Dynamic response analysis method of piezoelectric hybrid functionally graded sandwich hyperbolic plates

Abstract

The present invention discloses a dynamic response analysis method for a piezoelectric hybrid functionally graded sandwich hyperbolic plate. Taking into account the deformation compensation control effect of the piezoelectric material in the FGM layer, the relevant dynamic equilibrium equations are established based on the constitutive equations of the piezoelectric material regarding the force, electricity, and thermal coupling of stress, strain, electric displacement, and electric field, as well as the relationship between the electric field and potential. The Galerkin method and Newmark-β method are then used to linearize and discretize the motion differential control equations of the piezoelectric hybrid FGM sandwich hyperbolic plate with a wrinkling effect under the action of a moving load. In the numerical results, the relationships between the temperature rise pattern and action range of the nonlinear temperature field, the action form and movement speed of the moving load, the wrinkle morphology and deformation range, the material performance parameters, the structural geometric parameters, the applied voltage, and the active control on the nonlinear dynamic response of the FGM sandwich hyperbolic plate are discussed.

Description

Technical Field

[0001] The present invention relates to the technical field of composite material structure dynamics, and in particular to a dynamic response analysis method of a piezoelectric hybrid functional gradient sandwich hyperbolic plate. Background Art

[0002] Functionally graded materials (FGMs) were initially developed to address thermal stress mitigation in rocket propulsion systems. They are a new class of materials developed to meet the specialized material requirements of high-tech fields such as aerospace and currently hold broad application prospects in engineering. The basic FGM components consist of a highly heat-resistant ceramic matrix and a high-strength metal matrix. These two matrices serve as the surfaces of the FGM, gradually blending and transitioning into each other in a gradient-like manner. This smooth transition of the material's building blocks enables a gradient change in the structure's physical properties. In particular, the smooth matching of thermal expansion parameters effectively mitigates thermal stress concentrations caused by sudden cooling and rapid temperature rise. This material design concept has driven the emergence and rapid technological development and application of new FGMs with various specific components. Piezoelectric sensing materials, through the interactive conversion of their deformation and electric potential energy, can produce specific mechanical effects, thereby enhancing and enriching the functionality of FGMs. Research results indicate that the design and application of functionally graded piezoelectric materials (FGPMs) and their structures can offer significant technological value in the field of smart sensors and related equipment.

[0003] Hyperbolic plates are a common engineering structure that, through curvature adjustment, can be transformed into diverse engineering structures, including flat plates, cylindrical shells, and variable-curvature shells. Due to the bidirectional curvature of the structure, these structures exhibit geometric nonlinearity in mechanical principles. This is particularly true in nonlinear dynamics, where factors such as local structural reinforcement, nonlinear elastic base compensation, material parameter nonlinearity, and nonlinear load effects are considered, leading to complex nonlinear dynamic problems. Due to the demands of practical engineering applications, the influence of elastic bases on plate and shell structures has also attracted attention.

[0004] The intelligent structure system integrates sensing, execution and control. With the development of intelligent technology, it is widely used in various engineering structures. It can provide equipment structures with multiple intelligent functions such as self-health diagnosis, adaptive execution output, correction and calibration of input signals, and even self-learning and evolution, to obtain more powerful performance and more secure installation guarantees. This intelligent structure system is of great significance for the design performance evaluation and interactive control of multi-purpose composite materials structures in various occasions.

[0005] Based on the design concept of intelligent structural systems, according to the constitutive equations of piezoelectric materials regarding the force, electricity and thermal coupling of stress, strain, electric displacement and electric field, as well as the relationship between electric field and electric potential, the nonlinear dynamic response and active control of the piezoelectric hybrid FGPM composite sandwich hyperbolic plate with pleating effect are analyzed and calculated through the constant gain velocity negative feedback control law. The structural dynamics problem of elastic bodies under moving loads also has a wide range of engineering application backgrounds in the fields of intelligent equipment, aerospace, ships and warships, rail transportation, wind power and new energy. Summary of the Invention

[0006] The technical problem to be solved by the present invention is: the present invention provides a dynamic response analysis method for a piezoelectric hybrid functional gradient sandwich hyperbolic plate, which can efficiently and accurately perform calculation analysis to achieve the structural design and performance index response of the hybrid composite material sandwich hyperbolic plate in a complex working environment.

[0007] In order to solve the above technical problems, the technical solution proposed by the present invention is:

[0008] A dynamic response analysis method for a piezoelectric hybrid functionally gradient sandwich hyperbolic plate, comprising the following steps:

[0009] Step S1, constructing a mechanical model of a FGM sandwich hyperbolic plate with mixed morphology based on a corrugated structure;

[0010] The sandwich hyperbolic plate has three layers, including a core layer and isotropic surface layers on both sides of the core layer. The core layer is a piezoelectric hybrid functional gradient material layer. The constitutive and geometric relationships of the piezoelectric hybrid functional gradient sandwich hyperbolic plate with wrinkle effect are constructed.

[0011] Step S2, constructing the dynamic equilibrium equation of the corresponding problem;

[0012] Step S3, solving and analyzing mechanical problems based on the dynamic parameters of the fold geometry;

[0013] According to the morphology setting of the piezoelectric hybrid functionally graded hyperbolic plate, the stiffness coefficient of the hyperbolic plate is selected differently at different spatial positions, and a constructor is constructed to characterize the hybrid wrinkle morphology characteristics, thereby expanding the mechanical information contained in the dynamic equilibrium equation of the hyperbolic plate model.

[0014] Step S4, constructing a mathematical expression of the piezoelectric active control system;

[0015] Analyze the nonlinear dynamic characteristics of piezoelectric hybrid functionally graded sandwich hyperbolic plates with wrinkle effect under moving loads;

[0016] Step S5: solving and analyzing the dynamic response based on the Galerkin method, the Newmark method and the iterative method.

[0017] As a further improvement of the above technical solution:

[0018] Preferably, in step S1, the material property P of the sandwich layer can be expressed as:

[0019]

[0020] Among them, P FM-1 and P FM-2 Respectively represent the material properties of the outer and inner layers of the sandwich layer; V FM-1 is the volume component of material type-1;

[0021] The constitutive relation of the piezoelectric field is described as follows:

[0022] {σ} (k) =[C] (k) ({ε} (k) -{α} (k) ΔT)-{e} (k) {E} (k)

[0023] {D} (k) ={e} (k) {ε} (k) +{K} (k) {E} (k) (k=0, N+1)

[0024] Among them, {e} is the piezoelectric constant matrix of the material, {D} is the electric displacement matrix, {K} is the dielectric constant matrix, and {E} is the electric field intensity matrix;

[0025] The expression of temperature field T(z) is:

[0026]

[0027] Among them, h p is the thickness of the hyperbolic plate; K k is the thermal conductivity coefficient of the kth layer material of the sandwich structure; i is the cumulative process variable, for example, when k = 3, K k represents the thermal conductivity of the third layer of material, then i = 1, 2, 3;

[0028] Relying on the mid-surface stress and strain components and bending strain components Get strain (ε x , ε y , ε xy ) is as follows:

[0029]

[0030]

[0031]

[0032] (-h / 2≤z≤h / 2)

[0033] Among them, the mid-surface strain component It can be expressed by the mid-plane displacement (u0, v0, w0) as follows

[0034]

[0035]

[0036]

[0037] The expression for bending strain is:

[0038]

[0039] Preferably, in step S2, the nonlinear dynamic control equation of the sandwich hyperbolic plate in the thermal environment is expressed as follows:

[0040]

[0041]

[0042]

[0043] Where I P Represents the inertia of the sandwich hyperbolic plate, which is expressed as follows

[0044]

[0045] Among them, ρ k (k=1, 2, 3) represents the density of each material layer of the sandwich hyperbolic plate; (N x , N y , N xy ) and (M x , M y , M xy ) represent the in-plane force and bending moment respectively; P(x, y) represents the load amplitude according to the coordinate change; V load Indicates the speed of moving load; A load (x) represents the range of the mobile load action area; δ v and δ region are Dirac functions, taking values ​​of 1 or 0, which are used to constrain the moving trajectory and action range of the moving load, i.e., δ v =1 means the load moves to the corresponding coordinate, δ region =1 indicates that the load is within the corresponding coordinate area.

[0046] Preferably, in step S2, the expressions for stress and moment components are as follows:

[0047]

[0048]

[0049]

[0050]

[0051]

[0052] in

[0053]

[0054] The forces and moments associated with the thermal environment are expressed as follows

[0055]

[0056] The force and torque are expressed by the components related to the piezoelectric effect as follows

[0057]

[0058] Preferably, in step S3, the tensile stiffness coefficient and the bending stiffness coefficient of the hyperbolic plate are expressed as follows:

[0059]

[0060]

[0061] Among them, A ij is the tensile stiffness coefficient, B ij is the bending coupling stiffness coefficient, D ij is the bending stiffness coefficient, E ij , F ij and H ij and higher-order stiffness coefficients.

[0062] Preferably, in step S4, assuming that the piezoelectric layer is polarized only along the thickness direction and no voltage is applied to the outer surface of the sensing layer, the electric displacement of the sensing layer can be expressed as:

[0063]

[0064] The amount of charge output by the sensing layer is:

[0065]

[0066] In the above formula, A eis the effective electrode on the surface. The current of the sensor is expressed as

[0067]

[0068] After the current amplifier gain G e and negative speed feedback control gain G V After the action, the feedback voltage applied to the bottom of the actuating layer is

[0069] V(t)=G V (G e i(t))=G*i(t)

[0070] Where G is the gain coefficient.

[0071] Preferably, in step S5, the control equilibrium equation and boundary conditions are satisfied, and the relevant displacement parameter function is:

[0072]

[0073]

[0074]

[0075] in, is the unknown displacement amplitude, m and n are the half-wave numbers in the x- and y-axis directions respectively;

[0076] The displacement parameters are obtained using the Galerkin method. The system of dynamic differential governing equations with respect to the time variable t;

[0077] In the time domain, the Newmark method is used to obtain the acceleration term w in the control equations ，tt , speed term w ，t , the expression of the displacement term w at time point t+Δt;

[0078] described The nonlinear dynamic control equations about the time variable t are transformed into a linear equation of the corresponding variables and solved iteratively.

[0079] The dynamic response analysis method of the piezoelectric hybrid functionally gradient sandwich hyperbolic plate provided by the present invention has the following advantages over the prior art:

[0080] (1) The dynamic response analysis method of the piezoelectric hybrid functional gradient sandwich hyperbolic plate of the present invention is based on the dynamic characteristics and active control of the piezoelectric hybrid FGM sandwich hyperbolic plate with wrinkle effect under the action of moving load in nonlinear temperature field. A hyperbolic plate structural model based on corrugated structure that can describe arbitrary wrinkle sharpness is constructed. At the same time, the deformation compensation control effect of the piezoelectric material of the FGM layer is considered. According to the constitutive equations of the piezoelectric material for force, electricity and thermal coupling of stress, strain, electric displacement and electric field, as well as the relationship between electric field and potential, the relevant dynamic equilibrium equations are established. The Galerkin method and Newmark-β are used to realize the linear discretization of the motion differential control equation of the piezoelectric hybrid FGM sandwich hyperbolic plate with wrinkle effect under the action of moving load. In the numerical results, the relationship between the temperature rise mode and action range of the nonlinear temperature field, the action form and moving speed of the moving load, the wrinkle morphology and deformation range, the material performance parameters, the structural geometric parameters, the applied voltage and the active control on the nonlinear dynamic response of the FGM sandwich hyperbolic plate is discussed.

[0081] (2) The dynamic response analysis method of the piezoelectric hybrid functional gradient sandwich hyperbolic plate of the present invention proposes a mechanical model of the FGM sandwich hyperbolic plate with a mixed morphology based on a corrugated structure, so as to achieve an accurate and independent description of the structural geometric relationship, reflecting a more realistic mechanical problem.

[0082] (3) The dynamic response analysis method of the piezoelectric hybrid functionally graded sandwich hyperbolic plate of the present invention considers an active control strategy for the piezoelectric effect when the structure is moving, thereby more efficiently suppressing the deformation of the structure and enhancing the structure's load resistance performance. By selecting an appropriate control gain to enhance the mechanical properties of the piezoelectric composite structure, the dynamic performance of the functionally graded sandwich hyperbolic plate under moving loads and the active control of the intelligent structural system can be efficiently and accurately performed to achieve the structural design and performance index response of the hybrid composite sandwich hyperbolic plate in complex working environments. BRIEF DESCRIPTION OF THE DRAWINGS

[0083] Figure 1 It is a schematic diagram of the piezoelectric hybrid functional gradient sandwich hyperbolic plate of the present invention.

[0084] FIG2( a ) is a schematic diagram of the circular distribution wrinkle morphology in the embodiment.

[0085] FIG2( b ) is a schematic diagram of the trapezoidal corrugated fold morphology in the embodiment.

[0086] FIG2( c ) is a schematic diagram of the mixed corrugated morphology in the embodiment.

[0087] FIG3( a ) is a dimensionless displacement curve diagram of coordinate sampling points of a hyperbolic plate in an embodiment of the present invention.

[0088] FIG3( b ) is a stress response curve diagram of the coordinate sampling points of the hyperbolic plate in an embodiment of the present invention. DETAILED DESCRIPTION

[0089] The following is a detailed description of the specific embodiments of the present invention. It should be understood that the specific embodiments described herein are only used to illustrate and explain the present invention and are not intended to limit the present invention.

[0090] like Figure 1 As shown in Figure 3, the dynamic response analysis method of the piezoelectric hybrid functional gradient sandwich hyperbolic plate of the present invention is based on the corrugation configuration theory to describe the elastic wrinkle effect, and the corrugation configuration shown in Figure 2 is used to describe the wrinkle geometric model. The equivalent mechanical performance analysis of the wrinkle effect is achieved by combining the corrugation configuration geometric morphology parameters with the hybrid model; the impact of wrinkles is limited to the hyperbolic plate structure, and does not cause changes in the material properties such as cracking and delamination of the functional gradient material layer, aging damage, and material plasticity; the hyperbolic plate structure applicable to this application has a small curvature, and the influence of the curvature on the wrinkle geometric model in the thickness direction can be ignored. In the geometric equation, the structural curvature of the hyperbolic plate is used to express the effect on the mechanical properties.

[0091] The dynamic response analysis method of this embodiment includes the following steps:

[0092] Step S1: Construct a mechanical model of a FGM sandwich hyperbolic plate with a mixed morphology based on a corrugated structure

[0093] Step S1-1: Establishing the geometric parameters of the sandwich hyperbolic plate

[0094] like Figure 1 As shown in the figure, the sandwich hyperbolic plate has three layers, including a core layer and isotropic surface layers on both sides of the core layer. The core layer is a piezoelectric hybrid functional gradient material layer (Exponential Law-FGM). The main curvature radii of the piezoelectric hybrid functional gradient sandwich hyperbolic plate are R1 and R2, the side lengths are a and b, and the thickness of the hyperbolic plate is h. The hyperbolic plate structure is subjected to a moving transverse nonlinear load P(x, y), which moves along the surface of the hyperbolic plate structure at a constant initial velocity V. load The nonlinear load P(x, y) moves regularly along the predetermined trajectory in the y-axis direction. The load action range during the movement is A. region (x), the hyperbolic plate structure is placed in a nonlinear temperature field T(x, y, z).

[0095] The core layer has a general material property P, with a mass density of ρ and an elastic modulus of E. The material property P varies along the thickness of the hyperbolic plate, while the Poisson's ratio v is a constant. In this example, the outer surface of the hyperbolic plate (z = -h / 2) is rich in material type -1, while the inner surface of the hyperbolic plate (z = h / 2) is rich in material type -2.

[0096] Step S1-2: Constructing a mechanical model of a sandwich hyperbolic plate

[0097] The material property function of each component material expressed by the volume component is expressed as:

[0098] P=P FM-1 V FM-1 +P FM-2 V FM-2 (1)

[0099] Among them, P FM-1 and P FM-2 Respectively represent the material properties of the outer and inner layers of the sandwich layer; V FM-1 and V FM-2 They are the volume components of material type-1 and material type-2, and their relationship is:

[0100] V FM-1 +V FM-2 =1 (2)

[0101] Volume component V FM-1 It changes along the normal direction of the plate thickness in the form of an exponential function, and the functional relationship expression is:

[0102]

[0103] The general material property P can be expressed as:

[0104]

[0105] The FGM material properties change smoothly from P to P along the thickness of the hyperbolic plate according to the volume parameter n. FM-1 (z=-h core / 2) Transition to P FM-2 (z=h core / 2).

[0106] Sandwich layer thickness h core , the inner and outer layers are The constitutive relation of the piezoelectric field is described as follows

[0107] {σ} (k) =[C] (k) ({ε} (k) -{α} (k) ΔT)-{e} (k) {E} (k)

[0108] {D} (k) ={e} (k) {ε} (k) +{K} (k) {E} (k)(k=0,N+1) (5)

[0109] Where {e} is the material's piezoelectric constant matrix, {D} is the electric displacement matrix, {K} is the dielectric constant matrix, and {E} is the electric field strength matrix. If the plate is thin and the positive and negative electrodes are on the surface, the voltage applied to the piezoelectric layer can be set parallel to the plate thickness direction. Then, only the electric field strength component E in the plate thickness direction exists. z .

[0110] Further knowledge

[0111]

[0112] The stiffness coefficient is expressed as follows:

[0113]

[0114] Where, (σ xx , σ yy , τ xy ) is the membrane stress, (τ xz , τ yz ) is the transverse shear stress, represents the stiffness coefficient; and represents the thermal expansion coefficient along the x and y directions, ΔT is the temperature increment relative to the initial state, and ΔT = T-T0 is defined, where T0 is the initial stress state of the hyperbolic plate and T0 = 0; is the piezoelectric coefficient of the material, E z is the external electric field strength, E z =V / h core , V and h core represent the applied voltage and the thickness of the piezoelectric layer, respectively.

[0115] The temperature change is limited to the thickness direction of the hyperbolic plate, and the initial temperature of the piezoelectric enhanced hybrid functional gradient sandwich hyperbolic plate with wrinkle effect is T0. The one-dimensional steady-state heat conduction equation and the corresponding thermal boundary conditions are as follows

[0116]

[0117] Among them, K k (z) is the thermal conductivity of the k-th layer material (k=1, 2, 3).

[0118] The thermal boundary conditions are as follows:

[0119] ΔT(z) z＝h / 2 =T1-T0, ΔT(z) z＝-h / 2 =T2-T0 (8)

[0120] Where T1 and T2 are the upper and lower surface temperatures of the piezoelectrically enhanced hybrid functionally gradient sandwich hyperbolic plate with wrinkle effect, respectively. According to the thermal boundary conditions, the expression of the temperature field T(z) is:

[0121]

[0122] Among them, h p is the thickness of the hyperbolic plate; K k is the thermal conductivity coefficient of the kth layer material of the sandwich structure; i is the cumulative process variable, for example, when k = 3, K k Represents the thermal conductivity coefficient of the third layer material, then i=1, 2, 3.

[0123] Based on Kirchhoff hypothesis (classical theory), relying on the mid-surface stress and strain components and bending strain components Get strain (ε x , ε y , ε xy ) is as follows:

[0124]

[0125]

[0126]

[0127] (-h / 2≤z≤h / 2)

[0128] Based on the Von Karman theory, the mid-surface strain component It can be expressed by the mid-plane displacement (u0, v0, w0) as follows

[0129]

[0130]

[0131]

[0132] The expression for bending strain is:

[0133]

[0134] Step S2: construct the dynamic equilibrium equation corresponding to the dynamic response problem of the piezoelectric hybrid functional gradient sandwich hyperbolic with wrinkle effect under the action of moving load in thermal environment

[0135] Based on Hamilton's variational principle, the dynamic equilibrium equation of the sandwich hyperbolic plate can be obtained as follows

[0136]

[0137] Among them, U represents the strain energy of the structure, V represents the thermal potential energy, W represents the work done by the external force, and K represents the kinetic energy.

[0138] The strain energy of the sandwich hyperbolic plate is:

[0139]

[0140] The potential energy of the functionally graded sandwich hyperbolic plate with wrinkle effect is mainly caused by thermal stress, and the corresponding expression is:

[0141]

[0142] The work done by the impact force is:

[0143]

[0144] The kinetic energy of the sandwich hyperbolic plate is expressed as follows

[0145]

[0146] Substituting the above derivation into formula (17), setting δu, δv, δw = 0, and ignoring the in-plane displacements u and v, the nonlinear dynamic control equation of the sandwich hyperbolic plate in a thermal environment can be expressed as follows:

[0147]

[0148]

[0149]

[0150] Where, ρ k (k=1, 2, 3) represents the density of each material layer of the sandwich hyperbolic plate; (N x , N y , N xy ) and (M x , M y , M xy ) represent the in-plane force and bending moment respectively; P(x, y) represents the load amplitude according to the coordinate change; V load Indicates the speed of moving load; A load (x) represents the range of the mobile load action area; δ v and δ region are Dirac functions, taking values ​​of 1 or 0, which are used to constrain the moving trajectory and action range of the moving load, i.e., δ v =1 means the load moves to the corresponding coordinate, δ region =1 indicates that the load is within the corresponding coordinate area.

[0151] IP They represent the inertia of the sandwich hyperbolic plate, and are expressed as follows:

[0152]

[0153] Among them, ρ k (k=1, 2, 3) represents the density of each material layer of the sandwich hyperbolic plate, I P represent the mass density and inertia of the sandwich hyperbolic plate respectively;

[0154] The expressions for the stress and moment components are as follows:

[0155]

[0156]

[0157]

[0158]

[0159]

[0160] in

[0161]

[0162] The forces and moments associated with the thermal environment are expressed as follows

[0163]

[0164] The force and torque are expressed by the components related to the piezoelectric effect as follows

[0165]

[0166] Tensile stiffness coefficient A ij , bending coupling stiffness coefficient B ij , bending stiffness coefficient D ij and the higher-order stiffness coefficient E ij , F ij and H ij The definition is as follows:

[0167]

[0168]

[0169] Considering the functionally graded sandwich hyperbolic plate as a simply supported constraint, the boundary condition relationship is expressed as follows

[0170] Four-sided simply supported boundary conditions:

[0171]

[0172]

[0173] Step S3: Solve and analyze the mechanical problem based on the dynamic parameters of the fold geometry

[0174] In this example, circular and trapezoidal corrugation configurations are used to describe the wrinkle effect on a piezoelectric hybrid functionally graded sandwich hyperbolic plate. A hybrid distribution closer to reality is achieved by combining the circular and trapezoidal distributions. The corresponding structural distribution diagram is shown in Figure 2.

[0175] The two forms of circular distribution and trapezoidal distribution, corresponding to the tensile stiffness coefficient and bending stiffness coefficient can be expressed as follows:

[0176] (1) As shown in FIG2(a), the half-length of the broken line of the circular distribution is l = πR + 2L, and the half-period is c = 2R.

[0177]

[0178]

[0179]

[0180]

[0181]

[0182]

[0183]

[0184]

[0185] I1=πR

[0186]

[0187] (2) As shown in Figure 2(b), the half-length of the broken line of the trapezoidal corrugation is

[0188]

[0189]

[0190]

[0191]

[0192]

[0193]

[0194]

[0195]

[0196]

[0197]

[0198] According to the morphology of the piezoelectric hybrid functional gradient hyperbolic plate, the stiffness coefficient of the hyperbolic plate is selected differently at different spatial positions to construct and The morphological characteristics of quasi-hybrid wrinkles are characterized by the function of the coordinate value (x, y), and the mechanical information contained in the dynamic equilibrium equation of the hyperbolic plate model is further expanded to realize the solution and analysis of the mechanical problems corresponding to the geometric dynamic parameters of the wrinkles, and analyze the influence of the associated parameters on the dynamic performance.

[0199] Step S4, constructing the mathematical expression of the piezoelectric active control system

[0200] The collaborative deformation of the overall structure based on active control drive is introduced, that is, the hybrid functional gradient piezoelectric layer can not only deform synchronously with the structural body, but also drive its own deformation under the stimulation of external conditions and even guide the morphological changes of the structural body with the help of external degrees of freedom.

[0201] Active control and stiffness compensation are the integration of actuators, deformable structures, and external load bearers. The structure can not only bear loads, but also sense changes in the internal and external environment and respond by changing its own physical properties or shape. It has the advantages of fast response, adaptability, self-diagnosis, and self-repair. In response to the special functional requirements of adaptive performance enhancement and displacement compensation in engineering equipment design, the external loads in the real environment are complex, so nonlinear moving loads are introduced to analyze the nonlinear dynamic characteristics of a piezoelectric hybrid functional gradient sandwich hyperbolic plate with wrinkling effect under moving loads. This fills the gap in the corresponding technical field and obtains an efficient and accurate analytical solution to the corresponding problem, which is used for the quantitative and precise design of high-performance hybrid composite structures in complex working environments.

[0202] In order to construct a piezoelectric control system, the upper piezoelectric layer is used as the sensing layer, and the lower piezoelectric layer is used as the actuating layer. During the control process, the signal of the sensing layer comes from the deformation of the hybrid piezoelectric functional gradient material layer after being subjected to the moving load. The deformation generates a voltage = current signal. The signal collected by the sensing layer is first amplified by the amplifier and then transmitted to the controller. The negative velocity feedback control principle is adopted to convert the collected signal into a control voltage for the actuating layer, thereby realizing active control of the nonlinear vibration of the entire structure. The present invention aims to analyze the dynamic response method. Regarding the control system, it is to analyze the calculation method for design support. Conventional technicians in the field of communications can think of the design method of the control system based on the control content recorded in the present invention. This is relatively easy to implement. The design of specific electrical components and air systems is not within the scope of protection of the present invention, so it is not described in detail.

[0203] In this embodiment, assuming that the piezoelectric layer is polarized only along the thickness direction and no voltage is applied to the outer surface of the sensing layer, the electric displacement of the sensing layer can be expressed as:

[0204]

[0205] The amount of charge output by the sensing layer is

[0206]

[0207] Where A is the effective surface electrode and the current of the sensing layer is

[0208]

[0209] After the current amplifier gain G e and negative speed feedback control gain G v After the action, the feedback voltage applied to the lower surface of the actuating layer is:

[0210] V(t)=G v (G e i(t))=G*i(t) (34)

[0211] Where G is the gain coefficient. When the structure is actively controlled, the upper surface of the actuating layer is grounded.

[0212] Step S5: solving and analyzing the dynamic response based on the Galerkin method, the Newmark method and the iterative method.

[0213] In order to satisfy the control balance equation (21) and the boundary condition (28), the relevant displacement parameters can be taken to satisfy the function:

[0214]

[0215]

[0216]

[0217] in, is the unknown displacement amplitude, m and n are the half-wave numbers in the x and y directions, respectively.

[0218] The Galerkin method is used to calculate the displacement parameter Replace the displacement variables in the equilibrium equation (21).

[0219] In the time domain, the Newmark method is used to control the acceleration term w in the equations. ，tt , speed term w ，t , the displacement term w at time point t+Δt can be expressed as

[0220]

[0221] (w ，t ) J =(w ，t ) J-1 +[O2(w ，tt ) J +(1-O2)(w ，tt ) J-1 ](Δτt) (37b)

[0222]

[0223] Among them, variables O1 and O2 are the damping and stability parameters used to control the entire computing system. The quadratic extrapolation method is used to process the nonlinear terms of the dynamic control equations to determine the value of the initial trial iteration step. The nonlinear dynamic control equations about the time variable t are transformed into a linear equation of the corresponding variables and solved iteratively.

[0224] The influence of wrinkle morphology parameters on the dynamic response of a functionally graded sandwich hyperbolic plate under moving loads was considered. Based on the theoretical model of the wrinkled hyperbolic plate derived previously and the modified stiffness coefficients, a comprehensive analysis and evaluation of the correlation between the wrinkle height morphology parameters and the calculated results under various nonlinear moving loads was performed. Curves of the plate's deformation and stress over time were obtained, and the relevant interaction relationships are shown in Figure 3.

[0225] Figure 3(a) shows the range of the moving lateral load in an environment with a temperature change of 300K (0, a) / load intensity (P d (x) = 1000sin(πx / a) N / m 2) / movement speed (5 m / s) and an applied constant voltage of 200 V affect the nonlinear dynamic response of a piezoelectric hybrid functionally graded curved plate. A constant voltage of 200 V is applied to both the upper and lower surfaces of the piezoelectric hybrid functionally graded sandwich hyperbolic plate. The figure shows the dimensionless displacement and stress response curves at the (0.1a, 0.1b) / (0.2a, 0.2b) / (0.3a, 0.3b) / (0.4a, 0.4b) / (0.5a, 0.5b) coordinate sampling points of the hyperbolic plate under the same lateral load conditions.

[0226] The above examples are merely preferred embodiments of the present invention and are not intended to limit the present invention in any way. Although the present invention has been disclosed above with reference to preferred embodiments, they are not intended to limit the present invention. Therefore, any simple modifications, equivalent variations, and modifications to the above examples that do not depart from the technical solution of the present invention and are based on the technical essence of the present invention shall fall within the scope of protection of the technical solution of the present invention.

Claims

1. A dynamic response analysis method for a piezoelectric hybrid functionally graded sandwich hyperbolic plate, characterized in that: The dynamic response analysis method comprises the following steps: Step S1, constructing a mechanical model of a FGM sandwich hyperbolic plate with mixed morphology based on a corrugated structure; The sandwich hyperbolic plate has three layers, including a core layer and isotropic surface layers on both sides of the core layer. The core layer is a piezoelectric hybrid functional gradient material layer. The constitutive and geometric relationships of the piezoelectric hybrid functional gradient sandwich hyperbolic plate with wrinkle effect are constructed. Step S2, constructing the dynamic equilibrium equation of the corresponding problem; Step S3, solving and analyzing mechanical problems based on the dynamic parameters of the fold geometry; According to the morphology setting of the piezoelectric hybrid functionally graded hyperbolic plate, the stiffness coefficient of the hyperbolic plate is selected differently at different spatial positions, and a constructor is constructed to characterize the hybrid wrinkle morphology characteristics, thereby expanding the mechanical information contained in the dynamic equilibrium equation of the hyperbolic plate model. Step S4, constructing a mathematical expression of the piezoelectric active control system; Analyze the nonlinear dynamic characteristics of piezoelectric hybrid functionally graded sandwich hyperbolic plates with wrinkle effect under moving loads; Step S5, solving and analyzing the dynamic response based on the Galerkin method, the Newmark method and the iterative method; In step S1, the material property P of the core layer can be expressed as: Among them, P FM-1 and P FM-2 Respectively represent the material properties of the outer and inner layers of the sandwich layer; V FM-1 is the volume component of material type-1; The constitutive relation of the piezoelectric field is described as follows: {s} (k) =[C] (k) ({e} (k) -{a} (k) ΔT)-{e} (k) {E} (k) {D} (k) ={and} (k) {ε} (k) +{K} (k) {AND} (k) (k=0,N+1) Among them, {e} is the piezoelectric constant matrix of the material, {D} is the electric displacement matrix, {K} is the dielectric constant matrix, and {E} is the electric field intensity matrix; The expression of temperature field I(z) is: Among them, h p is the thickness of the hyperbolic plate; K k is the thermal conductivity coefficient of the kth layer material of the sandwich structure; i is the cumulative process variable, for example, when k = 3, K k represents the thermal conductivity of the third layer of material, then i=1,2,3; Relying on the mid-surface stress and strain components and bending strain components Get strain (ε x ,ε y ,ε xy ) is as follows: (-h / 2≤z≤h / 2) Among them, the mid-surface strain component It can be expressed by the mid-plane displacement (u0, v0, w0) as follows The expression for bending strain is: In step S2, the nonlinear dynamic control equation of the sandwich hyperbolic plate in the thermal environment is expressed as follows: Where I P Represents the inertia of the sandwich hyperbolic plate, which is expressed as follows Among them, ρ k (k=1,2,3) represents the density of each material layer of the sandwich hyperbolic plate; (N x ,N y ,N xy ) and (M x ,M y ,M xy ) represent the in-plane force and bending moment respectively; P(x,y) represents the load amplitude according to the coordinate change; V load Indicates the speed of moving load; A load (x) represents the range of the mobile load action area; δ v and δ region are Dirac functions, taking values ​​of 1 or 0, which are used to constrain the moving trajectory and action range of the moving load, i.e., δ v =1 means the load moves to the corresponding coordinate, δ region =1 indicates that the load is within the corresponding coordinate area.

2. The dynamic response analysis method of the piezoelectric hybrid functionally gradient sandwich hyperbolic plate according to claim 1 is characterized in that: In step S2, the expressions for stress and moment components are as follows: in The forces and moments associated with the thermal environment are expressed as follows The force and torque are expressed by the components related to the piezoelectric effect as follows 3. The dynamic response analysis method of the piezoelectric hybrid functionally gradient sandwich hyperbolic plate according to claim 1 is characterized in that: In step S3, the tensile stiffness coefficient and the bending stiffness coefficient of the hyperbolic plate are expressed as follows: Among them, A ij is the tensile stiffness coefficient, B ij is the bending coupling stiffness coefficient, D ij is the bending stiffness coefficient, E ij ,E ij and H ij and higher-order stiffness coefficients.

4. The dynamic response analysis method of the piezoelectric hybrid functionally gradient sandwich hyperbolic plate according to claim 1 is characterized in that: In step S4, assuming that the piezoelectric layer is polarized only along the thickness direction and no voltage is applied to the outer surface of the sensing layer, the electric displacement of the sensing layer can be expressed as: The amount of charge output by the sensing layer is: In the above formula, A e is the effective electrode on the surface, and the current of the sensor is expressed as After the current amplifier gain G e and negative speed feedback control gain G V After the action, the feedback voltage applied to the bottom of the actuating layer is V(t) = G V (G e i(t))=G*i(t) Where G is the gain coefficient.

5. The dynamic response analysis method of the piezoelectric hybrid functionally gradient sandwich hyperbolic plate according to claim 4 is characterized in that: In step S5, the control equilibrium equation and boundary conditions are satisfied, and the relevant displacement parameter function is: in, is the unknown displacement amplitude, m and n are the half-wave numbers in the x- and y-axis directions respectively; The displacement parameters are obtained using the Galerkin method. The system of dynamic differential governing equations with respect to the time variable t; In the time domain, the Newmark method is used to obtain the acceleration term w in the control equations ,tt , speed term w ,t , the expression of the displacement term w at time point t+Δt; described The nonlinear dynamic control equations about the time variable t are transformed into a linear equation of the corresponding variables and solved iteratively.

Images