Method for determining the nonlinear response of a shell structure to external excitations of multi-physical field coupling

Abstract

Embodiments of the present application relate to the technical field of structural dynamics characteristic analysis, and particularly relate to a method suitable for determining nonlinear response of a plate-shell structure to external excitation of multi-physical field coupling, comprising: determining a relationship between stress and strain of the plate-shell structure; determining a relationship between strain energy, kinetic energy and work done by external force of the plate-shell structure according to the determined relationship; determining the external excitation according to multi-physical fields applied to the plate-shell structure; updating the relationship between the strain energy, kinetic energy and work done by external force of the plate-shell structure according to the external excitation; transforming the updated relationship by using Hamilton's variational principle, and determining the response of the plate-shell structure to the external excitation. The method provided by the embodiments of the present application is beneficial to the study of 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-physical field coupling with high precision and strong universality, which has great significance for aerospace structural dynamics modeling analysis, vibration reduction and buffering, etc.

Description

TECHNICAL FIELD

[0001] Embodiments of the present application relate to the technical field of structural dynamics characteristic analysis, and in particular, to a method for determining nonlinear response of a plate-shell structure to external excitation of multi-physical field coupling. BACKGROUND

[0002] The statements herein are merely provided to give a basic understanding of the present application, and do not necessarily constitute the prior art. The plate-shell structure is a common load-bearing component, which has the advantages of good mechanical properties, simple design and analysis, less material consumption, and easy processing, and 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 process, plate-shell structures are widely used in the field of aerospace. However, the increasingly complex extreme service environment continuously improves 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 of different materials will generate different electric charges or magnetic fields after mechanical stress is applied, and corresponding structural stresses will be generated under different electromagnetic fields. At the same time, changes in temperature will affect the material properties of the plate-shell structure and generate thermal stresses. The multi-field coupling environment will directly affect the dynamics behavior of the plate-shell structure. Therefore, when modeling and analyzing the structural dynamics of the plate-shell structure, the multi-field coupling environment needs to be considered for vibration reduction and cushioning design. SUMMARY

[0004] In the following, a brief summary of the present application is given to provide a basic understanding of some aspects of the present application. It should be understood that this summary is not an exhaustive overview of the present application. It is not intended to identify 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 some concepts in a simplified form as a prelude to a more detailed description to be discussed later.

[0005] Embodiments of the present application provide a method for determining nonlinear response of a plate-shell structure to external excitation of multi-physical field coupling, 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 force work 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 the relationship between strain energy, kinetic energy, and external force work of the plate-shell structure according to the external excitation determined in step S30; S50: transforming the relationship determined in step S40 using the Hamilton variational principle; S60: determining the response of the plate-shell structure to the external excitation according to the transformed relationship obtained in step S50.

[0006] The method provided by the embodiment of the application can determine the relationship among the strain energy, kinetic energy and external force work of the plate shell structure through the relationship between the stress and strain of the plate shell structure, and then update the determined relationship among the strain energy, kinetic energy and external force work and transform it by using the Hamilton variation principle according to the external excitation, so as to determine the response of the plate shell structure to the external excitation. The method can more accurately determine the unstable region in the plate shell structure, thereby facilitating the research on the strong nonlinear vibration characteristics of the plate shell structure, so as to obtain the nonlinear dynamic response result of the plate shell structure under the multi-physical field coupling with high precision and strong universality, and further ensure the safety of the plate shell structure after the structure vibration reduction design is applied to the fields of aerospace, vehicles, chemical industry and the like.

[0007] These and other advantages of the application will become more apparent in light of the following detailed description of preferred embodiments of the application. BRIEF DESCRIPTION OF DRAWINGS

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

[0009] Figure 1 is a schematic flowchart of a method for determining the nonlinear response of a plate shell structure to external excitation under multi-physical field coupling according to an embodiment of the application;

[0010] Figure 2 is a schematic diagram of a plate shell structure under multi-physical field coupling according to an embodiment of the application;

[0011] Figure 3 is Figure 2 is a top view of the plate shell structure shown in FIG. 1;

[0012] Figure 4a is a nonlinear amplitude-frequency response curve of the plate shell structure under multi-physical field coupling with an initial external electric potential of 100 V varying with the electric field according to an embodiment of the application;

[0013] Figure 4b is a nonlinear amplitude-frequency response curve of the plate shell structure under multi-physical field coupling with an initial external electric potential of 200 V varying with the electric field according to an embodiment of the application;

[0014] Figure 5a is a nonlinear amplitude-frequency response curve of the plate shell structure under multi-physical field coupling with an initial external magnetic potential of 50 A varying with the magnetic field according to an embodiment of the application;

[0015] Figure 5b is a nonlinear amplitude-frequency response curve of the plate-shell structure according to the embodiment of the present application under the multi-physical field coupling of the initial external magnetic potential of 300 A and the change of the magnetic field;

[0016] Figure 6a is a nonlinear amplitude-frequency response curve of the plate-shell structure according to the embodiment of the present application under the multi-physical field coupling of the temperature change of 50 ℃ and the change of the temperature;

[0017] Figure 6b is a nonlinear amplitude-frequency response curve of the plate-shell structure according to the embodiment of the present application under the multi-physical field coupling of the temperature change of 100 ℃ and the change of the temperature.

[0018] It should be noted that the accompanying drawings are not necessarily drawn to scale, but are merely intended to show illustrative aspects in a schematic manner. DETAILED DESCRIPTION

[0019] In the following, exemplary embodiments of the present application will be described with reference to the accompanying drawings. In the description, not all of the features of the actual implementation are described in order to avoid obscuring the application with unnecessary detail. It should be appreciated that in the development of any such actual implementation, numerous implementation-specific decisions must be made in order to achieve the developer's specific goals, such as compliance with system- and business-related constraints, which will vary from one implementation to another. Moreover, it should be appreciated that such a development effort might be complex and time-consuming, but would nevertheless be a routine undertaking of design, fabrication, and manufacture for those of ordinary skill having the benefit of this disclosure.

[0020] It should also be noted that, in the specification, only the device structures and / or processing steps closely related to the scheme according to the present application are shown in the drawings, and other details not closely related to the present application are omitted.

[0021] It should be noted that, unless otherwise defined, technical or scientific terms used in the present application should be understood as their common meaning to those skilled in the art to which the present application pertains.

[0022] In the description of the embodiments of the present application, the meaning of "a plurality of" is at least two, for example, two, three, etc., unless otherwise explicitly and specifically limited.

[0023] Under the action of multi-physical field coupling, the nonlinear large amplitude vibration of the plate and shell structure is increased, and the problems such as nonlinear resonance generated thereby will affect the normal operation of the equipment, and even cause danger to the operating personnel. In the related art, the dynamic performance analysis of a high-dimensional nonlinear system is usually solved by relying on a traditional approximate analytic method or simulation. For the traditional approximate analytic method, in the solving process, a function form given in advance is needed to be relied on, and only a low-dimensional system can be solved. For finite element simulation, in the solving process, a long time is consumed, the calculation amount is large, and complex chaos, bifurcation and other phenomena cannot be predicted, so the strong nonlinear vibration behavior of the plate and shell structure cannot be accurately analyzed.

[0024] In view of the above technical problems, the embodiment of the present application provides a method suitable for determining the nonlinear response of a plate and shell structure to external excitation of multi-physical field coupling, Figure 1 is a flowchart of the method suitable for determining the nonlinear response of a plate and shell structure to external excitation of multi-physical field coupling according to the embodiment of the present application, as Figure 1 shown, which includes the following steps S10 to S60.

[0025] S10: Determine the relationship between the stress and the strain of the plate and shell structure.

[0026] S20: According to the relationship determined in step S10, determine the relationship between the strain energy, kinetic energy and external force work of the plate and shell structure.

[0027] S30: According to the multi-physical field applied to the plate and shell structure, determine the external excitation.

[0028] S40: According to the external excitation determined in step S30, update the relationship between the strain energy, kinetic energy and external force work of the plate and shell structure.

[0029] S50: Transform the relationship determined in step S40 by using the Hamiltonian variation principle.

[0030] S60: According to the transformed relationship obtained in step S50, determine the response of the plate and shell structure to the external excitation.

[0031] The method provided by the embodiment of the application can determine the relationship among the strain energy, kinetic energy and external force work of the plate-shell structure through the relationship between the stress and strain of the plate-shell structure, and then update the determined relationship among the strain energy, kinetic energy and external force work according to the external excitation and transform the relationship by using the Hamilton variation principle, so as to determine the response of the plate-shell structure to the external excitation. The method can more accurately determine the unstable region in the plate-shell structure, thereby facilitating the research on the strong nonlinear vibration characteristics of the plate-shell structure, so as to obtain the nonlinear dynamic response result of the plate-shell structure under the multi-physical field coupling with high precision and strong universality, and further ensure the safety of the plate-shell structure after the structure vibration reduction design is applied to the field of aerospace and the like.

[0032] In some embodiments, the plate-shell structure can be a flat plate or a plate structure with curvature, for example, a cylindrical shell structure.

[0033] In some embodiments, the multi-physical field includes but is not limited to an electric field, a magnetic field or a varying temperature field.

[0034] In some embodiments, in the step S10, the stress and the strain satisfy the following expressions (1) to (3):

[0035] (1);

[0036] (2);

[0037] (3).

[0038] wherein, represents a stress component; represents an elastic constant; represents a strain component; represents a thermal modulus constant; represents a temperature change; represents a piezoelectric constant; represents a magnetic field component; represents a piezoelectric constant; represents an electric field component; represents an electric displacement component; represents a piezoelectric constant; represents a pyroelectric constant; represents a magnetoelectric constant; represents a dielectric constant; represents a magnetic induction component; represents a piezomagnetic constant; represents a pyromagnetic constant; represents a magnetic constant; represents a magnetoelectric constant.

[0039] In some embodiments, when external excitation acts on the shell structure, the stress field at the action point depends not only on the strain of the point, but also on the strain of all other external force fields.

[0040] In some embodiments, in the step S40, taking the cylindrical shell as an example, the following relationships between strain energy, kinetic energy and work done by external force satisfy the following expressions (4) to (6):

[0041] (4);

[0042] (5);

[0043] (6).

[0044] wherein, represents kinetic energy; represents the length of the shell structure; represents an inertia term; represents the coordinate in the length direction of the shell structure; represents the coordinate in the circumferential direction of the shell structure; represents the coordinate in the thickness direction of the shell structure; represents the displacement of the mid-plane point of the shell structure in the direction; represents the displacement of the shell structure in the direction perpendicular to the direction and the direction; represents the displacement of the mid-plane point of the shell structure in the direction; represents a partial derivative symbol; represents the radius of the shell structure; represents potential energy; represents the thickness of the shell structure; represents the stress in the direction; represents the strain in the direction; represents the stress in the direction; represents the strain in the direction; represents the shear stress in the direction; represents the shear strain in the direction; represents the electric displacement in the direction; represents the electric displacement in the direction; represents the electric displacement in the direction; represents an electric field in the direction represents an electric field in the direction represents an electric field in the direction represents a magnetic induction in the direction represents a magnetic induction in the direction represents a magnetic induction in the direction represents a magnetic field in the direction represents a magnetic field in the direction represents a magnetic field in the direction represents the area of a section of a plate shell structure represents a force in the direction represents a force in the direction represents a force in the direction represents a middle surface strain in the direction represents a middle surface strain in the direction represents a middle surface strain in the direction represents a bending moment in the direction represents a bending moment in the direction represents a bending moment in the direction represents a curvature and a torsion in the direction represents a curvature and a torsion in the direction represents a curvature and a torsion in the direction represents the work of an external force represents the force of an electric field in the direction represents the force of an electric field in the direction represents the force of an electric field in the direction represents the force of an electric field in the direction , represents different piezoelectric constants represents an initial voltage represents the force of a temperature field in the direction represents the force of a temperature field in the direction represents the force of a temperature field in the direction represents the force of a temperature field in the direction represents temperature change, represents different parameters; represents the force generated by the magnetic field in the direction; represents the force generated by the magnetic field in the direction; represents determined different parameters; represents initial magnetic potential; represents external mechanical point load, represents external excitation amplitude, represents the frequency of external excitation, represents time, represents function relationship, represents the coordinate value of external force, represents the coordinate value of external force.

[0045] The method provided by the embodiment of the application determines the relationship among the strain energy, kinetic energy and work done by the external force of the plate shell structure through the above expressions (4) to (6), and the calculation result is reliable.

[0046] Figure 2 is a schematic diagram of a plate shell structure under multi-physical field coupling according to the embodiment of the application, Figure 3 is a top view of the plate shell structure shown in Figure 2 , as shown in Figure 2 and Figure 3 , wherein represents a spatial coordinate system in which the plate shell structure is located, represents a coordinate in the length direction of the plate shell structure, represents a coordinate in the thickness direction of the plate shell structure, represents a coordinate of the plate shell structure in a direction perpendicular to the direction and the direction, the plate shell structure is in a multi-physical field coupling environment with an electric field , a magnetic field and a temperature field that changes, wherein the electric field changes between and , and the temperature field changes from an initial temperature to .

[0047] ​​​​Under the satisfaction of Maxwell equations, the distribution of the magnetic potential and the electric potential along the thickness direction of the plate shell structure satisfies the following expressions (7) and (8):

[0048] (7);

[0049] (8).

[0050] wherein, represents the distribution of the magnetic potential along the thickness direction of the plate shell structure; represents the coordinate in the length direction of the plate shell structure; represents the coordinate in the circumferential direction of the plate shell structure; represents the coordinate in the thickness direction of the plate shell structure; represents time; represents the distribution of the electric potential along the thickness direction of the plate shell structure; represents a set parameter, , represents the thickness of the plate shell structure; represents the change of the magnetic potential in space; represents the change of the electric potential in space; represents the initial external magnetic potential; represents the initial external electric potential.

[0051] The electric field component and the magnetic field component in the coupled multi-physical field can be determined by the above expressions (7) and (8).

[0052] In some embodiments, the electric field component satisfies the following expressions (9) to (11):

[0053] (9);

[0054] (10);

[0055] (11).

[0056] wherein, represents the coordinate in the length direction of the plate shell structure; represents the coordinate in the circumferential direction of the plate shell structure; represents the coordinate in the thickness direction of the plate shell structure; represents the electric field component in the direction; represents the distribution of the electric potential along the thickness direction of the plate shell structure; represents a set parameter, , represents the thickness of the plate shell structure; represents the radius of the plate shell structure; represents a change in electric potential in space; represents a component of electric field in the direction; represents a component of electric field in the direction.

[0057] In some embodiments, a component of magnetic field satisfies the following expressions (12) to (14):

[0058] (12);

[0059] (13);

[0060] (14).

[0061] wherein, represents a coordinate in the length direction of the shell structure; represents a coordinate in the circumferential direction of the shell structure; represents a coordinate in the thickness direction of the shell structure; represents a component of magnetic field in the direction; represents a distribution of magnetic potential along the thickness direction of the shell structure; represents a set parameter, , represents a thickness of the shell structure; represents a radius of the shell structure; represents a change in magnetic potential in space; represents a component of magnetic field in the direction; represents a component of magnetic field in the direction.

[0062] In some embodiments, a component of strain at a position where an arbitrary point on the shell structure is located satisfies the following expression (15):

[0063] (15).

[0064] wherein, represents a strain in the direction; represents a strain in the direction; represents a shear strain in the direction; represents a strain in the mid-plane direction, represents a strain in the mid-plane direction; represents a strain in the mid-plane Strain in the direction of movement; , Indicates different mid-surface curvatures; This indicates a mid-surface twist.

[0065] In some embodiments, based on the constitutive relation of the plate and shell structure under the action of an electromagnetic field, the following expressions (16) to (18) can be obtained:

[0066] (16);

[0067] (17);

[0068] (18).

[0069] in, express Stress in the direction; express Strain in the direction of movement; express Stress in the direction; express Strain in the direction of movement; express Shear stress in the direction; express Shear strain in the direction; Indicates temperature change; express Magnetic induction in direction; express Magnetic induction in direction; express Magnetic induction in direction; express Electric field in the direction; express Electric field in the direction; express Electric field in the direction; express Electric displacement in the direction; express Electric displacement in the direction; express Electric displacement in the direction; express Magnetic field direction; express Magnetic field direction; express Magnetic field direction; , , , , , , , and They respectively satisfy the following expressions:

[0070]

[0071]

[0072]

[0073]

[0074]

[0075] in, Represents the elastic coefficient; Indicates piezoelectric coefficient, Indicates dielectric constant, Indicates the piezomagnetic coefficient, Indicates magnetoelectric coefficient, Indicates magnetic coefficient, Indicates the coefficient of thermal modulus, Indicates the thermoelectric coefficient; This represents the thermomagnetic coefficient. It is understood that in such an embodiment, , These represent the same or different ordinal numbers. Physical quantities with different ordinal numbers correspond to values ​​under different conditions. For example, , This indicates different piezoelectric coefficients.

[0076] In some embodiments, for a changing temperature field, the temperature typically varies nonlinearly along the thickness direction of the shell structure. In the absence of a heat source, the one-dimensional steady-state heat conduction equation satisfies the following expression (19):

[0077] (19).

[0078] Where d represents the derivative symbol; Represents coordinate values; Indicates thermal conductivity; Indicates temperature; the boundary conditions of the plate and shell structure should satisfy the following expression (20):

[0079] (20).

[0080] in, Indicates the thickness of the plate and shell structure A changing temperature function; Indicates the external temperature; Indicates the internal temperature; This indicates the thickness of the plate / shell structure.

[0081] In some embodiments, the power law index is used. Thermal conductivity is expressed in the form of The non-uniformity satisfies the following expression (21):

[0082] (twenty one).

[0083] By solving expression (21), the relationship between temperature change and the thickness of the plate shell structure can be obtained as follows: expression (22):

[0084] (twenty two).

[0085] in, Indicates the thickness of the plate and shell structure A changing temperature function; Indicates the internal temperature; Indicates the thickness of the plate / shell structure; Indicates temperature change; Indicates the external thermal conductivity; Indicates the internal thermal conductivity; Indicates the power-law exponent; This represents the given parameter; its specific expression is shown in formula (23), and the parameter... and temperature change The following expressions (23) and (24) are satisfied respectively:

[0086] (twenty three);

[0087] (twenty four).

[0088] Figure 4a This is a nonlinear amplitude-frequency response curve of a plate and shell structure under multiphysics coupling with an initial external potential of 100V, according to an embodiment of this application, as a function of the electric field. Figure 4b This is a nonlinear amplitude-frequency response curve of a plate-shell structure under multiphysics coupling with an initial external potential of 200V, according to an embodiment of this application, as shown in the figure. Figure 4a and 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 dashed line represents the unstable solution determined by the method of the embodiment of this application, and the solid line represents the stable solution.

[0089] Figure 5ais a nonlinear amplitude-frequency response curve of a plate-shell structure according to an embodiment of the present application, which changes with a magnetic field under multi-physical field coupling when an initial external magnetic potential is 50 A, as shown in FIG. 5A, Figure 5b is a nonlinear amplitude-frequency response curve of a plate-shell structure according to an embodiment of the present application, which changes with a magnetic field under multi-physical field coupling when an initial external magnetic potential is 300 A, as shown in FIG. 5B, Figure 5a and Figure 5b , wherein the abscissa represents a dimensionless excitation frequency, the ordinate represents a dimensionless amplitude of the plate-shell structure, the dashed line is an unstable solution determined by using the method according to an embodiment of the present application, and the solid line is a stable solution.

[0090] Figure 6a is a nonlinear amplitude-frequency response curve of a plate-shell structure according to an embodiment of the present application, which changes with a temperature under multi-physical field coupling when a temperature change is 50℃, as shown in FIG. 6A, Figure 6b is a nonlinear amplitude-frequency response curve of a plate-shell structure according to an embodiment of the present application, which changes with a temperature under multi-physical field coupling when a temperature change is 100℃, as shown in FIG. 6B, Figure 6a and Figure 6b , wherein the abscissa represents a dimensionless excitation frequency, the ordinate represents a dimensionless amplitude of the plate-shell structure, the dashed line is an unstable solution determined by using the method according to an embodiment of the present application, and the solid line is a stable solution.

[0091] Referring to Figures 4a to 6b , the nonlinear response of the plate-shell structure to the external excitation of the multi-physical field coupling can be determined by using the method according to an embodiment of the present application to obtain a continuous curve, and further to determine the change rule of the nonlinear response of the plate-shell structure to the external excitation of the multi-physical field coupling.

[0092] In some embodiments, in the step S50, the transformed relationship determined in the step S40 is further transformed by using a nonlinear fractional order damping model, which is beneficial to the research and analysis of the damping nonlinearity of the plate-shell structure.

[0093] In some embodiments, the nonlinear fractional order damping model satisfies the following expression (25):

[0094] (25).

[0095] wherein, represents a stress; represents a Young's modulus of the plate-shell structure; represents a strain; represents a constant related to the characteristics of the material or structure; represents a relaxation time, and , represents a damping coefficient, which is used to reflect the dissipation characteristics of the material; represents time; represents order fractional derivative.

[0096] In such embodiments, The order fractional derivative satisfies the following expression (26):

[0097] (26).

[0098] wherein, denotes an order fractional derivative operator, and , ; denotes time; denotes a function related to time; denotes a frequency component, is an integer for indicating an index of the frequency component; denotes temperature; denotes an imaginary unit.

[0099] It can be determined by the above expression (26) that: Further, The order fractional derivative also satisfies the following expressions (27) to (29):

[0100] (27);

[0101] (28);

[0102] (29).

[0103] In such embodiments, when = 1, a classical first order derivative can be obtained; when = 0, the original relationship can be obtained.

[0104] In some embodiments, the relationship obtained in the step S50 satisfies the following expressions (30) to (34):

[0105] (30);

[0106] (31);

[0107] (32);

[0108] (33);

[0109] (34).

[0110] wherein, Represents the coordinates along the length of the plate and shell structure; Represents the coordinates in the circumferential direction of the plate and shell structure; Represents the coordinates along the thickness direction of the plate / shell structure; Indicates the plate and shell structure along Displacement of the midplane point in the direction; Indicates the plate and shell structure along with direction and Displacement in a direction perpendicular to the direction; Indicates the plate and shell structure along Displacement of the midplane point in the direction; express Force in the direction; Indicates the radius of the plate and shell structure; express Force in the direction; Represents the inertial term; Indicates time; express Bending moment in the direction; express Bending moment in the direction; express Force in the direction; express Bending moment in the direction; Indicates the electric field along The force generated by the direction, Indicates the electric field along The force generated by the direction, , Representing different piezoelectric constants, Indicates the initial voltage; Indicates the temperature field at The force generated in the direction, Indicates the temperature field at The force generated in the direction, , Indicates temperature change, , Indicate different parameters; Indicates structural damping; Indicates external mechanical point load. , Indicates the amplitude of external incentives. Indicates the frequency of external stimuli. Represents a functional relationship. Indicates the location of external forces Coordinate values Indicates the location of external forces Coordinate values; Indicates the thickness of the plate / shell structure; represents an electric displacement in the direction; represents a given symbol, ; represents an electric displacement in the direction; represents an electric displacement in the direction; represents a magnetic induction in the direction; represents a magnetic induction in the direction; represents a magnetic induction in the direction.

[0111] In some embodiments, based on the simply supported boundary condition, the displacement field satisfies the following expressions (35) to (39):

[0112] (35);

[0113] (36);

[0114] (37);

[0115] (38);

[0116] (39).

[0117] wherein, represents a middle surface displacement in the direction; represents a middle surface displacement in the direction; represents a middle surface displacement in the direction; represents an electric displacement; represents a magnetic displacement; represents a circumferential truncation coefficient; represents an axial truncation coefficient; represents an axial half-wave number; represents a circumferential wave number; represents an excitation mode subscript; represents a concomitant mode subscript; represents an established excitation mode function in the direction; represents an established concomitant mode function in the direction; represents an established axially symmetric mode function in the direction; represents an established excitation mode function in the direction; representing the set-up of the excitation modal functions in the direction; representing the set-up of the excitation modal functions in the direction; representing the set-up of the excitation modal functions in the direction; representing the set-up of the axisymmetric modal functions in the direction; representing the set-up of the excitation modal functions in the direction of the electric field; representing the set-up of the excitation modal functions in the direction of the magnetic field; representing the set-up of the axisymmetric modal functions in the direction of the magnetic field.

[0118] In some embodiments, in the step S60, the transformed relationship is converted from the partial differential equation set to the ordinary differential equation set by using the multi-modal Galerkin method, which is of high calculation efficiency and high reliability.

[0119] In some embodiments, in the step S60, the following steps are further included: S61: establishing the mode shape function of the plate and shell structure under the external excitation; S62: converting the transformed relationship obtained in the step S50 according to the mode shape function; and S63: weighting and integrating the relationship obtained in the step S62 to obtain the ordinary differential equation set.

[0120] The method provided by the embodiments of the present application converts the transformed relationship by using the mode shape function and performs weighting and integration, so that the ordinary differential equation set can be obtained, thereby improving the solving accuracy of the response of the plate and shell structure to the external excitation.

[0121] In some embodiments, the transformed nonlinear partial differential equation set is converted into the multi-modal coupled ordinary differential equation set by using the weighted multi-modal Galerkin method. That is, the original equation is sequentially weighted by using appropriate functions and is integrated to the middle surface of the plate and shell structure, wherein the weighting function satisfies the following expressions (40) to (44):

[0122] (40);

[0123] (41);

[0124] (42);

[0125] (43);

[0126] (44).

[0127] wherein, represents a weighting function in the direction of; represents a variable symbol for distinguishing different weighting functions; represents a weighting function in the direction of; represents a weighting function in the direction of; represents a weighting function in the direction of; represents a weighting function in the direction of; represents an axial half-wave number; represents a circumferential wave number; represents a length of the plate-shell structure.

[0128] In some embodiments, in the step S61, the mode shape functions include a driving mode, a companion mode, and an axisymmetric mode, so as to more accurately simulate the response of the plate-shell structure under external excitation.

[0129] In some embodiments, in the step S63, the step further includes the following steps: S631: converting the ordinary differential equation set from a non-autonomous system to an autonomous system; and S632: solving the autonomous system.

[0130] Embodiments of the present application provide a method for converting an ordinary differential equation set from a non-autonomous system to an autonomous system, so as to solve the ordinary differential equation set.

[0131] In some embodiments, in the step S631, the ordinary differential equation is converted from a non-autonomous system to an autonomous system by using the following expression (45):

[0132] (45).

[0133] wherein, , respectively represent different periodic orbit radii; , respectively represent different periodic orbit functions; represents a system natural frequency.

[0134] In some embodiments, the solutions of the non-autonomous system can be regarded as periodic orbit motions along a circle with a radius of 1, and these solutions can cross the circle with a radius of 1 at a frequency of In the initial condition setting stage, the and can be set as thereby determining ; then, the exact solution of the next step (step ) is predicted from the solution of the current step (step ), whose solution satisfies the following expressions (46) and (47):

[0135] (46);

[0136] (47).

[0137] wherein, represents the current step or step ; represents the predicted solution; represents the current solution; represents the predicted solution increment; represents the corrected solution, that is, in the iteration process, the current solution is corrected in order to more accurately approach the true solution; represents the step number of iteration or the current state index; represents the predicted frequency; represents the current frequency; represents the predicted frequency increment; represents the corrected frequency, that is, represents the adjusted frequency value in each step of iteration.

[0138] In some embodiments, in the S632 step, the autonomous system is solved by using the quasi-arclength continuation method to determine the response of the plate shell structure to external excitation, which is conducive to improving the accuracy of the determined nonlinear dynamic frequency characteristics and variation law of the plate shell structure.

[0139] The implementation process of the quasi-arclength continuation method will be described below through an embodiment.

[0140] The initial value of the initial step length is determined, and the frequency point of the initial step can be selected in the area where the nonlinearity is not strong.

[0141] The predicted step length can provide an initial value for the next frequency point, and in the arclength continuation method, the predicted initial value satisfies the following expressions (48) and (49):

[0142] (48);

[0143] (49).

[0144] wherein, represents the current step length, represents the next step length; represents the predicted initial solution, that is, the system state or displacement under the current step length or frequency point; represents the current solution; represents a predicted solution increment, which is a change amount between the current solution and the predicted solution; represents a predicted initial frequency, which is a next frequency value predicted based on the current solution or state in the current step; represents a current frequency; represents a predicted frequency increment, which is a change amount between the predicted frequency and the current frequency.

[0145] The predicted initial value is generally in the tangential direction, and for the frequency-domain algebraic equation calculated by the harmonic balance method, the predicted solution increment and the predicted frequency increment satisfy the following expression (50):

[0146] (50).

[0147] wherein, represents a function with respect to the system state and the frequency.

[0148] In the above expression (50), it can be limited by adding a step condition, which satisfies the following expression (51):

[0149] (51).

[0150] wherein, represents an arc length increment, which is used to describe a change amount along the arc length path in the parameter space; the symbol || || represents a norm of a vector.

[0151] By combining the above expressions (50) and (51), the predicted solution increment and the predicted frequency increment satisfy the following expressions (52) and (53), respectively:

[0152] (52);

[0153] (53).

[0154] Since the tangential direction is two, when determining the predicted solution increment and the predicted frequency increment , the direction also needs to be determined to ensure that the continuous direction can always move forward, and the judgment process satisfies the following expression (54):

[0155] (54).

[0156] wherein, represents a transpose symbol.

[0157] In some embodiments, the predicted solution obtained along the tangent direction may deviate from the actual exact solution. By correcting the predicted solution once or multiple times, an exact solution that satisfies the nonlinear algebraic equation can be obtained.

[0158] In some embodiments, the nonlinear characteristics of the plate and shell structure are more pronounced at certain excitation frequencies, making it difficult to obtain the accurate corrected solution within a limited number of iterations when using the arc-length extension method 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, thereby enabling the rapid acquisition of the accurate solution on the amplitude-frequency curve in continuous directions.

[0159] If a deviation exists between the predicted solution and the exact solution due to a large step size, the step size needs to be reduced and the current step size re-estimated and corrected until an accurate solution is obtained. The current step size and the previous step size satisfy the following expression (55):

[0160] (55).

[0161] in, Indicates the current step size; Indicates the previous step was longer; This indicates the minimum step size set.

[0162] 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 should be 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):

[0163] (56).

[0164] in, Indicates the step size in the arc-length extension process; This indicates the size of the current step, which is the first step in the arc length extension process. The step size used in the step; This indicates the maximum step size set.

[0165] In some embodiments, the stability of the response obtained in step S60 is determined using multivariate Floquet theory, which has high reliability.

[0166] In some embodiments, for nonlinear systems that satisfy the system equilibrium equations, perturbations are added to the equations. The stability of the solution can be studied. The perturbation equation satisfies the following expression (57):

[0167] (57).

[0168] wherein, denotes the generalized force; denotes the second order derivative of the generalized coordinate; denotes the first order derivative of the generalized coordinate; denotes the generalized coordinate.

[0169] Based on the stability of the steady-state solution, the stability of the above expression (57) can be determined.

[0170] For the embodiments of the present application, it also needs to be explained that, in the case of no conflict, the embodiments and the features in the embodiments of the present application can be combined with each other to obtain new embodiments.

[0171] The above is only a specific implementation of the present application, but the protection scope of the present application is not limited thereto, and the protection scope of the present application shall be subject to the protection scope of the claims.

Claims

1. A method for determining the nonlinear response of a plate and shell structure to external excitations coupled with multiphysics, characterized in that, Includes the following steps: S10: Determine the relationship between stress and strain in the plate and shell structure; S20: Based on the relationship determined in step S10, determine the relationship between the strain energy, kinetic energy, and work done by external forces of the plate and shell structure; S30: Determine the external excitation based on the multiphysics field applied to the shell structure; S40: Based on the external excitation determined in step S30, update and determine the relationship between the strain energy, kinetic energy and work done by the external force of the plate and shell structure; S50: Transform the relationship determined in step S40 using Hamilton's variational principle; S60: Based on the transformed relationship obtained in step S50, determine the response of the plate and shell structure to external excitation, thereby identifying the unstable region in the plate and shell structure and obtaining the nonlinear dynamic response result of the plate and shell structure under the multi-physics coupling. In step S10, the stress and the strain satisfy the following expression: ； ； ； in, Represents stress components; Represents the elastic constant; Represents strain components; Represents the thermal modulus constant; Indicates temperature change; Represents the piezoelectric constant; Indicates the magnetic field component; Represents the piezoelectric constant; Indicates the electric field components; Represents the electric displacement component; Represents the piezoelectric constant; Represents the thermoelectric constant; Represents the magnetoelectric constant; Indicates the dielectric constant; Indicates the magnetic induction component; Indicates the piezomagnetic constant; Represents the thermomagnetic constant; Represents the magnetic constant; Represents the magnetoelectric constant; The multiphysics fields include: electric field, magnetic field, and changing temperature field.

2. The method according to claim 1, characterized in that, In step S60, the transformed relation is transformed using the multimodal Galerkin method, converting it from a system of partial differential equations into a system of ordinary differential equations.

3. The method according to claim 2, characterized in that, Step S60 also includes the following steps: S61: Establish the mode shape function of the plate and shell structure under the external excitation; S62: Based on the mode shape function, transform the transformed relationship obtained in step S50; S63: Take the weighted integral of the relation obtained in step S62 to obtain the set of ordinary differential equations.

4. The method according to claim 3, characterized in that, In step S61, the mode shape function includes driving mode, accompanying mode, and axisymmetric mode.

5. The method according to claim 4, characterized in that, Step S63 also includes the following steps: S631: Transform the system of ordinary differential equations 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 the pseudo-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 multivariate Flokai theory.

8. The method according to claim 1, characterized in that, Step S50 also includes transforming the relationship determined in step S40 using a nonlinear fractional damping model.

9. The method according to claim 1, characterized in that, In step S40, the relationship between the kinetic energy, strain energy, and work done by external forces of the plate and shell structure satisfies the following expression: ； ； ； in, Indicates kinetic energy; This indicates the length of the plate shell structure; Represents the inertial term; This represents the coordinates along the length of the plate shell structure; This represents the coordinates in the circumferential direction of the plate shell structure; This represents the coordinates along the thickness direction of the plate shell structure; Indicates that the plate shell structure is along the Displacement of the midplane point in the direction; This indicates that the plate shell structure is along the same line as the... Direction and the stated Displacement in a direction perpendicular to the direction; Indicates that the plate shell structure is along the Displacement of the midplane point in the direction; Indicates the partial derivative sign; Indicates the radius of the plate shell structure; Represents potential energy; This indicates the thickness of the plate shell structure; Indicates the Stress in the direction; Indicates the Strain in the direction of movement; express Stress in the direction; Indicates the Strain in the direction of movement; express Shear stress in the direction; Indicates the Shear strain in the direction; Indicates the Electric displacement in the direction; Indicates the Electric displacement in the direction; Indicates the Electric displacement in the direction; Indicates the Electric field in the direction; Indicates the Electric field in the direction; express Electric field in the direction; Indicates the Magnetic induction in direction; Indicates the Magnetic induction in direction; Indicates the Magnetic induction in direction; Indicates the Magnetic field direction; Indicates the Magnetic field direction; Indicates the Magnetic field direction; This represents the area of ​​the cross-section of the plate / shell structure. Indicates the Force in the direction; Indicates the Force in the direction; Indicates the Force in the direction; 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 the direction; Indicates the Bending moment in the direction; Indicates the Bending moment in the direction; Indicates the Curvature and torsion in direction; Indicates the Curvature and torsion in direction; Indicates the Curvature and torsion in direction; This indicates that work has been done by an external force. Indicates the electric field along the described The force generated by the direction, Indicates the electric field along the described The force generated by the direction, , Representing different piezoelectric constants, Indicates the initial voltage; Indicates the temperature field in the The force generated in the direction, Indicates the temperature field in the The force generated in the direction, , Indicates temperature change, , Indicate different parameters; Indicates the magnetic field is in The force generated in the direction; Indicates the magnetic field is in The force generated in the direction; , Indicates distinct parameters; Indicates the initial magnetic potential; Indicates external mechanical point load. , Indicates the amplitude of external incentives. Indicates the frequency of external stimuli. Indicates time, Represents a functional relationship. Indicates the location of external forces Coordinate values Indicates the location of external forces Coordinate values.

10. The method according to claim 1, characterized in that, The relation obtained in step S50 satisfies the following expression: ； ； ； ； ； in, This represents the coordinates along the length of the plate shell structure; This represents the coordinates in the circumferential direction of the plate shell structure; This represents the coordinates along the thickness direction of the plate shell structure; Indicates that the plate shell structure is along the Displacement of the midplane point in the direction; This indicates that the plate shell structure is along the same line as the... Direction and the stated Displacement in a direction perpendicular to the direction; The plate shell structure is indicated along Displacement of the midplane point in the direction; express Force in the direction; Indicates the radius of the plate shell structure; express Force in the direction; Represents the inertial term; Indicates time; Indicates the Bending moment in the direction; Indicates the Bending moment in the direction; Indicates the Force in the direction; Indicates the Bending moment in the direction; Indicates the electric field along the described The force generated by the direction, Indicates the electric field along the described The force generated by the direction, , Representing different piezoelectric constants, Indicates the initial voltage; Indicates the temperature field in the The force generated in the direction, Indicates the temperature field in the The force generated in the direction, , Indicates temperature change, , Indicate different parameters; Indicates structural damping; Indicates external mechanical point load. , Indicates the amplitude of external incentives. Indicates the frequency of external stimuli. Represents a functional relationship. Indicates the location of external forces Coordinate values Indicates the location of external forces Coordinate values; This indicates the thickness of the plate shell structure; Indicates the Electric displacement in the direction; Represents a given symbol, ; Indicates the Electric displacement in the direction; Indicates the Electric displacement in the direction; Indicates the Magnetic induction in direction; Indicates the Magnetic induction in direction; Indicates the Magnetic induction in direction.

11. The method according to claim 5, characterized in that, In step S631, the ordinary differential equation is transformed from the non-autonomous system to the autonomous system using the following expression: ； in, , These represent the radii of different periodic orbits; , These represent different periodic orbital functions; This indicates the system's inherent frequency.

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