Method suitable for determining non-linear response of plate shell structure to external excitation of multi-physics field coupling

Abstract

The embodiment of the invention relates to the technical field of structural dynamic characteristic analysis, in particular to a method suitable for determining the nonlinear response of a plate shell structure to external excitation of multi-physics field coupling, and the method comprises the steps: determining the relation between the stress and strain of the plate shell structure; according to the determined relationship, determining the relationship among the strain energy, the kinetic energy and the external force acting of the plate shell structure; external excitation is determined according to the multiple physical fields applied to the plate shell structure; updating and determining the relationship among the strain energy, the kinetic energy and the external force work of the plate shell structure according to the external excitation; and converting the updated relationship by using the Hamiltonian variation principle, and determining the response of the plate shell structure to external excitation. The method provided by the embodiment of the invention is beneficial to research on strong nonlinear vibration characteristics of the plate shell structure so as to obtain a nonlinear dynamic response result of the plate shell structure under multi-physics coupling, which is relatively high in precision and relatively high in universality, and has relatively great significance in aspects of aerospace structure dynamics modeling analysis, vibration reduction and buffering and the like.

Description

Technical Field

[0001] The embodiments of the present application relate to the technical field of structural dynamic characteristics analysis, and specifically to a method suitable for determining the nonlinear response of a plate-shell structure to an external excitation coupled with multiple physical fields. Background Art

[0002] The statements herein merely provide background information related to the present application and do not necessarily constitute prior art. Plate-shell structure is a common load-bearing component, which has the advantages of good mechanical properties, simple design and analysis, less consumables, and easy processing. It is widely used in the design and manufacture of aerospace equipment, marine engineering equipment, high-tech ships, advanced rail transportation equipment and other industries.

[0003] In recent years, with the acceleration of modernization construction, plate-shell structures have been widely used in the aerospace field. However, the increasingly complex extreme service environment has continuously increased the service requirements of plate-shell structures, and most plate-shell structures are in a complex multi-field coupling environment. For example, plate-shell structures made of different materials will generate different charges or magnetic fields after mechanical stress is applied, and corresponding structural stresses will be generated under the action of different electromagnetic fields. At the same time, temperature changes will also affect the material properties of the plate-shell structure and generate thermal stress. The multi-field coupling environment will directly affect the dynamic behavior of the plate-shell structure. Therefore, the multi-field coupling environment needs to be considered when performing structural dynamics modeling analysis and vibration reduction and buffering design for plate-shell structures. Summary of the invention

[0004] A brief overview of the present application is provided below in order to provide a basic understanding of certain aspects of the present application. It should be understood that this overview is not an exhaustive overview of the present application. It is not intended to identify the key or important parts of the present application, nor is it intended to limit the scope of the present application. Its purpose is merely to present certain concepts in a simplified form as a prelude to a more detailed description discussed later.

[0005] An embodiment of the present application provides a method suitable for determining the nonlinear response of a plate-shell structure to an external excitation coupled with multiple physical fields, comprising the following steps: S10: determining the relationship between stress and strain of the plate-shell structure; S20: determining the relationship between strain energy, kinetic energy and external work done by the plate-shell structure based on the relationship determined in step S10; S30: determining the external excitation based on the multiple physical fields applied to the plate-shell structure; S40: updating and determining the relationship between strain energy, kinetic energy and external work done by the plate-shell structure based on the external excitation determined in step S30; S50: transforming the relationship determined in step S40 using the Hamiltonian variational principle; S60: determining the response of the plate-shell structure to the external excitation based on the transformed relationship obtained in step S50.

[0006] The method provided in the embodiment of the present application can determine the relationship between the strain energy, kinetic energy and external work done by the plate-shell structure through the relationship between the stress and strain of the plate-shell structure, and then update the determined relationship between the strain energy, kinetic energy and external work done by the external force according to the external excitation and transform it using the Hamiltonian variational principle, so as to determine the response of the plate-shell structure to the external excitation. This method can relatively accurately determine the unstable area in the plate-shell structure, which is conducive to the study of the strong nonlinear vibration characteristics of the plate-shell structure, so as to obtain the nonlinear dynamic response results of the plate-shell structure under multi-physical field coupling with higher accuracy and stronger universality, thereby ensuring the safety of the plate-shell structure after the structural vibration reduction design is applied to the fields of aerospace, vehicles, chemical industry, etc.

[0007] These and other advantages of the present application will become more apparent through the following detailed description of the preferred embodiments of the present application in conjunction with the accompanying drawings. BRIEF DESCRIPTION OF THE DRAWINGS

[0008] In order to further illustrate the above and other advantages and features of the present application, the specific embodiments of the present application are further described in detail below in conjunction with the accompanying drawings. The accompanying drawings together with the following detailed description are included in this specification and form a part of this specification. Elements with the same function and structure are represented by the same reference numerals. It should be understood that these drawings only describe typical examples of the present application and should not be regarded as limiting the scope of the present application.

[0009] Figure 1 is a schematic flow chart of a method for determining a nonlinear response of a plate-shell structure to an external excitation coupled with multiple physical fields according to an embodiment of the present application; Figure 2 is a schematic diagram of a plate-shell structure under multi-physics field coupling according to an embodiment of the present application; Figure 3 yes Figure 2 A top view of the plate and shell structure shown; Figure 4a is a nonlinear amplitude-frequency response curve of a plate-shell structure changing with electric field under multi-physical field coupling with an initial external potential of 100V according to an embodiment of the present application; Figure 4b It is a nonlinear amplitude-frequency response curve of the plate-shell structure changing with the electric field under multi-physical field coupling with an initial external potential of 200V according to an embodiment of the present application; Figure 5a It is a nonlinear amplitude-frequency response curve of the plate-shell structure with the change of magnetic field under multi-physics field coupling with an initial external magnetic potential of 50A according to an embodiment of the present application; Figure 5b It is a nonlinear amplitude-frequency response curve of a plate-shell structure with a change of magnetic field under multi-physics field coupling with an initial external magnetic potential of 300A according to an embodiment of the present application; Figure 6a It is a nonlinear amplitude-frequency response curve of a plate-shell structure with temperature change under multi-physics field coupling with a temperature change of 50° C. according to an embodiment of the present application; Figure 6b It is a nonlinear amplitude-frequency response curve of a plate-shell structure with temperature change under multi-physical field coupling with a temperature change of 100° C. according to an embodiment of the present application.

[0010] It should be noted that the drawings are not necessarily drawn to scale, but are merely shown in a schematic manner that does not affect the reader's understanding. DETAILED DESCRIPTION

[0011] Exemplary embodiments of the present application will be described below in conjunction with the accompanying drawings. For the sake of clarity and conciseness, not all features of the actual implementation are described in the specification. However, it should be understood that many implementation-specific decisions must be made in the process of developing any such actual implementation in order to achieve the developer's specific goals, such as meeting those constraints related to the system and business, and these constraints may vary from implementation to implementation. In addition, it should be understood that although the development work may be very complex and time-consuming, it is only a routine task for those skilled in the art who benefit from the content of this application.

[0012] It is also necessary to explain here that, in order to avoid obscuring the present application due to unnecessary details, only the device structure and / or processing steps closely related to the scheme according to the present application are shown in the accompanying drawings, while other details that are not very relevant to the present application are omitted.

[0013] It should be noted that, unless otherwise defined, the technical terms or scientific terms used in this application should have the common meanings understood by persons with ordinary skills in the field to which this application belongs.

[0014] In the description of the embodiments of the present application, “plurality” means at least two, such as two, three, etc., unless otherwise clearly and specifically defined.

[0015] Under the action of multi-physical field coupling, the nonlinear large vibration of the plate and shell structure will be increased, and the resulting nonlinear resonance and other problems will affect the normal operation of the equipment and even cause danger to the operator. In the related technology, the dynamic performance analysis of high-dimensional nonlinear systems usually relies on traditional approximate analytical methods or simulations to solve. For traditional approximate analytical methods, it is necessary to rely on a pre-given function form during the solution process, and only low-dimensional systems can be solved. For finite element simulation, the solution process is time-consuming and computationally intensive, and it is impossible to predict complex chaos, bifurcation and other phenomena, and it is impossible to accurately analyze the strong nonlinear vibration behavior of such plate and shell structures.

[0016] In view of the above technical problems, an embodiment of the present application provides a method suitable for determining the nonlinear response of a plate-shell structure to an external excitation coupled with multiple physical fields. Figure 1 is a flow chart of a method for determining a nonlinear response of a plate-shell structure to an external excitation coupled with multiple physical fields according to an embodiment of the present application, such as Figure 1 As shown, it includes the following steps S10 to S60.

[0017] S10: Determine the relationship between stress and strain in plate and shell structures.

[0018] S20: According to the relationship determined in step S10, the relationship between the strain energy, kinetic energy and work done by the external force of the plate and shell structure is determined.

[0019] S30: Determine external excitation according to the multi-physics field applied to the plate and shell structure.

[0020] S40: According to the external excitation determined in step S30, the relationship between the strain energy, kinetic energy and work done by the external force of the plate and shell structure is updated and determined.

[0021] S50: Using the Hamiltonian variational principle, transform the relationship determined in step S40.

[0022] S60: Determine the response of the plate and shell structure to the external excitation according to the transformed relationship obtained in step S50.

[0023] The method provided in the embodiment of the present application can determine the relationship between the strain energy, kinetic energy and external work done by the plate-shell structure through the relationship between the stress and strain of the plate-shell structure, and then update the determined relationship between the strain energy, kinetic energy and external work done by the external force according to the external excitation and transform it using the Hamiltonian variational principle, so as to determine the response of the plate-shell structure to the external excitation. This method can more accurately determine the unstable area in the plate-shell structure, which is conducive to the study of the strong nonlinear vibration characteristics of the plate-shell structure, so as to obtain the nonlinear dynamic response results of the plate-shell structure under multi-physical field coupling with higher accuracy and stronger universality, thereby ensuring the safety of the plate-shell structure after the structural vibration reduction design is applied to aerospace and other fields.

[0024] In some embodiments, the plate shell structure may be a flat plate or a plate structure with curvature, such as a cylindrical shell structure.

[0025] In some embodiments, the multi-physics field includes, but is not limited to, an electric field, a magnetic field, or a varying temperature field.

[0026] In some embodiments, in step S10, the stress and strain satisfy the following expressions (1) to (3): (1); (2); (3).

[0027] in, represents the stress component; represents the elastic constant; represents the strain component; represents the thermal modulus constant; Indicates temperature change; represents the piezoelectric constant; represents the magnetic field component; represents the piezoelectric constant; represents the electric field component; represents the electric displacement component; represents the piezoelectric constant; represents the thermoelectric constant; represents the magnetoelectric constant; represents the dielectric constant; represents the magnetic induction component; represents the piezomagnetic constant; represents the thermomagnetic constant; represents the magnetic constant; Represents the magnetoelectric constant.

[0028] In some embodiments, when external excitation acts on the plate and shell structure, the stress field at the point of action depends not only on the strain at the point, but also on the strains of all other external force fields.

[0029] In some embodiments, in step S40, taking the plate shell structure as a cylindrical shell as an example, the following relationship between strain energy, kinetic energy and work done by external force is given to satisfy the following expressions (4) to (6): (4); (5); (6).

[0030] in, Indicates kinetic energy; Represents the length of the plate and shell structure; represents the inertia term; Represents the coordinates in the length direction of the plate and shell structure; Represents the coordinates of the plate and shell structure in the circumferential direction; Represents the coordinates in the thickness direction of the plate and shell structure; Indicates that the plate and shell structure Displacement of the mid-plane point in the direction; Indicates that the plate and shell structure Direction and Displacement in a direction perpendicular to the direction; Indicates that the plate and shell structure Displacement of the mid-plane point in the direction; represents the symbol of partial derivative; Represents the radius of the plate and shell structure; represents potential energy; Indicates the thickness of the plate and shell structure; express Stress in direction; express Strain in direction; express Stress in direction; express Strain in direction; express Shear stress in the direction; express Shear strain in the direction; express Direction of electric displacement; express Direction of electric displacement; express Direction of electric displacement; express Direction of the electric field; express Direction of the electric field; express Direction of the electric field; express Directional magnetic induction; express Directional magnetic induction; express Directional magnetic induction; express Direction of magnetic field; express Direction of magnetic field; express Direction of magnetic field; Represents the cross-sectional area of ​​the plate and shell structure; express Directional force; express Directional force; express Directional force; express mid-surface strain in the direction; express mid-surface strain in the direction; express mid-surface strain in the direction; express Bending moment in direction; express Bending moment in direction; express Bending moment in direction; express Curvature and twist in direction; express Curvature and twist in direction; express Curvature and twist in direction; Indicates work done by external force; Indicates that the electric field The force generated by the direction, Indicates that the electric field The force generated by the direction, , Represents different piezoelectric constants, represents the initial voltage; Indicates the temperature field The force generated in the direction, Indicates the temperature field The force generated in the direction, , Indicates temperature change, , Indicates different parameters; Indicates that the magnetic field The force generated in the direction; Indicates that the magnetic field The force generated in the direction; , Indicates the different parameters determined; represents the initial magnetic potential; represents the external mechanical point load, , represents the external excitation amplitude, represents the frequency of external excitation, Indicates time, represents the functional relationship, Indicates the location of external forces Coordinate values, Indicates the location of external forces Coordinate value.

[0031] The method provided in the embodiment of the present application determines the relationship between the strain energy, kinetic energy and work done by external forces of the plate and shell structure through the above expressions (4) to (6), and the calculation results are relatively reliable.

[0032] Figure 2 is a schematic diagram of a plate-shell structure under multi-physics field coupling according to an embodiment of the present application, Figure 3 yes Figure 2 The top view of the plate and shell structure shown in Figure 2 and Figure 3 As shown, represents the space coordinate system where the plate and shell structure is located, represents the coordinates in the length direction of the plate and shell structure, represents the coordinates in the thickness direction of the plate and shell structure, Indicates that the plate and shell structure Direction and The coordinates in the direction perpendicular to the direction, the plate-shell structure is in an electric field ,magnetic field and changing temperature field The multi-physics coupled environment, where the electric field exist and Changes between From the initial temperature Changes to .

[0033] Under the Maxwell equations, the distribution of magnetic potential and electric potential along the thickness direction of the plate and shell structure satisfies the following expressions (7) and (8): (7); (8).

[0034] in, It represents the distribution of magnetic potential along the thickness direction of the plate and shell structure; Represents the coordinates in the length direction of the plate and shell structure; Represents the coordinates of the plate and shell structure in the circumferential direction; Represents the coordinates in the thickness direction of the plate and shell structure; Indicates time; Represents the distribution of electric potential along the thickness direction of the plate and shell structure; Indicates the set parameters. , Indicates the thickness of the plate and shell structure; Represents the change of magnetic potential in space; Represents the change of electric potential in space; represents the initial external magnetic potential; represents the initial external potential.

[0035] The electric field component and the magnetic field component in the coupled multi-physics field can be determined by the above expressions (7) and (8).

[0036] In some embodiments, the electric field component The following expressions (9) to (11) are satisfied: (9); (10); (11).

[0037] in, Represents the coordinates in the length direction of the plate and shell structure; Represents the coordinates of the plate and shell structure in the circumferential direction; Represents the coordinates in the thickness direction of the plate and shell structure; express The electric field component in the direction; Represents the distribution of electric potential along the thickness direction of the plate and shell structure; Indicates the set parameters. , Indicates the thickness of the plate and shell structure; Represents the radius of the plate and shell structure; Represents the change of electric potential in space; express The electric field component in the direction; express The electric field component in the direction.

[0038] In some embodiments, the magnetic field component The following expressions (12) to (14) are satisfied: (12); (13); (14).

[0039] in, Represents the coordinates in the length direction of the plate and shell structure; Represents the coordinates of the plate and shell structure in the circumferential direction; Represents the coordinates in the thickness direction of the plate and shell structure; express Directional magnetic field component; It represents the distribution of magnetic potential along the thickness direction of the plate and shell structure; Indicates the set parameters. , Indicates the thickness of the plate and shell structure; Represents the radius of the plate and shell structure; Represents the change of magnetic potential in space; express Directional magnetic field component; express Directional magnetic field component.

[0040] In some embodiments, the strain component at any point on the plate and shell structure satisfies the following expression (15): (15).

[0041] in, express Strain in direction; express Strain in direction; express Shear strain in the direction; Represents the midplane Strain in direction, Represents the midplane Strain in direction; Represents the midplane Strain in direction; , represents different mid-surface curvatures; Indicates midface torsion.

[0042] In some embodiments, based on the constitutive relationship of the plate-shell structure under the action of the electromagnetic field, the following expressions (16) to (18) can be obtained: (16); (17); (18).

[0043] in, express Stress in direction; express Strain in direction; express Stress in direction; express Strain in direction; express Shear stress in the direction; express Shear strain in the direction; Indicates temperature change; express Directional magnetic induction; express Directional magnetic induction; express Directional magnetic induction; express Direction of the electric field; express Direction of the electric field; express Direction of the electric field; express Direction of electric displacement; express Direction of electric displacement; express Direction of electric displacement; express Direction of magnetic field; express Direction of magnetic field; express Direction of magnetic field; , , , , , , , and They satisfy the following expressions respectively:

[0044]

[0045]

[0046]

[0047]

[0048] in, represents the elastic modulus; represents the piezoelectric coefficient, Represents the dielectric constant, represents the piezomagnetic coefficient, represents the magnetoelectric coefficient, Represents the magnetic coefficient, represents the thermal modulus coefficient, represents the thermoelectric coefficient; It is understood that in such an embodiment, , They represent the same or different ordinals respectively. Physical quantities with different ordinals correspond to values ​​under different conditions. For example, , Represents different piezoelectric coefficients.

[0049] In some embodiments, for a changing temperature field, the temperature usually changes nonlinearly along the thickness direction of the plate and shell structure. When there is no heat source, the one-dimensional steady-state heat conduction equation satisfies the following expression (19): (19).

[0050] Wherein, d represents the derivative symbol; Indicates coordinate value; represents thermal conductivity; represents the temperature; the boundary conditions of the plate and shell structure should satisfy the following expression (20): (20).

[0051] in, Indicates the thickness of the plate and shell structure Varying temperature function; Indicates the external temperature; Indicates the internal temperature; Represents the thickness of the plate and shell structure.

[0052] In some embodiments, the power law exponent Thermal conductivity expressed in the form The inhomogeneity satisfies the following expression (21): (twenty one).

[0053] By solving expression (21), the relationship between the temperature change and the thickness of the plate and shell structure can be obtained to satisfy the following expression (22): (twenty two).

[0054] in, Indicates the thickness of the plate and shell structure Varying temperature function; Indicates the internal temperature; Indicates the thickness of the plate and shell structure; Indicates temperature change; represents the external thermal conductivity; represents the internal thermal conductivity; represents the power law exponent; represents a given parameter; its specific expression is as shown in formula (23), and the parameter and temperature changes The following expressions (23) and (24) are satisfied respectively: (twenty three); (twenty four).

[0055] Figure 4a is a nonlinear amplitude-frequency response curve of a plate-shell structure changing with electric field under multi-physical field coupling with an initial external potential of 100V according to an embodiment of the present application, Figure 4b is a nonlinear amplitude-frequency response curve of the plate-shell structure with the change of the electric field under the multi-physical field coupling with an initial external potential of 200V according to the embodiment of the present application, such as Figure 4aand Figure 4b As shown, the horizontal axis represents the dimensionless excitation frequency, the vertical axis represents the dimensionless amplitude of the plate and shell structure, the dotted line is the unstable solution determined by the method of the embodiment of the present application, and the solid line is the stable solution.

[0056] Figure 5a is a nonlinear amplitude-frequency response curve of a plate-shell structure with a change in magnetic field under multi-physics field coupling with an initial external magnetic potential of 50A according to an embodiment of the present application, Figure 5b is a nonlinear amplitude-frequency response curve of the plate-shell structure with the change of magnetic field under the multi-physics field coupling with the initial external magnetic potential of 300A according to the embodiment of the present application, such as Figure 5a and Figure 5b As shown, the horizontal axis represents the dimensionless excitation frequency, the vertical axis represents the dimensionless amplitude of the plate and shell structure, the dotted line is the unstable solution determined by the method of the embodiment of the present application, and the solid line is the stable solution.

[0057] Figure 6a is a nonlinear amplitude-frequency response curve of a plate-shell structure with temperature variation under multi-physics field coupling with a temperature variation of 50° C. according to an embodiment of the present application, Figure 6b is a nonlinear amplitude-frequency response curve of a plate-shell structure with temperature variation under multi-physics field coupling with a temperature variation of 100° C. according to an embodiment of the present application, such as Figure 6a and Figure 6b As shown, the horizontal axis represents the dimensionless excitation frequency, the vertical axis represents the dimensionless amplitude of the plate and shell structure, the dotted line is the unstable solution determined by the method of the embodiment of the present application, and the solid line is the stable solution.

[0058] Reference Figures 4a to 6b By using the method of the embodiment of the present application to determine the nonlinear response of the plate-shell structure to the external excitation of the multi-physical field coupling, a continuous curve can be obtained, and then the changing law of the nonlinear response of the plate-shell structure to the external excitation of the multi-physical field coupling can be determined.

[0059] In some embodiments, step S50 further includes transforming the relationship determined in step S40 after the transformation using a nonlinear fractional-order damping model, which is beneficial for studying and analyzing the damping nonlinearity of the plate-shell structure.

[0060] In some embodiments, the nonlinear fractional-order damping model satisfies the following expression (25): (25).

[0061] in, Indicates stress; It represents the Young's modulus of the plate and shell structure; Indicates strain; Represents a constant related to the properties of a material or structure; represents the relaxation time, and , Represents the damping coefficient, which is used to reflect the dissipation characteristics of the material; Indicates time; express Fractional derivative of order.

[0062] In such an embodiment, The fractional derivatives satisfy the following expression (26): (26).

[0063] in, express is a fractional derivative operator, and , ; Indicates time; represents a time-dependent function; represents the frequency component, is an integer used to represent the index of the frequency component; Indicates temperature; Represents an imaginary unit.

[0064] From the above expression (26), we can determine: . Further, The fractional derivatives also satisfy the following expressions (27) to (29): (27); (28); (29).

[0065] In such an embodiment, when = 1, the classical first-order derivative can be obtained; when =0, the original relationship can be obtained.

[0066] In some embodiments, the relationship obtained in step S50 satisfies the following expressions (30) to (34): (30); (31); (32); (33); (34).

[0067] in, Represents the coordinates in the length direction of the plate and shell structure; Represents the coordinates of the plate and shell structure in the circumferential direction; Represents the coordinates in the thickness direction of the plate and shell structure; Indicates that the plate and shell structure Displacement of the mid-plane point in the direction; Indicates that the plate and shell structure Direction and Displacement in a direction perpendicular to the direction; Indicates that the plate and shell structure Displacement of the mid-plane point in the direction; express Directional force; Represents the radius of the plate and shell structure; express Directional force; represents the inertia term; Indicates time; express Bending moment in direction; express Bending moment in direction; express Directional force; express Bending moment in direction; Indicates that the electric field The force generated by the direction, Indicates that the electric field The force generated by the direction, , Represents different piezoelectric constants, represents the initial voltage; Indicates the temperature field The force generated in the direction, Indicates the temperature field The force generated in the direction, , Indicates temperature change, , Indicates different parameters; represents the structural damping; represents the external mechanical point load, , represents the external excitation amplitude, represents the frequency of external excitation, represents the functional relationship, Indicates the location of external forces Coordinate values, Indicates the location of external forces Coordinate value; Indicates the thickness of the plate and shell structure; express Direction of electric displacement; Represents a given symbol, ; express Direction of electric displacement; express Direction of electric displacement; express Directional magnetic induction; express Directional magnetic induction; express Directional magnetic induction.

[0068] In some embodiments, based on simply supported boundary conditions, the displacement field satisfies the following expressions (35) to (39): (35); (36); (37); (38); (39).

[0069] in, express mid-plane displacement in the direction; express mid-plane displacement in the direction; express mid-plane displacement in the direction; represents electric displacement; represents magnetic displacement; represents the circumferential truncation coefficient; represents the axial truncation coefficient; represents the axial half-wave number; Indicates the circumferential wave number; denotes the excited mode subscript; denotes an adjoint modal subscript; Indicates the establishment Excitation mode functions in direction; Indicates the establishment adjoint mode functions in direction; Indicates the establishment Axisymmetric mode functions in the direction; Indicates the establishment Excitation mode functions in direction; Indicates the establishment adjoint mode functions in direction; Indicates the establishment Excitation mode functions in direction; Indicates the establishment adjoint mode functions in direction; Indicates the establishment Axisymmetric mode functions in the direction; represents the excited mode function in the direction of the established electric field; represents the adjoint mode function in the direction of the established electric field; represents the axisymmetric mode function in the direction of the established electric field; represents the excited mode function in the direction of the established magnetic field; represents the axisymmetric mode function in the direction of the established magnetic field; represents the axisymmetric mode functions in the direction of the established magnetic field.

[0070] In some embodiments, in step S60, the transformed relationship is converted from a set of partial differential equations to a set of ordinary differential equations using a multimodal Galerkin method, which has high computational efficiency and reliability.

[0071] In some embodiments, step S60 also includes the following steps: S61: establishing the vibration mode function of the plate and shell structure under external excitation; S62: converting the transformed relationship obtained in step S50 according to the vibration mode function; S63: weighted integrating the relationship obtained in step S62 to obtain a group of ordinary differential equations.

[0072] The method provided in the embodiment of the present application utilizes the transformed relationship of the vibration mode function conversion and performs weighted integration on it, so as to obtain a group of ordinary differential equations to improve the solution accuracy of the response of the plate and shell structure to external excitation.

[0073] In some embodiments, a weighted multi-mode Galerkin method is used to transform the transformed nonlinear partial differential equations into a multi-mode coupled ordinary differential equations. That is, the original equations are weighted with appropriate functions in sequence and integrated to the middle surface of the plate and shell structure, wherein the weighted function satisfies the following expressions (40) to (44): (40); (41); (42); (43); (44).

[0074] in, express Directional weighting function; Indicates the variable symbol, used to distinguish different weighting functions; express Directional weighting function; express Directional weighting function; express Directional weighting function; express Directional weighting function; represents the axial half-wave number; Indicates the circumferential wave number; Represents the length of the plate and shell structure.

[0075] In some embodiments, in step S61, the vibration mode function includes a driving mode, a companion mode, and an axisymmetric mode to more accurately simulate the response of the plate and shell structure when subjected to external excitation.

[0076] In some embodiments, in step S63, the following steps are also included: S631: converting the set of ordinary differential equations from a non-autonomous system to an autonomous system; S632: solving the autonomous system.

[0077] The method provided in the embodiment of the present application converts a set of transformed ordinary differential equations from a non-autonomous system to an autonomous system, so as to solve the set of transformed ordinary differential equations.

[0078] In some embodiments, in step S631, the following expression (45) is used to convert the ordinary differential equation from a non-autonomous system to an autonomous system: (45).

[0079] in, , They represent different periodic orbit radii respectively; , They represent different periodic orbit functions respectively; Represents the natural frequency of the system.

[0080] In some embodiments, the solutions of a non-self-consistent system can be viewed as periodic orbits along a circle of radius 1, and these solutions can be expressed as The frequency of crosses a circle with a radius of 1. In the initial condition setting stage, and Set to , thereby determining ; Then, according to the current step ( step) to predict the next step ( The exact solution of the step) satisfies the following expressions (46) and (47): (46); (47).

[0081] in, Indicates the current step or step; represents the predicted solution; represents the current solution; represents the predicted solution increment; represents the corrected solution, that is, during the iteration process, the current solution is used for correction in order to more accurately approximate the true solution; Indicates the number of iteration steps or the current state index; represents the frequency of prediction; Indicates the current frequency; represents the predicted frequency increment; Represents the corrected frequency, that is, the frequency value adjusted in each iteration.

[0082] In some embodiments, in step S632, the quasi-arc length extension method is used to solve the autonomous system and determine the response of the plate-shell structure to external excitation, which is beneficial to improving the accuracy of the nonlinear dynamic frequency characteristics and change laws of the determined plate-shell structure.

[0083] The implementation process of the pseudo-arc length extension method will be described below through examples.

[0084] Determine the initial value of the initial step length, and the frequency point of the initial step can be selected in the area where the nonlinearity is not strong.

[0085] The prediction step can provide an initial value for the next frequency point. In the arc length extension method, the prediction initial value satisfies the following expressions (48) and (49): (48); (49).

[0086] in, represents the current step length, Indicates that the next step is long; Represents the initial solution of the prediction, that is, the system state or displacement at the current step size or frequency point; represents the current solution; Indicates the predicted solution increment, which is the change from the current solution to the predicted solution; Represents the predicted initial frequency, that is, the next frequency value predicted based on the current solution or state in the current step; Indicates the current frequency; Represents the predicted frequency increment, which is the change between the predicted frequency and the current frequency.

[0087] The predicted initial value is generally along the tangent direction. For the frequency domain algebraic equation calculated by the harmonic balance method, the predicted solution increment and the predicted frequency increment The following expression (50) is satisfied: (50).

[0088] in, Represents a function of system state and frequency.

[0089] In the above expression (50), it can be restricted by adding a step condition restriction, which satisfies the following expression (51): (51).

[0090] in, represents the arc length increment, which is used to describe the change along the arc length path in the parameter space; the symbol ‖‖ represents the norm of the vector.

[0091] Combining the above expressions (50) and (51), we can obtain the predicted solution increment and the predicted frequency increment The following expressions (52) and (53) are satisfied respectively: (52); (53).

[0092] Since there are two tangent directions, the solution increment is determined and the predicted frequency increment It is also necessary to judge the direction to ensure that the continuous direction can always move forward. The judgment process satisfies the following expression (54): (54).

[0093] in, Represents the transpose symbol.

[0094] In some embodiments, there may be a certain deviation between the predicted solution obtained along the tangent direction and the actual exact solution. The predicted solution is corrected once or multiple times to obtain an exact solution that satisfies the nonlinear algebraic equation.

[0095] In some embodiments, since the nonlinear characteristics of the plate-shell structure are more obvious at certain excitation frequencies, it is difficult to obtain the corrected accurate solution under a limited number of iterations when the arc length continuation method is used for correction. In this case, the deviation between the estimated solution and the accurate solution can be shortened by controlling the continuous step size of the arc length, so as to quickly obtain the accurate solution on the amplitude-frequency curve in the continuous direction.

[0096] When there is a deviation between the predicted solution and the exact solution due to a large step size, it is necessary to reduce the step size and re-estimate and correct the current step size until an accurate solution is obtained. The current step size and the previous step size satisfy the following expression (55): (55).

[0097] in, Indicates the current step length; Indicates the previous step length; Indicates the minimum step size set.

[0098] When the predicted solution is close to the exact solution, the step size of the next step can be increased to speed up the calculation of the amplitude-frequency curve. When increasing the step size, the increased step size is less than or equal to the set maximum step size to avoid missing the resonance peak of the amplitude-frequency curve. The increased step size satisfies the following expression (56): (56).

[0099] in, Indicates the step length of the arc length extension process; Indicates the current step size, that is, the first step in the arc length extension process. The step size used by the step; Indicates the maximum step size that is set.

[0100] In some embodiments, the stability of the response obtained in step S60 is determined using multivariable Floquet theory, which has high reliability.

[0101] In some embodiments, for a nonlinear system that satisfies the system equilibrium equation, by adding a perturbation to the equation , the stability of the solution can be studied, and the perturbation equation satisfies the following expression (57): (57).

[0102] in, represents generalized force; represents the second derivative of generalized coordinates; Represents the first derivative of generalized coordinates; represents generalized coordinates.

[0103] Based on the stability of the steady-state solution, the stability of the above expression (57) can be determined.

[0104] Regarding the embodiments of the present application, it should also be noted that, in the absence of conflict, the embodiments of the present application and the features therein can be combined with each other to obtain new embodiments.

[0105] The above description is only a specific implementation manner of the present application, but the protection scope of the present application is not limited thereto. The protection scope of the present application shall be based on the protection scope of the claims.

Claims

1. A method for determining the nonlinear response of a plate-shell structure to an external excitation coupled with multiple physical fields, characterized in that: The following steps are involved: S10: determining the relationship between stress and strain of the plate-shell structure; S20: Determine the relationship between the strain energy, kinetic energy and work done by the external force of the plate-shell structure according to the relationship determined in step S10; S30: determining the external excitation according to the multi-physical field applied to the plate-shell structure; S40: updating and determining the relationship between the strain energy, kinetic energy and work done by the external force of the plate-shell structure according to the external excitation determined in step S30; S50: transforming the relationship determined in step S40 using the Hamiltonian variational principle; S60: Determine the response of the plate-shell structure to external excitation according to the transformed relationship obtained in step S50.

2. The method according to claim 1, characterized in that In step S60, the transformed relationship is converted using a multimodal Galerkin method, from a set of partial differential equations to a set of ordinary differential equations.

3. The method according to claim 2, characterized in that In step S60, the following steps are also included: S61: establishing a vibration mode function of the plate-shell structure under the external excitation; S62: converting the transformed relationship obtained in step S50 according to the vibration mode function; S63: Perform weighted integration on the relationship obtained in step S62 to obtain the group of ordinary differential equations.

4. The method according to claim 3, characterized in that: In step S61, the vibration mode function includes a driving mode, a companion mode, and an axisymmetric mode.

5. The method according to claim 4, characterized in that In step S63, the following steps are also included: S631: Convert the ordinary differential equation group so that it is converted from a non-autonomous system to an autonomous system; S632: Solve the autonomous system.

6. The method according to claim 5, characterized in that In step S632, the autonomous system is solved using a quasi-arc length extension method.

7. The method according to claim 5, characterized in that The stability of the response obtained in the S60 step was determined using the multivariable Floquet theory.

8. The method according to claim 1, characterized in that: In step S50, the method further includes transforming the relationship determined in step S40 using a nonlinear fractional-order damping model.

9. The method according to claim 1, characterized in that: In step S10, the stress and the strain satisfy the following expressions: ； ； ； in, represents the stress component; represents the elastic constant; represents the strain component; represents the thermal modulus constant; Indicates temperature change; represents the piezoelectric constant; represents the magnetic field component; represents the piezoelectric constant; represents the electric field component; represents the electric displacement component; represents the piezoelectric constant; represents the thermoelectric constant; represents the magnetoelectric constant; represents the dielectric constant; represents the magnetic induction component; represents the piezomagnetic constant; represents the thermomagnetic constant; represents the magnetic constant; Represents the magnetoelectric constant.

10. The method according to claim 1, characterized in that In step S40, the relationship between the kinetic energy, strain energy and work done by the external force of the plate-shell structure satisfies the following expression: ； ； ； in, Indicates kinetic energy; represents the length of the plate and shell structure; represents the inertia term; represents the coordinates in the length direction of the plate and shell structure; represents the coordinates of the plate and shell structure in the circumferential direction; represents the coordinates in the thickness direction of the plate and shell structure; Indicates that the plate shell structure is along the Displacement of the mid-plane point in the direction; Indicates that the plate and shell structure is along the Directions and Displacement in a direction perpendicular to the direction; Indicates that the plate shell structure is along the Displacement of the mid-plane point in the direction; represents the symbol of partial derivative; represents the radius of the plate and shell structure; represents potential energy; represents the thickness of the plate and shell structure; Indicates the Stress in direction; Indicates the Strain in direction; express Stress in direction; Indicates the Strain in direction; express Shear stress in the direction; Indicates the Shear strain in the direction; Indicates the Direction of electric displacement; Indicates the Direction of electric displacement; Indicates the Direction of electric displacement; Indicates the Direction of the electric field; Indicates the Direction of the electric field; express Direction of the electric field; Indicates the Directional magnetic induction; Indicates the Directional magnetic induction; Indicates the Directional magnetic induction; Indicates the Direction of magnetic field; Indicates the Direction of magnetic field; Indicates the Direction of magnetic field; Represents the cross-sectional area of ​​the plate and shell structure; Indicates the Directional force; Indicates the Directional force; Indicates the Directional force; Indicates the mid-surface strain in the direction; Indicates the mid-surface strain in the direction; Indicates the mid-surface strain in the direction; Indicates the Bending moment in direction; Indicates the Bending moment in direction; Indicates the Bending moment in direction; Indicates the Curvature and twist in direction; Indicates the Curvature and twist in direction; Indicates the Curvature and twist in direction; Indicates work done by external force; The electric field along the The force generated by the direction, The electric field along the The force generated by the direction, , Represents different piezoelectric constants, represents the initial voltage; Indicates the temperature field in The force generated in the direction, Indicates the temperature field in The force generated in the direction, , Indicates temperature change, , Indicates different parameters; Indicates that the magnetic field is The force generated in the direction; Indicates that the magnetic field is The force generated in the direction; , Indicates the different parameters determined; represents the initial magnetic potential; represents the external mechanical point load, , represents the external excitation amplitude, represents the frequency of external excitation, Indicates time, represents the functional relationship, Indicates the external force Coordinate values, Indicates the external force Coordinate value.

11. The method according to claim 1, characterized in that The relationship obtained in step S50 satisfies the following expression: ； ； ； ； ； in, represents the coordinates in the length direction of the plate and shell structure; represents the coordinates of the plate and shell structure in the circumferential direction; represents the coordinates in the thickness direction of the plate and shell structure; Indicates that the plate shell structure is along the Displacement of the mid-plane point in the direction; Indicates that the plate and shell structure is along the Directions and Displacement in a direction perpendicular to the direction; Indicates that the plate and shell structure Displacement of the mid-plane point in the direction; express Directional force; represents the radius of the plate and shell structure; express Directional force; represents the inertia term; Indicates time; Indicates the Bending moment in direction; Indicates the Bending moment in direction; Indicates the Directional force; Indicates the Bending moment in direction; The electric field along the The force generated by the direction, The electric field along the The force generated by the direction, , Represents different piezoelectric constants, represents the initial voltage; Indicates the temperature field in The force generated in the direction, Indicates the temperature field in The force generated in the direction, , Indicates temperature change, , Indicates different parameters; represents the structural damping; represents the external mechanical point load, , represents the external excitation amplitude, represents the frequency of external excitation, represents the functional relationship, Indicates the external force Coordinate values, Indicates the external force Coordinate value; represents the thickness of the plate and shell structure; Indicates the Direction of electric displacement; Represents a given symbol, ; Indicates the Direction of electric displacement; Indicates the Direction of electric displacement; Indicates the Directional magnetic induction; Indicates the Directional magnetic induction; Indicates the Directional magnetic induction.

12. The method according to claim 5, characterized in that In step S631, the ordinary differential equation is converted from the non-autonomous system to the autonomous system using the following expression: ； in, , They represent different periodic orbit radii respectively; , They represent different periodic orbit functions respectively; Represents the natural frequency of the system.

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