuring efficient transmission of the excitation signal to the specimen. The heating chamber is equipped with heating molybdenum rods and a thermocouple controller for real-time monitoring of the temperature inside the chamber and feedback of the data to the temperature controller to achieve precise temperature control. The experiment is carried out at four temperature points of 50°C, 100°C, 150°C, and 200°C. After setting the target temperature through the temperature controller and waiting for it to stabilize, perform sweep-frequency and fixed-frequency experiments in sequence. Use the PDV-100 Doppler laser vibrometer to capture the velocity response signal of the specimen, and use the LMS data acquisition instrument to collect the excitation and response signals in real time to obtain the natural frequency and resonance response of the structure to be measured.

[0236] Step 6: Verification of the theoretical model based on experimental tests;

[0237] Step 6.1: Theoretical and experimental comparison and verification of the dynamic characteristics of the functionally graded coating conical-cylindrical shell structure under normal temperature conditions. Based on the theoretical model established in Step 1 and the vibration test system built in Step 5, theoretical calculations and experimental tests were carried out on the conical-cylindrical shell structures with and without functionally graded coatings at normal temperature. First, compare and analyze the theoretical predictions and experimental test results of the modal vibration modes to verify the accuracy of the model in predicting modal characteristics; second, further confirm the reliability of the model by comparing the theoretical values and experimental values of the natural frequencies. To deeply verify the effectiveness of the model in predicting dynamic responses, for different fundamental harmonic excitation conditions, compare and analyze the theoretical resonance responses and experimental test results of the composite shell structures with and without coatings.

[0238] Step 6.2: Theoretical and experimental comparison verification of the dynamic characteristics of the functionally graded coating conical-cylindrical shell structure. Similar to Step 6.1, the theoretical calculations and experimental tests of the conical-cylindrical shell structures with and without functionally graded coatings under high-temperature conditions were further carried out for comparison verification. The specific contents include: obtaining the linear natural frequencies of the structure at different temperatures through sweep-frequency experiments and comparing them with the theoretical prediction results; using variable-amplitude excitation tests to compare the variation characteristics of the theoretical and tested nonlinear natural frequencies of the structure with temperature and amplitude, and comparing the theoretical and experimental results; comparing the resonance response characteristics of the structure at different temperatures to verify the prediction accuracy of the established model under dynamic load conditions.

[0239] Step 7: Study the dynamic performance mechanism of the functionally graded coating conical-cylindrical shell structure under high-temperature vibration environment

[0240] Based on the verified theoretical model, a parametric systematic analysis of the functionally graded coating-composite combined shell under high-temperature vibration environment was carried out. The key research contents include: the existence effect of the functionally graded coating (comparison with and without coating), the influence law of the coating thickness parameter, the regulation effect of the coating gradient index, and the influence mechanism of key parameters on the vibration characteristics of the structure. Through systematic parameter sensitivity analysis, the action mechanism of the functionally graded coating on the dynamic performance of the composite combined shell under high-temperature vibration conditions was revealed.

[0241] Specific example

[0242] The functionally graded coating conical-cylindrical shell structure consists of a conical-cylindrical shell matrix and a functionally graded protective coating. The matrix is prepared by continuous 3D printing technology and is five groups of carbon fiber reinforced polylactic acid resin composite specimens with the same geometric parameters. The functionally graded coating consists of two components provided by Jiangsu Junyuan New Materials Co., Ltd.: Component A is a 380 series ceramic coating, and Component B is an organic solvent binder added with high-temperature resistant fillers and functional inorganic fillers. This coating system can form a uniform protective film from room temperature to high-temperature environment, showing excellent high-temperature resistance and vibration and shock resistance characteristics. The specific material parameters of the specimen matrix material and the functionally graded coating are shown in Table 1, and all relevant parameters are provided and verified by the manufacturer.

[0243] Table 1 Specimen geometry and material parameters

[0244]

[0245] Based on the experimental verification process described in Step 6, Figure 4 and 5 respectively show the theoretical and experimental comparison results of the first four natural frequencies and resonance displacement responses of the functionally graded coating conical-cylindrical shell structure. Under the 50°C condition, the maximum deviation between the theoretical calculation and experimental test of the natural frequency of the specimen without coating is , the deviation after introducing the functionally graded coating is ; the maximum deviations of the corresponding resonance displacement responses are respectively and . Under the working condition of 100 °C, the maximum deviations of the natural frequencies of the uncoated and coated structures are respectively and ; the maximum deviations of the resonance displacement responses are respectively and . The theoretical and experimental errors under all test working conditions are within an acceptable range, verifying the effectiveness of the theoretical model.

[0246] Figures 6-9 The influence laws of the key parameters of the functionally graded coating (including the gradient index and thickness ratio) on the linear dynamic response characteristics (natural frequency and resonance response) of the structure are systematically analyzed under the temperature conditions of 50 °C and 100 °C; Figure 10 and 11 further study the action mechanism of the above parameters on the nonlinear dynamic response characteristics (nonlinear natural frequency and vibration response) of the structure under the high-temperature working condition of 150 °C.

Claims

1. A method for modeling and analyzing the high-temperature vibration dynamics of a conical-cylindrical shell with a functionally graded coating, characterized in that It includes the following steps: Step 1: Establish a linear vibration analysis model of a composite conical-cylindrical shell structure considering the thermodynamics characteristics of a functionally graded coating under a high-temperature vibration environment; Step 2: Solve the nonlinear vibration characteristics of the conical-cylindrical shell structure with a functionally graded coating under a thermal environment; Step 3: Identify the material parameters of the functionally graded coating-composite material combined shell and determine the corresponding fitting coefficients; Step 4: Prepare a functionally graded coating and coat it on the conical-cylindrical shell matrix structure, and measure the geometric and material parameters of the matrix and the coating; Step 5: Build a high-temperature vibration test system and measure the dynamic response characteristics of the specimen under normal temperature and multi-temperature conditions; Step 6: Verify the theoretical model based on experimental tests; Step 7: Study the dynamic performance mechanism of the conical-cylindrical shell structure with a functionally graded coating under a high-temperature vibration environment.

2. The high-temperature vibration dynamics modeling and analysis method for a conical-cylindrical shell with a functionally graded coating according to claim 1, characterized in that The said Step 1 includes: Step 1.1: Establish a coordinate system for the conical-cylindrical shell structure with a functionally graded coating; Step 1.2: Define the parameters related to the structure and materials; Step 1.3: Determine the equivalent material parameters of the composite material matrix and the functionally graded coating under a thermal environment; Step 1.4: Solve the linear vibration characteristics of the conical-cylindrical shell structure with a functionally graded coating under a thermal environment.

3. The high-temperature vibration dynamics modeling and analysis method for a conical-cylindrical shell with a functionally graded coating according to claim 1, characterized in that The said Step 2 modifies the material parameters discussed in Step 1.4 based on the Jones-Nelson nonlinear theory, and introduces the amplitude variable into the theoretical model, so that the structure has the characteristics of both temperature and amplitude dependence.

4. The high-temperature vibration dynamics modeling and analysis method of a functionally graded coating conical-cylindrical shell according to claim 1, characterized in that The said Step 3 includes: Step 3.1 Identify the material parameters; Take the natural frequency values of the specimens without coating and specimens with coating obtained by experimental tests under different excitation amplitudes and temperatures as the target values, and use the particle swarm algorithm to construct a natural frequency error function, so as to identify the elastic modulus of the composite material combined shell layer and the Young's modulus of the functionally graded coating under different excitation amplitudes and temperature conditions; Step 3.2 Determine the relevant fitting coefficients; Use the nonlinear material parameters obtained in Section 3.1, and combine with the Curve Fitting toolbox in MATLAB software to solve the fitting coefficients of the material parameters related to temperature and amplitude in the theoretical model of the functionally graded coating-composite material combined shell.

5. The high-temperature vibration dynamics modeling and analysis method of the functionally graded coating conical-cylindrical shell according to claim 1, characterized in that The said Step 4 includes: Step 4.1: Prepare the matrix sample: A conical-cylindrical shell specimen is prepared by using the continuous 3D printing technology, and its matrix material is composed of carbon fiber and polylactic acid resin; Step 4.2: Spray a functionally graded coating on the surface of the specimen by using the atomization deposition technology: This coating is a high-temperature resistant ceramic material, which can form a film with excellent flexibility, impact resistance and chemical stability at room temperature and high temperature environments. The coating is evenly sprayed on the surface of the specimen through an atomizer to ensure uniform coating thickness distribution; Step 4.3: The material parameters of the conical-cylindrical shell structure matrix and the functionally graded coating are determined by the inverse identification method.

6. The high-temperature vibration dynamics modeling and analysis method of the functionally graded coating conical-cylindrical shell according to claim 1, characterized in that The system of the said Step 5 mainly consists of the following equipment: a vibration table and its multi-functional controller, a power amplifier, a Polytec PDV-100 Doppler laser vibrometer, a heating box and a temperature controller, a thermocouple, a special fixture for the specimen, an LMS data acquisition instrument and a computer workstation.

7. The high-temperature vibration dynamics modeling and analysis method for a conical-cylindrical shell with a functionally graded coating according to claim 1, characterized in that The said Step 6 includes: Step 6.1: Theoretical and experimental comparison verification of the dynamic characteristics of the functionally graded coating conical-cylindrical shell structure under normal temperature conditions; Step 6.2: Theoretical and experimental comparison verification of the dynamic characteristics of the functionally graded coating conical-cylindrical shell structure under high temperature environment.

8. The high-temperature vibration dynamics modeling and analysis method for a functionally graded coating conical-cylindrical shell according to claim 1, characterized in that: Based on the verified theoretical model, step 7 conducts a parametric system analysis of the functionally graded coating-composite shell under high temperature vibration environment. The key research includes: the existence effect of the functionally graded coating (comparison with and without coating), the influence law of the coating thickness parameter, the regulation effect of the coating gradient index, and the influence law of key parameters on the vibration characteristics of the structure.

9. The high-temperature vibration dynamics modeling and analysis method of the functionally graded coating conical-cylindrical shell according to claim 1, characterized in that: Through systematic parameter sensitivity analysis in step 7, the action mechanism of the functionally graded coating on the dynamic performance of the composite shell under high temperature vibration conditions is revealed.

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