CN116384205B - Periodic orbit structure band gap calculation method based on energy method and Gaussian elimination method - Google Patents
Periodic orbit structure band gap calculation method based on energy method and Gaussian elimination method Download PDFInfo
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Abstract
The application discloses a periodic track structure band gap calculation method based on an energy method and a Gaussian elimination method, which separates boundary conditions and displacement shape functions under the framework of the energy method, and the displacement shape functions of a track structure can be represented by any series at the moment; the total energy functional of the periodic orbit structure is obtained, the boundary condition is processed by using a Gaussian elimination method, and the total energy functional variation is solved to obtain a motion equation of the orbit structure; scanning the wave number of the first irreducible Brillouin zone to obtain a dispersion curve of the orbit structure, and further obtaining the band gap characteristic of the orbit structure. Compared with the traditional energy method, the displacement shape function is simple in structure, convenient in boundary processing, good in convergence and high in calculation efficiency, and the periodic track structure can be calculated conveniently.
Description
Technical Field
The application belongs to the technical field of transportation engineering, and particularly relates to a periodic track structure band gap calculation method based on an energy method and a Gaussian elimination method.
Background
The track structure has a strong periodicity due to the way it is assembled and can therefore be considered as a kind of artificial periodic structure. The vibration wave has a special filter characteristic in the periodic structure, which is called a band gap characteristic. Studying the bandgap characteristics of the track structure helps to guide the vibration damping design of the track structure.
The band gap calculation method of the periodic structure comprises a numerical method, an analysis method and a semi-analysis method.
The numerical method is most typically a finite element method. The method involves the discrete structure into a plurality of cells, which are related by interpolation functions. The method has good geometric applicability and is widely applied to band gap solving of a periodic structure. Spectral finite elements, wave finite elements, and the like have been developed on the basis of finite elements. According to the method, grids are required to be divided again even if the model is slightly changed during solving, and the calculation efficiency during geometric parameter analysis and optimization is seriously affected.
Common analytical and semi-analytical methods include plane wave expansion, transmission matrix, multiple scattering, concentrated mass, and rotational radiation. These methods are well used in the respective fields of use, but are not satisfactory in dealing with the coupling problem. The energy method can convert the boundary value problem of the differential equation into the extremum problem of the functional, which has obvious advantages in the aspect of treating the coupling problem of the combined structure, thereby being beneficial to treating the plate-type ballastless track structure. However, conventional energy methods, such as the rayleigh law, require that the constructed displacement shape function satisfy the Bloch theorem, as well as other boundary conditions. Mathematically, the construction of the shape function is complex. Even if the shape function is constructed, the reconstructed displacement shape function contains wave numbers, and the mass matrix and the rigidity matrix need to be repeatedly calculated every time the wave numbers are scanned, which necessarily results in low calculation efficiency.
Disclosure of Invention
The application aims to solve the problems that a shape function is difficult to construct and the calculation efficiency is low when the periodic orbit structure is calculated by an energy method, and provides a periodic orbit structure band gap calculation method based on the energy method and a Gaussian elimination method.
In order to solve the problems, the technical scheme of the application is as follows: the periodic orbit structure band gap calculation method based on the energy method and the Gaussian elimination method comprises the following steps:
step S1: separating the displacement shape function of the track structure cell from the boundary condition, and establishing a displacement function of the track structure cell;
step S2: obtaining kinetic energy and strain energy of cells of the track structure according to material parameters of the periodic track structure without considering boundary conditions; forming a total energy functional equation according to the cell energy of the track structure;
step S3: forming a constraint condition matrix by the periodic boundary of the steel rail and the right-to-right boundary of the track slab;
step S4: processing constraint condition matrixes by using a Gaussian elimination method, representing original unknown coefficients by an auxiliary matrix and linear irrelevant column vectors, and introducing the original unknown coefficients into a total energy functional equation;
step S5: dividing the total energy functional function according to the Lagrangian equation to obtain a motion equation of the periodic orbit structure;
step S6: and obtaining a dispersion curve of the periodic orbit structure by scanning the wave number of the first irreducible Brillouin zone, and obtaining the band gap characteristic of the periodic orbit structure according to the dispersion curve.
Further preferably, the rail structure cells comprise steel rails, rail plates, CA mortar layers and fasteners, and plate seams are arranged among the rail structure cells.
It is further preferred that the rail and the rail plate are calculated using two coordinate systems, wherein the rail coordinate system is x r O r y r ，x r Is the abscissa of the steel rail, y r Is the ordinate of the steel rail, O r The origin of the rail coordinate system is that the rail plate coordinate system is x s O s y s ，x s Is the abscissa of the track slab, y s Is the ordinate of the track slab, O s The origin of the coordinate system of the track plate; the vertical displacement of the rail and the track slab is expressed as:
；
in the formula ：the vertical vibration displacement of the steel rail at the moment t is shown; />The rotation angle displacement of the steel rail at the moment t is shown; />The bending vibration displacement of the track plate at the moment t is represented; />The angular displacement of the track plate in the x-axis direction at the moment t is shown; />The angular displacement of the track plate in the y-axis direction at the moment t is shown; superscript T is the symbol of matrix transposition;representing the 1 st, 2 nd, 3 rd, 4 th, 5 th time-dependent unknown coefficients,/->Respectively representing 1 st, 2 nd, 3 rd, 4 th and 5 th time-dependent unknown coefficient matrixes; />Represents the j-th displacement shape function in the steel rail coordinate system, f represents the displacement shape function matrix of the steel rail coordinate system,/and a method for preparing the displacement shape function matrix>Represents the j-th displacement shape function in the track plate coordinate system, and g represents the displacement shape function matrix of the track plate coordinate system.
Further preferably, the strain energy and kinetic energy of the rails in the rail structure cells are expressed as:
；
in the formula ,Erail Representing strain energy of the steel rail; u (U) rail Represents the kinetic energy of the steel rail;，/>representing an unknown coefficient matrix array vector; />Is a partial guide symbol; the superscript H denotes a conjugate transpose; the superscript ". Cndot." represents the derivative with respect to time, representing the unknown coefficient matrix column vector +.>For time of dayIs a derivative of (2); />Representing the vertical vibration displacement of the rail +.>Representing the derivative of the vertical vibration displacement of the rail with respect to time,/->Indicating the angular displacement of the rail->Representing the derivative of the angular displacement of the rail with respect to time; g r Represents the shear modulus of the rail; /> and />Respectively representing the density, the sectional area, the elastic modulus, the moment of inertia and the shearing coefficient of the steel rail; />Representing a mass matrix of the rail; />Representing a stiffness matrix of the rail; />Represents the length of the track slab in a track structure cell,/->Representing the panel seam width.
Further preferably, the strain energy and kinetic energy of the rail plates in the rail structure cells are expressed as:
；
in the formula ,representing strain energy of the track slab; />Representing half of the track plate width; />Representing the kinetic energy of the track plate; /> and />Representing the density, thickness, area, bending stiffness, poisson's ratio, shear modulus and shear coefficient, respectively, of the rail plate,/->Representing bending vibration displacement of the track plate; />Angular displacement in the x-axis direction of the track plate is represented; />Indicating the angular displacement of the track plate in the y-axis direction; />Representing the derivative of the bending vibration displacement of the track plate with respect to time, is->The derivative of the angular displacement of the track plate in the x-axis direction with respect to time,/>Representing the derivative of the angular displacement of the track plate in the y-axis direction with respect to time; />Representing a mass matrix of the track slab; />Representing the stiffness of the track slabsA matrix.
Further preferably, the elastic potential energy of the fastener and the CA mortar layer in the cells of the track structure is expressed as:
；
in the formula ,representing elastic potential energy of the fastener; />Numbering the fasteners, wherein N represents the number of the fasteners; />Representing the elastic potential energy of the CA mortar layer; />Representing fastener stiffness; />Representing half of the width of the panel slit,/->Representing fastener spacing; />Representing the distance of the rail from the free boundary of the track slab; />Represents the average rigidity of the CA mortar layer; />Representing a stiffness matrix of the fastener;representing the stiffness matrix of the CA mortar layer.
Further preferably, the total energy functional equation in the cells of the track structure is:
；
wherein ,representing the total energy functional.
Further preferably, in step S3, the cycle boundaries are as follows:
；
wherein i is a complex number unit; k is wave number; e represents a natural constant;
the positive boundary of the track slab is expressed as:；
the constraint condition matrix is:
；
in the formula ,representing a displacement shape function matrix when the rail coordinates are zero; />Indicating the coordinates of the rail as +.>A displacement shape function matrix; />；/>Representing +.>A displacement shape function of the track plate; />Representing a constraint matrix.
Further preferably, in step S4, the constraint matrix is processed by gaussian elimination to obtain a full order matrix in the constraint matrix, and the full order matrix is then obtainedSplit into two-part added form, rewritten into:
；
in the formula ,is a full order matrix in the constraint condition matrix; />For P is the unknown coefficient matrix vector +.>Unknown coefficient matrix of corresponding position in the model; />A matrix formed by the residual column vectors in the constraint condition matrix; />Vector of matrix of unknown coefficients for Q +.>Unknown coefficient matrix of corresponding position in the model;
will be and />Vertically arranged to obtain a new column vector +.>Then->Can be represented by the unknown coefficient matrix vectors ∈ ->Obtained by primary equal-line transformation of (1), namely:
；
in the formula Is an elementary transformation matrix; />Representing an identity matrix, unknown coefficient matrix vectors +.>By->Linear representation:
；
wherein ,representing the matrix vector of unknown coefficients>By->An auxiliary matrix of the linear representation,the total energy functional equation is rewritten as:
；
in the formula ,representation->Derivative with respect to time, < >>Representation->Conjugate transpose of->Representation->Conjugate transpose of->Representation->Is a conjugate transpose of (a).
Further preferably, in step S5, the Lagrangian equation is usedAnd (3) obtaining a motion equation of the periodic orbit structure by dividing the total energy functional II:
。
the application has the beneficial effects that: the application is based on the framework of an energy method, can simply and quickly represent the displacement field of the cell of the track structure (without considering boundary conditions) by decoupling the shape function from the boundary conditions, obtains the total energy functional equation of the cell of the track structure according to the energy method, can change the total energy functional by processing constraint conditions through a Gaussian elimination method, further obtains the motion equation of the periodic track structure, and obtains the band gap characteristic of the track structure by scanning the first irreducible Brillouin zone. Compared with the traditional energy method that the shape function is difficult to construct and the calculation efficiency is low, the method decouples the shape function from the boundary conditions, so that the use difficulty of the energy method is greatly reduced, and the calculation efficiency of the energy method is remarkably improved.
Drawings
The application will be described in further detail with reference to the accompanying drawings and detailed description.
FIG. 1 is a flow chart of steps of a periodic orbit structure band gap calculation method based on an energy method and a Gaussian elimination method according to the application.
Fig. 2 is a schematic structural view of a periodic track structure.
FIG. 3 is a diagram of a computational model of a periodic orbit structure according to an embodiment of the present application.
Fig. 4 is a graph of periodic track structure dispersion calculated by the method of the present application.
Fig. 5 is a graph of periodic orbital structure dispersion calculated by the finite element method.
In the figure, 10-steel rail, 20-plate slit, 30-track structure cell, 40-track plate, 50-CA mortar layer and 60-fastener.
Detailed Description
The present application will be described in further detail with reference to the drawings and examples, in order to make the objects, technical solutions and advantages of the present application more apparent. It should be understood that the specific embodiments described herein are for purposes of illustration only and are not intended to limit the scope of the application.
As shown in fig. 1, a periodic track structure band gap calculation method based on an energy method and a gaussian elimination method includes the following steps S1 to S6, and the implementation of each step is described in detail below.
Step S1: and separating the displacement shape function of the track structure cell from the boundary condition, and establishing the displacement function of the track structure cell.
The periodic structure shown in fig. 2 is a three-plate ballastless track (CRTS iii), the track can be simplified by adopting a ironwood Xin Ke (Timoshenko) beam, the track plate 40 is simplified by adopting a Mindlin plate, the track structure cell 30 comprises a steel rail 10, a track plate 40, a CA mortar layer 50, a fastener 60, and the track structureBetween the cells 30 there are plate slits 20. As shown in fig. 3, the rail 10 and the track slab 40 are calculated using two coordinate systems, wherein the rail coordinate system is，/>Is the horizontal coordinate of the steel rail->Is the ordinate of the rail>The origin of the rail coordinate system is the rail plate coordinate system +.>，/>Is the abscissa of the track slab,/->Is the ordinate of the track slab,/>Is the origin of the orbital plate coordinate system. Only the vertical vibration of the vibration wave is considered in this embodiment. According to the principle of energy method calculation, the vertical displacement of the rail 10 and the track plate 40 can be expressed as:
；
in the formula ：the vertical vibration displacement of the steel rail at the moment t is shown; />The rotation angle displacement of the steel rail at the moment t is shown;watch (watch)The bending vibration displacement of the track plate at the moment t is shown; />The angular displacement of the track plate in the x-axis direction at the moment t is shown; />The angular displacement of the track plate in the y-axis direction at the moment t is shown; superscript T is the symbol of matrix transposition; />、Respectively represent the 1 st, 2 nd, 3 rd, 4 th and 5 th unknown coefficients related to time,respectively representing 1 st, 2 nd, 3 rd, 4 th and 5 th time-dependent unknown coefficient matrixes; />Represents the j-th displacement shape function in the steel rail coordinate system, f represents the displacement shape function matrix of the steel rail coordinate system,/and a method for preparing the displacement shape function matrix>Represents the j-th displacement shape function in the track plate coordinate system, and g represents the displacement shape function matrix of the track plate coordinate system.
Step S2: obtaining kinetic energy and strain energy of cells of the track structure according to material parameters of the periodic track structure without considering boundary conditions; and forming a total energy functional equation according to the cell energy of the track structure.
The strain energy and kinetic energy of the rail can be expressed as:
；
in the formula ,representing strain energy of the steel rail; />Represents the kinetic energy of the steel rail; />，/>Representing an unknown coefficient matrix array vector; />Is a partial guide symbol; the superscript H denotes a conjugate transpose; the superscript ". Cndot." denotes the derivative with respect to time, e.gRepresenting the unknown coefficient matrix vectors +.>Derivative with respect to time; />Representing the vertical vibration displacement of the rail +.>Representing the derivative of the vertical vibration displacement of the rail with respect to time,/->Indicating the angular displacement of the rail->Representing the derivative of the angular displacement of the rail with respect to time; g r Represents the shear modulus of the rail; /> and />Respectively representing the density, the sectional area, the elastic modulus, the moment of inertia and the shearing coefficient of the steel rail; />Representing a mass matrix of the rail; />Representing a stiffness matrix of the rail; />Represents the length of the track slab in a track structure cell,/->Representing the panel seam width.
The strain and kinetic energy of the track slab can be expressed as:
；
in the formula ,representing strain energy of the track slab; />Representing half of the track plate width; />Representing the kinetic energy of the track plate; /> and />Representing the density, thickness, area, bending stiffness, poisson's ratio, shear modulus and shear coefficient, respectively, of the rail plate,/->Representing bending vibration displacement of the track plate; />Angular displacement in the x-axis direction of the track plate is represented; />Indicating the angular displacement of the track plate in the y-axis direction; />Representing the derivative of the bending vibration displacement of the track plate with respect to time, is->The derivative of the angular displacement of the track plate in the x-axis direction with respect to time,/>Representing the derivative of the angular displacement of the track plate in the y-axis direction with respect to time;representing a mass matrix of the track slab; />Representing the stiffness matrix of the track slab.
9 fasteners are arranged between the steel rail and the track plate, and a Cement Asphalt (CA) mortar layer at the bottom of the track plate is set to be uniformly distributed with surface springs. The elastic potential energy of the fastener and the CA mortar layer is expressed as:
；
in the formula ,representing elastic potential energy of the fastener; />Numbering the fasteners, wherein N represents the number of the fasteners; />Representing the elastic potential energy of the CA mortar layer; />Representing fastener stiffness; />Representing panel seamsHalf of the width>Representing fastener spacing; />Representing the distance of the rail from the free boundary of the track slab; />Represents the average rigidity of the CA mortar layer; />Representing a stiffness matrix of the fastener; />Representing the stiffness matrix of the CA mortar layer.
The total energy functional equation can be expressed as:
；
wherein ,a basic principle for the variation of unknown coefficients according to the variation principle is that the total energy functional is represented: the unknown coefficients should be independent. />Because of the existence of constraints, there must be a linear correlation coefficient. In order to be able to vary the total energy functional ii and thus to obtain the frequency ω of the periodic orbit structure, a boundary condition treatment is required.
Step S3: and forming a constraint condition matrix by the periodic boundary of the steel rail and the right-to-right boundary of the track slab.
The cycle boundaries in this embodiment need to satisfy the Bloch (Bloch) theorem as follows:
；
wherein i is a complex number unit; k is wave number; e represents a natural constant.
The positive boundary of the track slab is expressed as:。
the constraint condition matrix is:
；
in the formula ,representing a displacement shape function matrix when the rail coordinates are zero; />Indicating the coordinates of the rail as +.>A displacement shape function matrix; />；/>Representing +.>A displacement shape function of the track plate; />Representing a constraint matrix.
Step S4: and processing the constraint condition matrix by using a Gaussian elimination method, representing the original unknown coefficient by an auxiliary matrix and a linear irrelevant column vector, and introducing the original unknown coefficient into a total energy functional equation.
The constraint condition matrix can be processed by a Gaussian elimination method to obtain a full order matrix in the constraint condition matrix. Will beSplitting intoThe form of the two-part addition, rewritten as:
；
in the formula ,is a full order matrix in the constraint condition matrix; />For P is the unknown coefficient matrix vector +.>Unknown coefficient matrix of corresponding position in the model; />A matrix formed by the residual column vectors in the constraint condition matrix; />Is->Vector in unknown coefficient matrix>Unknown coefficient matrix of the corresponding position in the model.
Will be and />Vertically arranged, a new column vector can be obtained>Then->Can be represented by the unknown coefficient matrix vectors ∈ ->Obtained by primary equal-line transformation of (1), namely:
；
in the formula Is an elementary transformation matrix; />Representing the identity matrix. Unknown coefficient matrix vectors +.>By->Linear representation:
；
wherein ,representing the matrix vector of unknown coefficients>By->An auxiliary matrix of the linear representation,. The total energy functional equation is rewritten as:
；
in the formula ,representation->Derivative with respect to time, < >>Representation->Conjugate transpose of->Representation->Conjugate transpose of->Representation->Is a conjugate transpose of (a).
Step S5: and obtaining a motion equation of the periodic orbit structure according to the Lagrangian equation and the total energy functional variation.
Are all unknown coefficients, according to Lagrange's equation +.>And (3) obtaining a motion equation of the periodic orbit structure by dividing the total energy functional II:
；
step S6: and obtaining a dispersion curve of the periodic orbit structure by scanning the wave number of the first irreducible Brillouin zone, and obtaining the band gap characteristic of the periodic orbit structure according to the dispersion curve.
TABLE 1 CRTS III slab track Material parameters
Parameters of the CRTS iii plate track in this example are shown in table 1, and in order to verify the accuracy of this example, the band gap characteristics of the track structure calculated by the method of the present application are compared with the band gap calculated by the finite element method. The comparison results are shown in fig. 4 and 5, and the results of the two are matched well. Three band gaps appear at the low-frequency position of the CRTS III type plate track, and the widths of the three band gaps are calculated to be 0-89.8 Hz, 106.5-115.7 Hz and 122.0-229.5 Hz by using an energy method. The numerical results obtained by the finite element method are 0-89.8 Hz, 106.5-115.8 Hz and 122.0 Hz-225.3 Hz.
The above examples are merely illustrative of the preferred embodiments of the present application and are not intended to limit the spirit and scope of the present application. Various modifications and improvements of the technical scheme of the present application will fall within the protection scope of the present application without departing from the design concept of the present application, and the technical content of the present application is fully described in the claims.
Claims (9)
1. The periodic orbit structure band gap calculation method based on the energy method and the Gaussian elimination method is characterized by comprising the following steps of:
step S1: separating the displacement shape function of the track structure cell from the boundary condition, and establishing a displacement function of the track structure cell;
step S2: obtaining kinetic energy and strain energy of cells of the track structure according to material parameters of the periodic track structure without considering boundary conditions; forming a total energy functional equation according to the cell energy of the track structure;
step S3: forming a constraint condition matrix by the periodic boundary of the steel rail and the right-to-right boundary of the track slab;
step S4: processing constraint condition matrixes by using a Gaussian elimination method, representing original unknown coefficients by an auxiliary matrix and linear irrelevant column vectors, and introducing the original unknown coefficients into a total energy functional equation;
step S5: dividing the total energy functional function according to the Lagrangian equation to obtain a motion equation of the periodic orbit structure;
step S6: obtaining a dispersion curve of the periodic orbit structure by scanning the wave number of the first irreducible Brillouin zone, and obtaining the band gap characteristic of the periodic orbit structure according to the dispersion curve;
in step S4, the constraint condition matrix is processed by Gaussian elimination to obtain a full order matrix in the constraint condition matrix,representing constraint matrix>Representing the matrix vector of unknown coefficients, will +.>Split into two-part added form, rewritten into:
；
in the formula ,is a full order matrix in the constraint condition matrix; />Is->Vector in unknown coefficient matrix>Unknown coefficient matrix of corresponding position in the model; />A matrix formed by the residual column vectors in the constraint condition matrix; />Is->Vector in unknown coefficient matrix>Unknown coefficient matrix of corresponding position in the model;
will be and />Vertically arranged to obtain a new column vector +.>Then->Can be represented by the unknown coefficient matrix vectors ∈ ->Obtained by primary equal-line transformation of (1), namely:
；
in the formula Is an elementary transformation matrix; />Representing an identity matrix, unknown coefficient matrix vectors +.>By->Linear representation:
；
wherein ,representing the matrix vector of unknown coefficients>By->An auxiliary matrix of the linear representation,the total energy functional equation is rewritten as:
；
in the formula ,representation->Derivative with respect to time, < >>Representation->Conjugate transpose of->Representation->Is used for the conjugate transpose of (a),representation of/>Conjugate transpose of->Representing a mass matrix of the rail; />Representing a mass matrix of the track slab;representing a stiffness matrix of the track slab; />Representing a stiffness matrix of the rail; />Representing a stiffness matrix of the fastener; />Representing the stiffness matrix of the CA mortar layer.
2. The periodic track structure band gap calculation method based on the energy method and the Gaussian elimination method according to claim 1, wherein the track structure cells comprise steel rails, track plates, CA mortar layers and fasteners, and plate gaps are formed between the track structure cells.
3. The method for periodic track structure band gap calculation based on energy method and Gaussian elimination method according to claim 2, wherein the steel rail and the track slab are calculated by adopting two coordinate systems, wherein the steel rail coordinate system isx r O r y r ，x r Is the horizontal coordinate of the steel rail,y r is the vertical coordinate of the steel rail,O r the origin of the rail coordinate system is that the rail plate coordinate system isx s O s y s ，x s In the abscissa of the track plate,y s in the ordinate of the track plate,O s the origin of the coordinate system of the track plate; the vertical displacement of the rail and the track slab is expressed as:
；
in the formula ：representation oftVertical vibration displacement of the steel rail at moment; />Representation oftMoment the rotation angle displacement of the steel rail; />Representation oftBending vibration displacement of the track plate at the moment; />Representation oftTime track platexAngular displacement in the axial direction; />Representation oftTime track plateyAngular displacement in the axial direction; superscript T is the symbol of matrix transposition;representing the 1 st, 2 nd, 3 rd, 4 th, 5 th time-dependent unknown coefficients,/->Respectively representing 1 st, 2 nd, 3 rd, 4 th and 5 th time-dependent unknown coefficient matrixes; />Representing the j-th displacement in the rail coordinate systemThe function of the shape is that,fdisplacement shape function matrix representing rail coordinate system, < ->Representing the j-th displacement shape function in the track slab coordinate system,grepresenting a displacement shape function matrix of the track plate coordinate system.
4. A periodic track structure band gap calculation method based on an energy method and a gaussian elimination method according to claim 3, wherein strain energy and kinetic energy of a steel rail in a track structure cell are expressed as:
；
；
in the formula ,E rail representing strain energy of the steel rail;U rail represents the kinetic energy of the steel rail;，/>representing an unknown coefficient matrix array vector; />Is a partial guide symbol; the superscript H denotes a conjugate transpose; the superscript ". Cndot." represents the derivative with respect to time, representing the unknown coefficient matrix column vector +.>Derivative with respect to time; />Representing the vertical vibration displacement of the rail +.>Representing the derivative of the vertical vibration displacement of the rail with respect to time,indicating the angular displacement of the rail->Representing the derivative of the angular displacement of the rail with respect to time; and />Respectively representing the density, the sectional area, the elastic modulus, the moment of inertia and the shearing coefficient of the steel rail; />Represents the shear modulus of the rail; />Represents the length of the track slab in a track structure cell,/->Representing the panel seam width.
5. The periodic orbit structure band gap calculation method based on the energy method and the gaussian elimination method according to claim 4, wherein the strain energy and the kinetic energy of the orbit plate in the orbit structure cell are expressed as:
；
；
in the formula ,representing strain energy of the track slab; />Representing half of the track plate width; />Representing the kinetic energy of the track plate; /> and />Representing the density, thickness, area, bending stiffness, poisson's ratio, shear modulus and shear coefficient, respectively, of the rail plate,/->Representing bending vibration displacement of the track plate; />Representing track slabsxAngular displacement in the axial direction; />Representing track slabsyAngular displacement in the axial direction; />Representing the derivative of the bending vibration displacement of the track plate with respect to time, is->Representing track slabsxDerivative of the angular displacement of the axis direction with respect to time, < >>Representing track slabsyThe derivative of the angular displacement in the axial direction with respect to time.
6. The periodic orbit structure band gap calculation method based on the energy method and the Gaussian elimination method according to claim 5, wherein the elastic potential energy of the fastener and the CA mortar layer in the orbit structure cell is expressed as:
；
；
in the formula ,representing elastic potential energy of the fastener; />Numbering the fasteners, wherein N represents the number of the fasteners; />Representing the elastic potential energy of the CA mortar layer; />Representing fastener stiffness; />Representing half of the width of the panel slit,/->Representing fastener spacing;representing the distance of the rail from the free boundary of the track slab; />Represents the average stiffness of the CA mortar layer.
7. The method for periodic orbit structure band gap calculation based on energy method and Gaussian elimination method according to claim 6, wherein the total energy functional equation in the orbit structure cell is:
；
wherein ,representing the total energy functional.
8. The periodic orbit structure band gap calculation method according to claim 7, wherein in step S3, the period boundary is as follows:
；
wherein i is a complex number unit;kwave number; e represents a natural constant;
the positive boundary of the track slab is expressed as:；
the constraint condition matrix is:
；
in the formula ,representing a displacement shape function matrix when the rail coordinates are zero; />Indicating the coordinates of the rail as +.>A displacement shape function matrix; />；/>Representing +.>And the displacement shape function of the track plate.
9. The periodic orbit structure band gap calculation method according to claim 8, wherein in step S5, according to the lagrangian equationFunctional on total energy>And (3) obtaining a motion equation of the periodic orbit structure by variation:
。
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