A method and device for analyzing nonlinear vibration characteristics of three-dimensional serpentine structures

Abstract

The present application discloses a method and device for analyzing the nonlinear vibration characteristics of a three-dimensional serpentine structure, the method comprising: modeling a three-dimensional piezoelectric serpentine ductile structure as a beam to obtain a three-dimensional serpentine beam model; wherein the three-dimensional piezoelectric serpentine ductile structure and the three-dimensional serpentine beam model are formed by a plurality of L-shaped units that are sequentially flipped along the direction of one side thereof, and sequentially connected end to end along the direction of the other side thereof to form a bow-shaped structure; determining the kinetic energy and potential energy of the three-dimensional serpentine beam model; determining the control equation of the three-dimensional serpentine beam model based on the kinetic energy and potential energy; determining the dynamic response equation based on the control equation to determine the vibration characteristics of the three-dimensional serpentine beam model. The method solves the problem of the lack of research on the vibration characteristics of the three-dimensional piezoelectric serpentine ductile structure in the prior art, and can analyze the vibration characteristics of the three-dimensional piezoelectric serpentine ductile structure, providing theoretical guidance for the application of the three-dimensional piezoelectric serpentine ductile structure.

Description

Technical Field

[0001] The present application relates to the technical field of mechanics and vibration analysis, and in particular to a method and device for analyzing the nonlinear vibration characteristics of a three-dimensional serpentine structure. Background Art

[0002] Stretchable flexible electronic devices based on the principle of mechanical buckling can overcome the defects of traditional inorganic electronic devices that are not stretchable and can meet the application requirements of various complex curved surfaces. Among them, the serpentine structure is often used in energy, medical health, aerospace and other fields because of its excellent ductility.

[0003] Theoretical research on serpentine structures is usually based on their applications in stretchable sensors, wearable electronic devices, and flexible optical devices. The existing technology has proposed a theoretical model for serpentine interconnected structures that can produce large deformations. The model can predict a linear model with 50% stretchability, and characterize nonlinear effects through approximate models, providing theoretical guidance for the application of serpentine structures in flexible electronic devices. However, most of these theoretical and applied studies are based on two-dimensional planar serpentine structures, such as building serpentine interconnected sensor arrays and laminated structures based on skin surfaces. At present, more engineering problems are based on the application of three-dimensional structures on flexible surfaces, which requires solving the problem of transferring three-dimensional structures from planar substrates to flexible substrates. The emergence of three-dimensional assembly technology has solved these problems well. It realizes the transformation of electronic device structures from two-dimensional planar structures to three-dimensional structures, which can not only meet the high storage ratio and application needs of thin-film spacecraft, but also meet the mechanical flexibility design requirements of thin-film spacecraft. The three-dimensional assembly technology based on compression buckling control proposed previously, this design method can not only give play to the excellent mechanical properties of extensible structures, but also realize reverse design and flexible regulation. The controllable three-dimensional extensible serpentine structure formed by this technology has new applications compared to the two-dimensional serpentine structure. However, there is a lack of systematic theoretical research on three-dimensional piezoelectric serpentine stretchable structures. Summary of the invention

[0004] The embodiments of the present application solve the problem of the lack of research on the vibration characteristics of three-dimensional piezoelectric serpentine extendable structures in the prior art by providing a method and device for analyzing the nonlinear vibration characteristics of a three-dimensional serpentine structure.

[0005] In a first aspect, an embodiment of the present application provides a method for analyzing the nonlinear vibration characteristics of a three-dimensional serpentine structure, comprising: modeling a three-dimensional piezoelectric serpentine stretchable structure as a beam to obtain a three-dimensional serpentine beam model; wherein the three-dimensional piezoelectric serpentine stretchable structure and the three-dimensional serpentine beam model are composed of a plurality of L-shaped units that are flipped in sequence along the direction of one side thereof, and are connected end to end in sequence along the direction of the other side thereof to form a bow-shaped structure; determining the kinetic energy and potential energy of the three-dimensional serpentine beam model; determining the control equation of the three-dimensional serpentine beam model based on the kinetic energy and the potential energy; and determining the dynamic response equation based on the control equation to determine the vibration characteristics of the three-dimensional serpentine beam model.

[0006] In combination with the first aspect, in a possible implementation method, before modeling the three-dimensional piezoelectric serpentine stretchable structure as a beam, it also includes: pre-stretching the flexible substrate, and adhering the piezoelectric serpentine structure to the pre-stretched flexible substrate to fix its two ends; releasing the pre-stretching applied to the flexible substrate so that the piezoelectric serpentine structure is compressed and buckled to form a curved three-dimensional piezoelectric serpentine stretchable structure.

[0007] In combination with the first aspect, in a possible implementation method, determining the kinetic energy of the three-dimensional serpentine beam model includes: obtaining the vibration mode of the three-dimensional serpentine beam model in the first-order buckling mode through finite element simulation; determining the displacement function of the L-shaped unit in the three-dimensional serpentine beam model based on the vibration mode; determining the displacement mode of the three-dimensional serpentine beam model in the first-order buckling mode; defining the theoretical deflection of the three-dimensional serpentine beam model in the first-order buckling mode; determining the deflection of the three-dimensional serpentine beam model according to the displacement mode of the first-order buckling mode, the displacement function of the L-shaped unit and the theoretical deflection; and determining the kinetic energy of the three-dimensional serpentine beam model based on the deflection of the three-dimensional serpentine beam model.

[0008] In combination with the first aspect, in a possible implementation manner, the deflection of the three-dimensional serpentine beam model is as follows:

[0009] ,

[0010] ;

[0011] In the formula, It means that the i-th L-type unit is in the global coordinate at time t The displacement function of the direction, It means that the i-th L-type unit is in the global coordinate at time t The displacement function of the direction, represents the length of the piezoelectric serpentine stretchable structure before releasing the pre-stretching, represents the global coordinates of the i-th L-type unit, represents the length of the three-dimensional serpentine beam model after the pre-stretching is released and buckling occurs, represents the modal coordinates of the three-dimensional serpentine beam model at time t, Indicates time, Indicates the length of the long side of the L-shaped unit.

[0012] In combination with the first aspect, in a possible implementation manner, the control equation is as follows:

[0013] ;

[0014] In the formula, represents the modal coordinates of the three-dimensional serpentine beam model at time t, represents the second-order derivative of the modal coordinates of the three-dimensional serpentine beam model with respect to time t, represents external incentives, Indicates time, , , , and Represents a dimensionless parameter.

[0015] In combination with the first aspect, in a possible implementation manner, the dynamic response equation is as follows:

[0016] ;

[0017] In the formula, represents the dimensionless modal coordinate amplitude, represents the dimensionless external excitation amplitude, represents dimensionless time, represents the dimensionless external excitation frequency, , , and represents a dimensionless parameter, Indicates time, Represents the thickness of the 3D serpentine beam model.

[0018] In combination with the first aspect, in a possible implementation manner, the external stimulus is as follows:

[0019] ;

[0020] In the formula, represents external incentives, Indicates time, represents the amplitude of the external excitation, represents the frequency of external excitation.

[0021] In the second aspect, an embodiment of the present application provides a three-dimensional serpentine structure nonlinear vibration characteristic analysis device, including: a modeling module, used to model the three-dimensional piezoelectric serpentine stretchable structure as a beam to obtain a three-dimensional serpentine beam model; wherein the three-dimensional piezoelectric serpentine stretchable structure and the three-dimensional serpentine beam model are composed of multiple L-shaped units that are flipped in sequence along the direction of one side and are connected end to end in sequence along the direction of the other side to form a bow-shaped structure; a first determination module, used to determine the kinetic energy and potential energy of the three-dimensional serpentine beam model; a second determination module, used to determine the control equation of the three-dimensional serpentine beam model based on the kinetic energy and the potential energy; a vibration characteristic analysis module, used to determine the dynamic response equation based on the control equation to determine the vibration characteristics of the three-dimensional serpentine beam model.

[0022] In a third aspect, an embodiment of the present application provides a device, comprising: a processor; a memory for storing processor-executable instructions; when the processor executes the executable instructions, it implements the method described in the first aspect or any possible implementation method of the first aspect.

[0023] In a fourth aspect, an embodiment of the present application provides a non-volatile computer-readable storage medium, wherein the non-volatile computer-readable storage medium includes a medium for storing a computer program or instructions, which, when executed, enables the method described in the first aspect or any possible implementation method of the first aspect to be implemented.

[0024] One or more technical solutions provided in the embodiments of the present application have at least the following technical effects or advantages:

[0025] By constructing a three-dimensional serpentine beam model, the embodiment of the present application can intuitively understand the stress and deformation of the material; by constructing a control equation, the performance of the material can be described in a mathematical way, which is convenient for solving and simulating using a computer; through the dynamic response equation, the vibration characteristics of the three-dimensional serpentine beam model can be determined. This effectively solves the problem of the lack of research on the vibration characteristics of three-dimensional piezoelectric serpentine ductile structures in the prior art, and can further analyze the vibration characteristics of three-dimensional piezoelectric serpentine ductile structures, providing theoretical guidance for the application of three-dimensional piezoelectric serpentine ductile structures. BRIEF DESCRIPTION OF THE DRAWINGS

[0026] In order to more clearly illustrate the technical solutions of the embodiments of the present application, the drawings required for use in the embodiments of the present application or the description of the prior art will be briefly introduced below. Obviously, the drawings described below are some embodiments of the present invention. For ordinary technicians in this field, other drawings can be obtained based on these drawings without paying any creative work.

[0027] Figure 1A flowchart of a method for analyzing nonlinear vibration characteristics of a three-dimensional serpentine structure provided in an embodiment of the present application;

[0028] Figure 2 A schematic diagram of the structure of a three-dimensional serpentine structure nonlinear vibration characteristic analysis device provided in an embodiment of the present application;

[0029] Figure 3 A schematic diagram of the preparation process of the three-dimensional piezoelectric serpentine extensible structure provided in an embodiment of the present application;

[0030] Figure 4 A schematic diagram of a piezoelectric serpentine structure provided in an embodiment of the present application;

[0031] Figure 5 A schematic diagram of an L-shaped unit provided in an embodiment of the present application;

[0032] Figure 6 A schematic diagram of a three-dimensional serpentine beam model provided in an embodiment of the present application;

[0033] Figure 7 A comparison curve diagram of the vibration mode obtained in the present application and the finite element result when the strain provided in the embodiment of the present application is 0.1;

[0034] Figure 8 A comparison curve diagram of the vibration mode obtained in the present application and the finite element result when the strain is 0.2 provided in the embodiment of the present application;

[0035] Fig. 9 A comparison curve diagram of the vibration mode obtained in the present application and the finite element result when the strain is 0.3 provided in the embodiment of the present application;

[0036] Fig.10 A three-dimensional schematic diagram of the relationship between voltage and potential energy provided in an embodiment of the present application;

[0037] Fig.11 Provided in the embodiments of this application A graph showing the relationship between potential energy and amplitude with a value of 0.05;

[0038] Fig.12 Provided in the embodiments of this application A graph showing the relationship between potential energy and amplitude with a value of 0.1;

[0039] Fig.13 Provided in the embodiments of this application A graph showing the relationship between potential energy and amplitude for a value of 0.2;

[0040] Fig.14 Provided in the embodiments of this application Typical phase plane plot of rate of change of amplitude versus amplitude for a value of 0.05;

[0041] Fig.15 Provided in the embodiments of this application A typical phase plane plot of the rate of change of amplitude versus amplitude for a value of 0.1;

[0042] Fig.16 Provided in the embodiments of this application Typical phase plane plot of rate of change of amplitude versus amplitude for a value of 0.2;

[0043] Fig.17 A periodic vibration curve diagram with an initial condition of 1 provided in an embodiment of the present application;

[0044] Fig.18 A periodic vibration curve diagram with an initial condition of 35 provided in the embodiment of the present application;

[0045] Fig.19 A chaotic vibration curve diagram with an initial condition of 10 provided in an embodiment of the present application;

[0046] Fig. 20 Trajectory diagrams on the phase plane for different initial conditions provided in the embodiments of the present application;

[0047] Fig.21 Corresponding trajectory diagrams of Poincare mappings with different initial conditions provided in the embodiments of the present application;

[0048] Fig. 22 A chaos effect curve diagram showing the sensitivity of initial conditions provided in the embodiments of the present application;

[0049] Fig.23 A graph showing the change of the Lyapunov exponent over time provided in an embodiment of the present application;

[0050] Fig.24 Provided in the embodiments of this application , , Typical phase plane diagram of rate of change of amplitude versus amplitude;

[0051] Fig.25 Provided in the embodiments of this application , , Typical phase plane diagram of rate of change of amplitude versus amplitude;

[0052] Fig.26 Provided for the embodiments of this application , , Typical phase plane diagram of rate of change of amplitude versus amplitude;

[0053] Fig. 27 Provided for the embodiments of this application , , Typical phase plane diagram of rate of change of amplitude versus amplitude;

[0054] Fig.28 A diagram showing the dependence of the chaos degree of the monostable potential provided in an embodiment of the present application on the external excitation amplitude and the external excitation frequency;

[0055] Fig.29 A diagram showing the dependence of the chaos degree of the bistable potential provided in an embodiment of the present application on the external excitation amplitude and the external excitation frequency. DETAILED DESCRIPTION

[0056] The technical solutions in the embodiments of the present application will be described clearly and completely below in conjunction with the drawings in the embodiments of the present application. Obviously, the described embodiments are part of the embodiments of the present invention, not all of the embodiments. Based on the embodiments of the present invention, all other embodiments obtained by ordinary technicians in this field without creative work are within the scope of protection of the present invention.

[0057] The following describes some of the techniques involved in the embodiments of the present application to facilitate understanding, and they should be considered as merely exemplary. Therefore, it should be appreciated by those of ordinary skill in the art that various changes and modifications may be made to the embodiments described herein without departing from the scope and spirit of the present application. Similarly, for the sake of clarity and conciseness, some descriptions of well-known functions and structures are omitted in the following description.

[0058] The present application is directed to a three-dimensional piezoelectric serpentine ductile structure formed by a piezoelectric material PZT-4 on an elastic substrate. PZT-4 is a type of lead zirconate titanate piezoelectric ceramic, which is a material made by replacing part of the lead with calcium, strontium or barium, and replacing zirconium with tin. PZT is a composition based on a solid solution of lead titanate and lead zirconate, and PZT-4 belongs to its hard material class.

[0059] Figure 1 1 is a flowchart of a method for analyzing nonlinear vibration characteristics of a three-dimensional serpentine structure provided in an embodiment of the present application, including steps 101 to 104. Figure 1 This is only an execution order shown in the embodiment of the present application, and does not represent the only execution order of a three-dimensional serpentine structure nonlinear vibration characteristic analysis method. In the case that the final result can be achieved, Figure 1 The steps shown may be performed in parallel or reversed.

[0060] Step 101: Model the three-dimensional piezoelectric serpentine ductile structure as a beam to obtain a three-dimensional serpentine beam model. Figure 3As shown, a schematic diagram of the preparation process of the three-dimensional piezoelectric serpentine ductile structure is given. represents the length of the three-dimensional serpentine beam model after the pre-tension is released and buckling occurs. Represents the static amplitude. First, the flexible substrate is pre-stretched, and then the piezoelectric serpentine structure is adhered to the pre-stretched flexible substrate, and the two ends of the piezoelectric serpentine structure are fixed to the flexible substrate. Finally, the pre-stretching applied to the flexible substrate is released, and the pre-stretching strain causes the piezoelectric serpentine structure to be compressed, and the compression produces buckling to form a curved three-dimensional piezoelectric serpentine extensible structure.

[0061] Based on the engineering beam theory (also known as the classical beam theory or the Euler-Bernoulli beam theory), the three-dimensional piezoelectric serpentine extendable structure is modeled as a beam, and a three-dimensional serpentine beam model is obtained, such as Figure 6 As shown. The three-dimensional piezoelectric serpentine extendable structure and the three-dimensional serpentine beam model in the present application are composed of a plurality of L-shaped units which are sequentially flipped along the direction of one side and sequentially connected end to end along the direction of the other side to form a bow-shaped structure. Figure 4 or Figure 6 As shown, the L-shaped units are flipped in sequence along the direction of their long sides, and are connected end to end in sequence along the direction of their short sides to form a bow-shaped structure. In the formed bow-shaped structure, the shapes of the odd-numbered L-shaped units are consistent, and the shapes of the even-numbered L-shaped units are consistent.

[0062] It should be noted that Figure 3-Figure 6 The shape of the L-shaped unit is only one embodiment of the present application. The present application does not distinguish between the long side and the short side by the length of the two sides of the L-shaped unit. Even if the lengths of the long side and the short side are interchanged, the three-dimensional piezoelectric serpentine extensible structure only changes in density, and the scheme of the present application is still applicable. The present application refers to the side of the L-shaped unit that is consistent with the length direction of the three-dimensional piezoelectric serpentine extensible structure as the long side, and the side perpendicular to the length direction of the three-dimensional piezoelectric serpentine extensible structure as the short side.

[0063] This application uses a Cartesian coordinate system, where the global coordinates are selected To describe the deformation of the entire three-dimensional piezoelectric serpentine stretchable structure, choose the local coordinates To describe the deformation in the L-type unit. Figure 4 As shown in FIG. , the origin O of the global coordinates of the three-dimensional piezoelectric serpentine extendable structure is located at its center. Figure 5 As shown, the coordinate origin o of the local coordinates of the L-shaped unit is located at the center of the long side of the L-shaped unit.

[0064] Step 102: Determine the kinetic energy and potential energy of the three-dimensional serpentine beam model. In the embodiment of the present application, in order to study the dynamic behavior of the three-dimensional piezoelectric serpentine stretchable structure in the flexible electronic device, it is first necessary to establish a theoretical model (three-dimensional serpentine beam model) and control equations of the three-dimensional piezoelectric serpentine stretchable structure.

[0065] In an embodiment of the present application, determining the kinetic energy of a three-dimensional serpentine beam model includes: obtaining the vibration mode of the three-dimensional serpentine beam model in a first-order buckling mode through finite element simulation. Determining the displacement function of the L-shaped unit in the three-dimensional serpentine beam model based on the vibration mode. Determining the displacement mode of the three-dimensional serpentine beam model in the first-order buckling mode. Defining the theoretical deflection of the three-dimensional serpentine beam model in the first-order buckling mode. Determining the deflection of the three-dimensional serpentine beam model based on the displacement mode of the first-order buckling mode, the displacement function of the L-shaped unit and the theoretical deflection. Determining the kinetic energy of the three-dimensional serpentine beam model based on the deflection of the three-dimensional serpentine beam model.

[0066] Specifically, the deformation of the three-dimensional serpentine beam model is described by the limited deformation beam theory. Assuming that the thickness of the three-dimensional serpentine beam model is is much smaller than other geometric parameters, i.e. (h represents the thickness of the three-dimensional serpentine beam model or L-shaped element, Represents the length of the long side of the L-shaped unit). The 3D serpentine beam model eliminates large deformations, and the membrane strain in the 3D serpentine beam model needs to consider the geometry. The 3D serpentine beam model causes geometric nonlinear vibrations due to large displacements (even if the strain is small), causing the equilibrium equations and geometric equations of the 3D serpentine beam model to become nonlinear. Therefore, according to the von Karman nonlinear beam theory, the strain in the X direction of the 3D serpentine beam model in the global coordinates is as follows:

[0067] .

[0068] In the formula, represents the strain in the X direction of the three-dimensional serpentine beam model in the global coordinate system. Represents the displacement function of the three-dimensional serpentine beam model in the Z direction of the global coordinate, It represents the displacement function of the three-dimensional serpentine beam model in the X direction of the global coordinate. represents the global coordinates of the i-th L-type unit, z represents the local coordinates, Indicates time.

[0069] In the embodiment of the present application, the constitutive relationship of the three-dimensional serpentine beam model is defined as follows:

[0070] ,

[0071] .

[0072] In the formula, It represents the membrane stress in the X direction of the three-dimensional serpentine beam model in the global coordinate system. represents the strain in the X direction of the three-dimensional serpentine beam model in the global coordinate system. Represents the three-dimensional serpentine beam model in global coordinates The electric displacement in the direction represents the elastic constant, and represents the piezoelectric coupling constant and dielectric constant of the three-dimensional serpentine beam model, Indicates the voltage value in the Z direction under the global coordinates, represents the electric potential and Z represents the global coordinate.

[0073] In the embodiment of the present application, the electric field conditions are as follows:

[0074] .

[0075] In the formula, represents the electric potential on the lower surface of the three-dimensional serpentine beam model, represents the electric potential on the upper surface of the three-dimensional serpentine beam model, represents the thickness of the 3D serpentine beam model, Represents the voltage applied between the upper and lower surfaces of the 3D serpentine beam model.

[0076] According to Gauss's law , the voltage value in the Z direction can be expressed as follows:

[0077] .

[0078] In the formula, Indicates the voltage value in the Z direction under the global coordinates, Z represents the global coordinates, and represents the piezoelectric coupling constant and dielectric constant of the three-dimensional serpentine beam model, represents the displacement function of the i-th L-type unit in the Z direction in the global coordinate at time t, represents the global coordinates of the i-th L-type unit, represents the thickness of the 3D serpentine beam model, represents the voltage applied between the upper and lower surfaces of the three-dimensional serpentine beam model. Indicates time.

[0079] For the three-dimensional serpentine beam model, according to its first-order buckling mode, the theoretical deflection of the buckled structure is assumed as follows:

[0080] ,

[0081] .

[0082] Among them, the displacement mode of the first-order buckling mode of the three-dimensional serpentine beam model refers to the displacement distribution state of the three-dimensional serpentine beam model when it reaches the critical point of buckling when subjected to external load. This application includes the displacement function in the X direction under the global coordinates and the displacement function in the Z direction of the first-order buckling mode under the global coordinates, which are described as follows:

[0083] ,

[0084] .

[0085] In order to separate the time and space variables, Galerkin discretization is used and and They are defined as follows:

[0086] ,

[0087] .

[0088] In the formula, represents the displacement function of the i-th L-type unit in the Z direction in the global coordinate at time t, represents the displacement function of the ith L-type unit in the X direction in the global coordinate at time t, represents the displacement function of the first-order buckling mode in the Z direction in the global coordinates, represents the displacement function of the first-order buckling mode in the X direction in global coordinates, represents the displacement function of the i-th L-type unit in the Z direction in the global coordinate system as time t changes, represents the displacement function of the ith L-type unit in the X direction in the global coordinate system as time t changes. represents the global coordinates of the i-th L-type unit, Indicates time, represents the length of the piezoelectric serpentine stretchable structure before releasing the pre-stretching, represents the length of the three-dimensional serpentine beam model after the pre-tension is released and buckling occurs. represents the modal coordinates of the three-dimensional serpentine beam model at time t, and Represents the first in-plane and out-of-plane vibration modes of a 3D serpentine beam model.

[0089] Among them, the first-order in-plane vibration mode refers to the form in which the three-dimensional serpentine beam model vibrates at the lowest non-zero frequency in the plane (such as the XY plane). In this mode, the vibration of the three-dimensional serpentine beam model mainly occurs in the plane, and the displacement perpendicular to the plane (such as the displacement in the Z direction) is relatively small or negligible. The first-order out-of-plane vibration mode refers to the form in which the three-dimensional serpentine beam model vibrates at the lowest non-zero frequency perpendicular to a plane (for example: the Z direction perpendicular to the XY plane). In this mode, the vibration of the three-dimensional serpentine beam model mainly occurs in the direction perpendicular to the plane.

[0090] Specifically, the first-order in-plane vibration mode and the first out-of-plane vibration mode It can be obtained through finite element simulation, and the vibration modes of the first-order in-plane and the first-order out-of-plane are as follows:

[0091] ,

[0092] .

[0093] Deflection is a description of the degree of deformation of a three-dimensional serpentine beam model after being subjected to force, especially the linear displacement perpendicular to the force direction. It is an important indicator for measuring the mechanical properties of piezoelectric materials. This application obtains the deflection of a three-dimensional serpentine beam model by substituting the first-order initial displacement mode and the vibration mode into the theoretical deflection, as follows:

[0094] ,

[0095] .

[0096] In the formula, and denote the first-order in-plane and out-of-plane vibration modes, respectively. represents the displacement function of the ith L-type unit in the X direction in the global coordinate at time t, represents the displacement function of the i-th L-type unit in the Z direction in the global coordinate at time t, represents the length of the piezoelectric serpentine stretchable structure before releasing the pre-stretching, represents the global coordinates of the i-th L-type unit, represents the length of the three-dimensional serpentine beam model after the pre-tension is released and buckling occurs. represents the modal coordinates of the three-dimensional serpentine beam model at time t, Indicates time, Indicates the length of the long side of the L-shaped unit.

[0097] In piezoelectric materials, electric field and mechanical stress interact with each other, and this interaction will cause the deformation of piezoelectric materials and changes in electric field distribution. In order to analyze the vibration characteristics of the three-dimensional serpentine beam model under periodic external excitation, it is necessary to derive the control equation of the three-dimensional serpentine beam model under periodic vibration loads, which is specifically expressed as follows:

[0098] .

[0099] in, ,

[0100] .

[0101] In the formula, and represent the kinetic energy and potential energy of the three-dimensional serpentine beam model, respectively. represents the modal coordinates of the three-dimensional serpentine beam model, represents external incentives, represents the material density of the 3D serpentine beam model, Indicates time, It represents the line width of the 3D serpentine beam model, that is, the width of the lines that make up the L-shaped unit. represents the length of the long side of the L-shaped unit, represents the thickness of the 3D serpentine beam model, represents the length of the piezoelectric serpentine stretchable structure before releasing the pre-stretching, represents the length of the three-dimensional serpentine beam model after the pre-tension is released and buckling occurs. Represents the three-dimensional serpentine beam model in global coordinates The electric displacement in the direction represents the displacement function of the ith L-type unit in the X direction in the global coordinate at time t, represents the displacement function of the i-th L-type unit in the Z direction in the global coordinate at time t, represents the global coordinates of the i-th L-type unit, It represents the membrane stress in the X direction of the three-dimensional serpentine beam model in the global coordinate system. represents the strain in the X direction of the three-dimensional serpentine beam model in the global coordinate system. represents the global coordinates, represents the global coordinates, Indicates the voltage value in the Z direction.

[0102] Step 103: Determine the control equation of the three-dimensional serpentine beam model according to the kinetic energy and potential energy. In the embodiment of the present application, the kinetic energy and potential energy of the three-dimensional serpentine beam model are substituted into the Lagrangian equation of motion to obtain the control equation, as follows:

[0103] .

[0104] In the formula, represents the modal coordinates of the three-dimensional serpentine beam model at time t, represents the second-order derivative of the modal coordinates of the three-dimensional serpentine beam model with respect to time t, represents external incentives, Indicates time, , , , and Represents a dimensionless parameter, as follows:

[0105] ,

[0106] ,

[0107] ,

[0108] ,

[0109] .

[0110] In the formula, represents the material density of the 3D serpentine beam model, represents the line width of the 3D serpentine beam model, represents the length of the long side of the L-shaped unit, represents the thickness of the 3D serpentine beam model, represents the length of the piezoelectric serpentine stretchable structure before releasing the pre-stretching, represents the length of the three-dimensional serpentine beam model after the pre-tension is released and buckling occurs. represents the elastic constant, and Represents the piezoelectric coupling constant and dielectric constant for the 3D serpentine beam model.

[0111] ,

[0112] .

[0113] in, represents the dimensionless modal coordinate amplitude, represents the dimensionless external excitation amplitude, represents dimensionless time, represents the dimensionless external excitation frequency.

[0114] Step 104: Determine the dynamic response equation based on the control equation to determine the vibration characteristics of the three-dimensional serpentine beam model. In the embodiment of the present application, the dimensionless parameters are substituted into the control equation to obtain the dynamic response equation, as follows:

[0115] .

[0116] In the formula, represents the dimensionless modal coordinate amplitude, represents the dimensionless external excitation amplitude, represents dimensionless time, represents the dimensionless external excitation frequency, , , and represents a dimensionless parameter, Indicates time, Represents the thickness of the 3D serpentine beam model.

[0117] Based on the above dynamic response equation, the three-dimensional serpentine beam model is subjected to periodic external excitation. When , the corresponding parameters of the three-dimensional serpentine beam model can be substituted, and the three-dimensional serpentine beam model under external excitation can be obtained through numerical solution. Vibration behavior under action.

[0118] The present application has carried out numerical simulation to study the nonlinear dynamic behavior of a three-dimensional serpentine beam model under periodic excitation. The method adopted in the present application is described in detail below with an actual example.

[0119] In order to verify the validity of the modal basis function in the global coordinate system, Figure 7 , Figure 8 and Fig. 9 The vibration modes obtained in this application and the finite element results are compared when the strain is 0.1, 0.2 and 0.3. represents the normalized displacement in the global coordinate Z direction, represents the normalized displacement in the global coordinate X direction, represents the global coordinates, represents the length of the piezoelectric serpentine stretchable structure before releasing the pre-stretching. The finite element results of the three-dimensional serpentine beam model are taken as the midpoint of the long side of each L-shaped unit and normalized. In the figure, the dotted line (blue) and the solid line (red) represent the global coordinates. Direction and global coordinates The results show that the three-dimensional serpentine beam model has good agreement with the finite element results when the strains are 0.1, 0.2 and 0.3, respectively, which proves that the vibration modes obtained in this application can well describe the vibration characteristics of the three-dimensional serpentine beam model in the global coordinate system.

[0120] When the three-dimensional serpentine beam model is not affected by external excitation, the system is a conservative system, and the external load amplitude is zero. In order to study the free vibration characteristics of the three-dimensional serpentine beam model, Fig.10 A three-dimensional schematic diagram of the relationship between voltage and potential energy of the three-dimensional serpentine beam model is drawn when the compressive strain is zero. represents the dimensionless modal coordinate amplitude, It represents the voltage applied between the upper and lower surfaces of the three-dimensional serpentine beam model. The results show that the increase of the voltage between the upper and lower surfaces of the three-dimensional serpentine beam model will lead to the generation of potential wells.

[0121] The shape of the potential energy has a strong influence on the dynamics. Typical shapes of potential energy are Fig.11 As shown, The value is 0.05. The potential well has a value of 0.05, which exhibits monostable behavior with one equilibrium point represented by a circle. Fig.12 and Fig.13 As shown, for For potential wells with values ​​of 0.1 and 0.2, the potential well exhibits bistable behavior, with the two equilibrium points represented by the two circles.

[0122] The phase structure can be revealed by using different numbers of initial conditions. For example, for monostable behavior ( value is 0.05), in Fig.14 Typical periodic trajectories are found around a single equilibrium point indicated by the “+” symbol in . Fig.15 They are shown separately Typical phase structure in the bistable case with a value of 0.1, where the two equilibrium points are represented by “+” and “★”, respectively. Fig.16 Shows Typical phase structure of the bistable case with a value of 0.2, where the two equilibrium points are represented by "+" and "★", and one of the stable potential wells ("+") becomes very small. represents the length of the long side of the L-shaped unit, represents the dimensionless modal coordinate amplitude, represents the length of the piezoelectric serpentine stretchable structure before releasing the pre-stretching, Indicates time.

[0123] Under the action of external forces, as the structure or external forces change, a variety of nonlinear dynamic behaviors will appear, such as periodic vibrations and even chaotic vibrations. Fig.17 Some typical dynamic behaviors of a bistable potential well are shown. Fig.17 and Fig.18 The initial conditions are shown respectively and initial conditions However, the initial condition Chaotic irregular vibrations may also occur, such as Fig.19 shown. Fig. 20 Three trajectories on the phase plane are shown, Fig.21 The corresponding trajectories of the Poincare map are shown, where the two periodic trajectories are represented by a finite number of points and the chaotic trajectory is represented by a wide region on the phase plane.

[0124] Here, the chaotic irregular vibration can be verified by its sensitivity to the initial conditions. Fig. 22 As shown in Figure 1, the two slightly different trajectories with initial conditions 10 and 10.01 first approach each other and soon become very different. In addition, when the time becomes larger, the Lyapunov exponent can approach the maximum Lyapunov exponent. Fig.23 Results for the Lyapunov exponent are shown, indicating that the largest Lyapunov exponent is about 0.1.

[0125] To further reveal the structure of various dynamics under different parameters. This application uses 20 uniformly selected initial conditions to reveal a specific structure with given parameters. For each initial condition, a relatively long trajectory is obtained, for example by using a longer duration . Thus, on the Poincare map, 5000 points are obtained for each trajectory. Then, all trajectories are displayed on the phase plane of the Poincare map. Fig.24 and Fig.25 The results of bistable potential are shown. Obviously, if Fig.24 As shown in Figure 2, chaotic vibrations appear near the separation line and when the strength of the force increases, it Fig.25 The area in will expand. For a monostable potential, the result is Fig.26 and 27 As shown in Figure 2, the emergence of chaos requires a larger force than that of bistable potential. When the strength of the force increases, the chaotic region Fig. 27 As shown in the figure, represents the dimensionless external excitation frequency, represents the length of the long side of the L-shaped unit, represents the dimensionless external excitation amplitude, represents the dimensionless modal coordinate amplitude, Indicates time.

[0126] When the value of the initial condition is relatively large, the resulting trajectory is periodic for both the bistable and monostable. This can be understood as follows: the resultant force is proportional to the slope of the potential energy. For large values ​​of the initial condition, the potential energy increases rapidly, and the resultant force may become greater than the external force and then become the dominant force.

[0127] There are various parameters that influence the dynamics. First, the nature of the external forces (including strength and frequency) has a large influence. Second, structural parameters also affect the dynamics, since parameters Directly present in the potential energy, this will have a strong effect.

[0128] This application first studies the effect of external forcing on the dynamics by means of 100 random initial conditions. For each trajectory, the final periodic or chaotic state is identified numerically. For example, for periodic vibrations, a state can repeat itself. Numerically, if a state can be repeated at least 5 times, then the resulting trajectory is periodic. Otherwise, the trajectory will be chaotic. In order to make the results robust, long trajectories with Poincare points are used. The chaos scores of these different trajectories are then obtained. This quantifies the prevalence of chaotic or periodic vibrations in the phase space under given parameters. The dependence of the degree of chaos on the amplitude of the external excitation and the frequency of the external excitation is shown in Fig.28 as well as Fig.29 As shown, the monostable potential is Fig.28 As shown, the bistable Fig.29 As shown, bistable potential is more prone to chaos.

[0129] Although the present application provides method operation steps as described in the embodiments or flowcharts, more or fewer operation steps may be included based on conventional or non-creative labor. The order of steps listed in this embodiment is only one way of executing the order of many steps and does not represent the only execution order. When the actual device or client product is executed, it can be executed in sequence or in parallel according to the method shown in this embodiment or the accompanying drawings (for example, in a parallel processor or multi-threaded processing environment).

[0130] like Figure 2 As shown, the embodiment of the present application also provides a three-dimensional serpentine structure nonlinear vibration characteristic analysis device 200. The device includes: a modeling module 201, a first determination module 202, a second determination module 203 and a vibration characteristic analysis module 204, as follows.

[0131] The modeling module 201 is used to model the three-dimensional piezoelectric serpentine ductile structure as a beam to obtain a three-dimensional serpentine beam model. The three-dimensional piezoelectric serpentine ductile structure and the three-dimensional serpentine beam model are formed by a plurality of L-shaped units being flipped in sequence along the direction of one side thereof and connected end to end in sequence along the direction of the other side thereof to form a bow-shaped structure.

[0132] The first determination module 202 is used to determine the kinetic energy and potential energy of the three-dimensional serpentine beam model.

[0133] The second determination module 203 is used to determine the control equation of the three-dimensional serpentine beam model according to the kinetic energy and the potential energy.

[0134] The vibration characteristic analysis module 204 is used to determine the dynamic response equation based on the control equation to determine the vibration characteristics of the three-dimensional serpentine beam model.

[0135] Some modules in the apparatus described in the present application can be described in the general context of computer executable instructions executed by a computer, such as program modules. Generally, program modules include routines, programs, objects, components, data structures, classes, etc. that perform specific tasks or implement specific abstract data types. The present application can also be practiced in distributed computing environments, in which tasks are performed by remote processing devices connected through a communication network. In a distributed computing environment, program modules can be located in local and remote computer storage media including storage devices.

[0136] The devices or modules described in the above application embodiments can be implemented by computer chips or entities, or by products with certain functions. For the convenience of description, the above devices are described in various modules according to their functions. When implementing the embodiments of the present application, the functions of each module can be implemented in the same or multiple software and / or hardware. Of course, the module that implements a certain function can also be implemented by combining multiple sub-modules or sub-units.

[0137] The methods, devices or modules described in this application can be implemented in the form of computer-readable program codes. The controller can be implemented in any appropriate manner. For example, the controller can take the form of a microprocessor or processor and a computer-readable medium storing computer-readable program codes (such as software or firmware) that can be executed by the (micro)processor, logic gates, switches, application-specific integrated circuits (English: Application Specific Integrated Circuit; Abbreviation: ASIC), programmable logic controllers and embedded microcontrollers. Examples of controllers include but are not limited to the following microcontrollers: ARC 625D, Atmel AT91SAM, Microchip PIC18F26K20 and Silicone Labs C8051F320. The memory controller can also be implemented as part of the control logic of the memory. Those skilled in the art also know that in addition to implementing the controller in the form of pure computer-readable program codes, the controller can be implemented in the form of logic gates, switches, application-specific integrated circuits, programmable logic controllers and embedded microcontrollers by logically programming the method steps. Therefore, this controller can be considered as a hardware component, and the devices included in it for implementing various functions can also be regarded as structures within the hardware component. Or even, the means for realizing various functions may be regarded as both a software module for realizing the method and a structure within a hardware component.

[0138] An embodiment of the present application further provides a device, comprising: a processor; a memory for storing processor executable instructions; when the processor executes the executable instructions, the method described in the embodiment of the present application is implemented.

[0139] The embodiments of the present application also provide a non-volatile computer-readable storage medium on which a computer program or instruction is stored. When the computer program or instruction is executed, the method described in the embodiments of the present application is implemented.

[0140] In addition, each functional module in each embodiment of the present invention may be integrated into one processing module, or each module may exist independently, or two or more modules may be integrated into one module.

[0141] The above storage media include but are not limited to random access memory (RAM), read-only memory (ROM), cache, hard disk drive (HDD) or memory card. The memory can be used to store computer program instructions.

[0142] It can be seen from the description of the above implementation methods that those skilled in the art can clearly understand that the present application can be implemented by means of software plus necessary hardware. Based on this understanding, the technical solution of the present application can be essentially or partly reflected in the prior art in the form of a software product, or it can be reflected in the implementation process of data migration. The computer software product can be stored in a storage medium, such as ROM / RAM, a magnetic disk, an optical disk, etc., including a number of instructions for a computer device (which can be a personal computer, a mobile terminal, a server, or a network device, etc.) to execute the methods described in the various embodiments of the present application or certain parts of the embodiments.

[0143] The various embodiments in this specification are described in a progressive manner, and the same or similar parts between the various embodiments can be referred to each other, and each embodiment focuses on the differences from other embodiments. All or part of this application can be used in many general or special computer system environments or configurations. For example: personal computers, server computers, handheld devices or portable devices, tablet devices, mobile communication terminals, multi-processor systems, microprocessor-based systems, programmable electronic devices, network PCs, minicomputers, mainframe computers, distributed computing environments including any of the above systems or devices, etc.

[0144] The above embodiments are only used to illustrate the technical solutions of the present application, rather than to limit the present application. Although the present application has been described in detail with reference to the aforementioned embodiments, a person of ordinary skill in the art should understand that the technical solutions described in the aforementioned embodiments may still be modified, or some or all of the technical features thereof may be replaced by equivalents. However, these modifications or replacements do not cause the essence of the corresponding technical solutions to deviate from the scope of the technical solutions of the present application.

Claims

1. A method for analyzing the nonlinear vibration characteristics of a three-dimensional serpentine structure, characterized in that: include: The three-dimensional piezoelectric serpentine ductile structure is modeled as a beam to obtain a three-dimensional serpentine beam model; wherein the three-dimensional piezoelectric serpentine ductile structure and the three-dimensional serpentine beam model are formed by a plurality of L-shaped units being flipped in sequence along the direction of one side thereof and connected end to end in sequence along the direction of the other side thereof to form a bow-shaped structure; Determine the kinetic energy and potential energy of the three-dimensional serpentine beam model; wherein the kinetic energy of the three-dimensional serpentine beam model is determined based on the deflection of the three-dimensional serpentine beam model; the deflection of the three-dimensional serpentine beam model is as follows: ， ； In the formula, It means that the i-th L-type unit is in the global coordinate at time t The displacement function of the direction, It means that the i-th L-type unit is in the global coordinate at time t The displacement function of the direction, represents the length of the piezoelectric serpentine stretchable structure before releasing the pre-stretching, represents the global coordinates of the i-th L-type unit, represents the length of the three-dimensional serpentine beam model after the pre-stretching is released and buckling occurs, represents the modal coordinates of the three-dimensional serpentine beam model at time t, Indicates time, represents the length of the long side of the L-shaped unit; Determine the control equation of the three-dimensional serpentine beam model according to the kinetic energy and the potential energy; A dynamic response equation is determined based on the control equation to determine the vibration characteristics of the three-dimensional serpentine beam model.

2. The method according to claim 1, characterized in that Before modeling the three-dimensional piezoelectric serpentine extendable structure as a beam, the method further includes: The flexible substrate is pre-stretched, and the piezoelectric serpentine structure is adhered to the pre-stretched flexible substrate to fix both ends thereof; The pre-stretching applied to the flexible substrate is released so that the piezoelectric serpentine structure is compressed and buckled to form a curved three-dimensional piezoelectric serpentine extensible structure.

3. The method according to claim 1, characterized in that Determining the kinetic energy of the three-dimensional serpentine beam model includes: Acquire the vibration mode of the three-dimensional serpentine beam model in the first-order buckling mode through finite element simulation; Determine the displacement function of the L-shaped unit in the three-dimensional serpentine beam model based on the vibration mode; determining a displacement mode of the three-dimensional serpentine beam model in a first-order buckling mode; defining the theoretical deflection of the three-dimensional serpentine beam model in the first-order buckling mode; Determining the deflection of the three-dimensional serpentine beam model according to the displacement mode of the first-order buckling mode, the displacement function of the L-shaped unit and the theoretical deflection; The kinetic energy of the three-dimensional serpentine beam model is determined based on the deflection of the three-dimensional serpentine beam model.

4. The method according to claim 1, characterized in that: The control equation is as follows: ； In the formula, represents the modal coordinates of the three-dimensional serpentine beam model at time t, represents the second-order derivative of the modal coordinates of the three-dimensional serpentine beam model with respect to time t, represents external incentives, Indicates time, , , , and Represents a dimensionless parameter.

5. The method according to claim 1, characterized in that The dynamic response equation is as follows: ； In the formula, represents the dimensionless modal coordinate amplitude, represents the dimensionless external excitation amplitude, represents dimensionless time, represents the dimensionless external excitation frequency, , , and represents a dimensionless parameter, Indicates time, Represents the thickness of the 3D serpentine beam model.

6. The method according to claim 4, characterized in that The external incentives are as follows: ； In the formula, represents external incentives, Indicates time, represents the amplitude of the external excitation, represents the frequency of external excitation.

7. A three-dimensional serpentine structure nonlinear vibration characteristic analysis device, characterized in that: include: A modeling module, used for modeling the three-dimensional piezoelectric serpentine ductile structure as a beam to obtain a three-dimensional serpentine beam model; wherein the three-dimensional piezoelectric serpentine ductile structure and the three-dimensional serpentine beam model are formed by a plurality of L-shaped units which are sequentially flipped along the direction of one side thereof and sequentially connected end to end along the direction of the other side thereof to form a bow-shaped structure; The first determination module is used to determine the kinetic energy and potential energy of the three-dimensional serpentine beam model; wherein the kinetic energy of the three-dimensional serpentine beam model is determined based on the deflection of the three-dimensional serpentine beam model; the deflection of the three-dimensional serpentine beam model is as follows: ， ； In the formula, It means that the i-th L-type unit is in the global coordinate at time t The displacement function of the direction, It means that the i-th L-type unit is in the global coordinate at time t The displacement function of the direction, represents the length of the piezoelectric serpentine stretchable structure before releasing the pre-stretching, represents the global coordinates of the i-th L-type unit, represents the length of the three-dimensional serpentine beam model after the pre-stretching is released and buckling occurs, represents the modal coordinates of the three-dimensional serpentine beam model at time t, Indicates time, represents the length of the long side of the L-shaped unit; A second determination module, used for determining a control equation of the three-dimensional serpentine beam model according to the kinetic energy and the potential energy; The vibration characteristic analysis module is used to determine the dynamic response equation based on the control equation to determine the vibration characteristic of the three-dimensional serpentine beam model.

8. A device for performing a method for analyzing the nonlinear vibration characteristics of a three-dimensional serpentine structure, characterized in that: include: processor; a memory for storing processor-executable instructions; When the processor executes the executable instructions, the method according to any one of claims 1 to 6 is implemented.

9. A non-volatile computer-readable storage medium, characterized in that: The device comprises a computer program or an instruction for storing the computer program or the instruction, which, when executed, enables the method according to any one of claims 1 to 6 to be implemented.

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