Analytical analysis method for free vibration characteristics of chain type beam structure containing conical beam

Abstract

The invention belongs to the technical field of dynamic modeling research, and particularly relates to an analytic method for free vibration characteristics of a chain type beam structure containing a conical beam. A conical beam transfer matrix expression based on a Bezier equation form is formed, the obtained transfer matrix expression has good ductility and can be connected with transfer matrixes of other beams with different characteristics, and the free vibration characteristic of the chain type beam structure containing the conical beam is solved based on typical boundary conditions of the chain type beam structure. The method comprises the following steps: acquiring a singular submatrix under a field transfer matrix, deducing to obtain a transcendental equation, solving the equation to obtain a free vibration frequency characteristic of the chain beam structure, obtaining an inherent frequency, reversely substituting the inherent frequency into the transfer matrix, and further solving to obtain a chain beam structure array. Therefore, the inherent characteristic solution of the chain type beam structure containing the conical beam is realized. The method is used for solving the problem that the chained beam structure containing the conical beam lacks analysis and expression.

Description

Technical Field

[0001] The invention belongs to the technical field of dynamic modeling research, and in particular relates to an analytical method for free vibration characteristics of a chain beam structure containing a tapered beam. Background Art

[0002] As the most basic structural component, beams are widely used in various engineering projects such as civil engineering, machinery, aerospace, etc. Therefore, the research on this type of components in the vibration and impact analysis of engineering structures is very common. At present, the vibration characteristics of uniform cross-section beams have been very perfect, and many scholars have done a lot of research on uniform cross-section beams using different numerical methods. However, with the advancement of engineering technology, variable cross-section beams have been widely used in actual engineering, such as chimneys, TV towers, bridges, etc., especially in ultrasonic vibration in precision machining, micro-electromechanical systems, vibration energy collection systems and acoustic black holes. Conical beams are widely used. At the same time, because the vibration equation of the variable cross-section beam is fundamentally different from that of the uniform cross-section cantilever beam. The resulting vibration natural frequency is very different, so it is very important to determine its bending vibration natural characteristics through calculation.

[0003] At present, there are four main methods for calculating the natural frequency of tapered beams: perturbation method, finite element semi-analytical method, Rayleigh-Ritz method, and analytical calculation. Among them, the frequency and deformation degree obtained by the perturbation method are not ideal. The use of finite element semi-analytical method is more advantageous for some complex structures, but for simple components and nonlinear vibrations that require precision, the process is too cumbersome and time-consuming. The Rayleigh-Ritz method is better for solving the natural frequency, but its vibration mode function is less consistent. Summary of the invention

[0004] The invention provides a method for analyzing the free vibration characteristics of a chain-type beam structure containing a tapered beam, so as to solve the problem that the transmission analysis of the chain-type structure containing the tapered beam lacks analytical expression.

[0005] The present invention is achieved through the following technical solutions:

[0006] A method for analyzing the free vibration characteristics of a chain beam structure containing a tapered beam, the method comprising the following steps:

[0007] Step 1: Based on the Euler-Bernoulli beam lateral vibration equation, considering the geometric characteristics of the tapered beam, the cross-sectional inertia moment and area of ​​the tapered beam varying with the axial length are taken as variables to obtain the tapered beam lateral vibration equation;

[0008] Step 2: Solve the lateral vibration equation of the tapered beam obtained in step 1 to obtain the theoretical analytical solution of the tapered beam deflection, rotation angle, bending moment and shear force containing Bessel functions;

[0009] Step 3: Solve the transfer relationship of the tapered tube parameters based on the deflection, rotation angle, bending moment and shear force of the tapered beam obtained in step 2, and further obtain the transfer relationship of the mechanical parameters of the beginning and end of the tapered beam;

[0010] Step 4: Based on the transfer relationship constructed in step 3, a transfer matrix expression of the tapered beam based on the Bessel equation is further formed. At the same time, the obtained transfer matrix expression has good ductility and can be connected with the transfer matrices of other beams with different characteristics to solve the free vibration characteristics of the chain beam structure containing the tapered beam;

[0011] Step 5: Based on the typical boundary conditions of the chain-beam structure, the transfer matrix of the tapered beam and other characteristic beams, the singular submatrix under the field transfer matrix can be obtained, and the transcendental equation can be derived. The free vibration frequency characteristics of the chain-beam structure are obtained by solving the equation, and the natural frequency is obtained. The natural frequency is substituted into the transfer matrix, and the chain-beam structure formation is further solved, thereby realizing the solution of the natural characteristics of the chain-beam structure containing the tapered beam.

[0012] Furthermore, the step 1 is specifically that the vibration equation for the Euler-Bernoulli beam is:

[0013]

[0014] Where x represents the axial position of the cross section, E represents the elastic modulus of the beam structure, I(x) is the moment of inertia at the x position, w is the main vibration frequency of the beam, and ρ is the density of the beam structure;

[0015] The length of the beam is L, the radius of the circular section on the left is r0, and the radius of the circular section on the right is r1. Therefore, the expressions of the cross-sectional area A(x) and moment of inertia I(x) of the beam are:

[0016]

[0017] Among them, x represents the axial position of the section, and the parameter Indicates the rate of change of the cross-sectional radius along the axial direction.

[0018] Furthermore, the step 2 is specifically to first solve the variable coefficient ordinary differential vibration equation, which can be simplified to:

[0019]

[0020] Where X = 1-(a / r0)x,

[0021] By introducing coordinate transformation:

[0022]

[0023] Finally, we can get the analytical solution containing Bessel equation:

[0024]

[0025] Substituting equation (4) into the relationship between the rotation angle, bending moment, shear force and deflection of the Euler-Bernoulli beam model, we can obtain:

[0026]

[0027] Among them, B1, B2, B3, B4 are constant coefficients, J n is the nth-order Bessel function, Y n is the n-order Neumann function,

[0028] I n is the nth order imaginary Bessel function, K n is the nth-order imaginary Neumann function, θ z (β) is the rotation angle of the beam structure at β, M z (β) is the bending moment of the beam structure at β, Q y (β) is the shear force of the beam structure at position β, and I(0) is the moment of inertia at position x=0.

[0029] Furthermore, the step three is specifically, according to formula (4)-(6), the transfer relationship is as follows:

[0030]

[0031] Where E1 = 1 / β 2 , E2=k 2 ΔR / (2Lβ 2 ), E3=-EI0dr 2 β 3 / (4L 2 k 2 ),

[0032] E4=EI0ΔR 3 β 2 / (8k 2 L 3 ).

[0033] Furthermore, the step three is specifically to obtain the transfer matrix:

[0034] T=Z(L)*Z(0) -1 (8)

[0035] Among them, Z(L) is the transfer relationship matrix at the position x=L, Z(0) -1 It is the inverse matrix of the transfer relation matrix at the cross section x=0.

[0036] Furthermore, the typical boundary conditions of the chain beam structure in step five are as follows: elastic modulus is 210, density is 7850, length is 3, left radius is 0.1, and right radius is 0.01.

[0037] Furthermore, the step 5 is specifically that, for a beam of uniform cross-section, the transfer relationship can be written as:

[0038]

[0039] Among them, p 4 =ρA / (EI)ω 2 ,

[0040] Then its transfer matrix is:

[0041] T T =Z T (L)*Z T (0) -1 (10)

[0042] For the chain beam structure, the transfer matrix can be written as:

[0043] T L-0 =T T ·...·T (11)

[0044] The order of the transfer matrices in the formula is determined by the left-to-right order in the chain beam structure.

[0045] Furthermore, after introducing typical boundary conditions, the transfer matrix can be written as follows:

[0046]

[0047] The final natural frequency is obtained by the corresponding transcendental equation:

[0048]

[0049] Among them, the superscript Z represents the zero quantity in the boundary condition, N represents the corresponding non-zero quantity in the boundary condition, and S is the natural frequency of the tapered beam at this order.

[0050] Furthermore, the step 5 is specifically to bring the natural frequency into the original transfer matrix and solve equation (10)

[0051] The linear relationship between the non-zero state variables can be obtained, and then the expression of the deflection at any position can be obtained by using the transfer relationship, and then the formation of the tapered beam at this order natural frequency can be obtained:

[0052]

[0053] Among them, S is the natural frequency of the tapered beam, T is the transfer matrix, N is the corresponding non-zero quantity in the boundary condition, L is the beam length, and Z is the zero quantity in the boundary condition.

[0054] A computer-readable storage medium stores a computer program, and the computer program implements the above method when executed by a processor.

[0055] The beneficial effects of the present invention are:

[0056] The invention can calculate the inherent characteristics of the tapered beam, and the method can be used for chain structure transmission analysis containing the tapered beam.

[0057] The method proposed in the invention has the characteristics of improving calculation efficiency and high calculation accuracy. It can accurately analyze and express the tapered beam part in the calculation process of the chain structure containing tapered beams, and provide technical support for vibration prediction and vibration optimization of tapered beams.

[0058] The present invention obtains an analytical solution to the lateral vibration of a tapered beam from a theoretical perspective, providing a more accurate calculation method for the lateral vibration analysis of a tapered beam. At the same time, the present invention can be used to calculate the vibration characteristics of beam structures involving other different characteristics, and has good continuity. BRIEF DESCRIPTION OF THE DRAWINGS

[0059] Figure 1 It is a flow chart of the method of the present invention.

[0060] Figure 2 It is a schematic diagram of the tapered beam theoretical analysis model of the present invention.

[0061] Figure 3 It is a comparison diagram of the formation results and the finite element results under the fixed-free boundary conditions in the implementation mode of the present invention, wherein (a) is a comparison diagram of the first-order natural frequency, (b) is a comparison diagram of the second-order natural frequency, and (c) is a comparison diagram of the third-order natural frequency. DETAILED DESCRIPTION

[0062] In the following description, specific details such as specific system structures, technologies, etc. are provided for the purpose of illustration rather than limitation, so as to provide a thorough understanding of the embodiments of the present application. However, it should be clear to those skilled in the art that the present application may also be implemented in other embodiments without these specific details. In other cases, detailed descriptions of well-known systems, devices, circuits, and methods are omitted to prevent unnecessary details from obstructing the description of the present application.

[0063] It should be understood that when used in this specification and the appended claims, the term "comprising" indicates the presence of described features, integers, steps, operations, elements and / or components, but does not exclude the presence or addition of one or more other features, integers, steps, operations, elements, components and / or groups thereof.

[0064] It should also be understood that the terms used in this application specification are only for the purpose of describing specific embodiments and are not intended to limit the application. As used in this application specification and the appended claims, the singular forms "a", "an" and "the" are intended to include plural forms unless the context clearly indicates otherwise.

[0065] The following is attached to this application specification Figure 1-3 , the technical solutions in the embodiments of the present application are clearly and completely described. Obviously, the described embodiments are only part of the embodiments of the present application, not all of the embodiments. Based on the embodiments in the present application, all other embodiments obtained by ordinary technicians in this field without creative work are within the scope of protection of the present application.

[0066] In the following description, many specific details are set forth to facilitate a full understanding of the present application, but the present application may also be implemented in other ways different from those described herein, and those skilled in the art may make similar generalizations without violating the connotation of the present application. Therefore, the present application is not limited to the specific embodiments disclosed below.

[0067] Implementation Method 1

[0068] This embodiment provides a method for analyzing the free vibration characteristics of a chain-type beam structure containing a tapered beam, the method comprising the following steps:

[0069] Step 1: Based on the Euler-Bernoulli beam lateral vibration equation, considering the geometric characteristics of the tapered beam, the cross-sectional inertia moment and area of ​​the tapered beam varying with the axial length are taken as variables to obtain the tapered beam lateral vibration equation;

[0070] Step 2: Solve the lateral vibration equation of the tapered beam obtained in step 1 to obtain the theoretical analytical solution of the tapered beam deflection, rotation angle, bending moment and shear force containing Bessel functions;

[0071] Step 3: Solve the transfer relationship of the tapered tube parameters based on the deflection, rotation angle, bending moment and shear force of the tapered beam obtained in step 2, and further obtain the transfer relationship of the mechanical parameters of the beginning and end of the tapered beam;

[0072] Step 4: Based on the transfer relationship constructed in step 3, a transfer matrix expression of the tapered beam based on the Bessel equation is further formed. At the same time, the obtained transfer matrix expression has good ductility and can be connected with the transfer matrices of other beams with different characteristics to solve the free vibration characteristics of the chain beam structure containing the tapered beam;

[0073] Step 5: Based on the typical boundary conditions of the chain-beam structure, the transfer matrix of the tapered beam and other characteristic beams, the singular submatrix under the field transfer matrix can be obtained, and the transcendental equation can be derived. The free vibration frequency characteristics of the chain-beam structure are obtained by solving the equation, and the natural frequency is obtained. The natural frequency is substituted into the transfer matrix, and the chain-beam structure formation is further solved, thereby realizing the solution of the natural characteristics of the chain-beam structure containing the tapered beam.

[0074] Furthermore, the step 1 is specifically that the vibration equation for the Euler-Bernoulli beam is:

[0075]

[0076] Where x represents the axial position of the cross section, E represents the elastic modulus of the beam structure, I(x) is the moment of inertia at the x position, w is the main vibration frequency of the beam, and ρ is the density of the beam structure.

[0077] For Figure 2 For the tapered beam structure shown, the beam length is L, the radius of the circular section on the left is r0, and the radius of the circular section on the right is r1. Therefore, the cross-sectional area A(x) and moment of inertia I(x) of the beam are expressed as:

[0078]

[0079] Among them, x represents the axial position of the section, and the parameter Indicates the rate of change of the cross-sectional radius along the axial direction.

[0080] Furthermore, the step 2 is specifically to first solve the variable coefficient ordinary differential vibration equation, which can be simplified to:

[0081]

[0082] Where X = 1-(a / r0)x,

[0083] By introducing coordinate transformation

[0084]

[0085] Finally, we can get the analytical solution containing Bessel equation:

[0086]

[0087] Substituting equation (4) into the relationship between the rotation angle, bending moment, shear force and deflection of the Euler-Bernoulli beam model, we can obtain:

[0088]

[0089] Among them, B1, B2, B3, B4 are constant coefficients, J n is the nth-order Bessel function, Y n is the nth-order Neumann function, I n is the nth order imaginary Bessel function, K n is the nth-order imaginary Neumann function, θ z (β) is the rotation angle of the beam structure at β, M z (β) is the bending moment of the beam structure at β, Q y (β) is the shear force of the beam structure at position β, and I(0) is the moment of inertia at position x=0.

[0090] Furthermore, the step three is specifically, according to formula (4)-(6), the transfer relationship is as follows:

[0091]

[0092] Where E1 = 1 / β 2 , E2=k 2 ΔR / (2Lβ 2 ), E3=-EI0dr 2 β 3 / (4L 2 k 2 ),

[0093] E4=EI0ΔR 3 β 2 / (8k 2 L 3 );

[0094] Furthermore, the transfer matrix obtained in step 4 is:

[0095] T=Z(L)*Z(0) -1 (8)

[0096] Among them, Z(L) is the transfer relationship matrix at the position x=L, Z(0) -1 It is the inverse matrix of the transfer relation matrix at the cross section x=0.

[0097] Furthermore, the typical boundary conditions of the chain beam structure in step 5 are free, simply supported or fixed. Since the transfer matrix of the uniform cross-section beam structure has been verified, the calculation of the present invention is mainly for the verification of the tapered beam part in the chain beam structure. The tapered beam parameters and boundaries used are shown in the following table:

[0098]

[0099] Furthermore, the step 5 is specifically as follows: for other characteristic cross-section beams such as uniform cross-section beams, the transfer relationship

[0100] The system can be written as:

[0101]

[0102] Among them, p 4 =ρA / (EI)ω 2 ,

[0103] Then its transfer matrix is:

[0104] T T =Z T (L)*Z T (0) -1 (10)

[0105] For the chain beam structure, the transfer matrix can be written as:

[0106] T L-0 =T T ·...·T (11)

[0107] The order of each transfer matrix in the formula is determined by the order from left to right in the chain beam structure;

[0108] After bringing in typical boundary conditions, the transfer matrix can be written as follows:

[0109]

[0110] The final natural frequency is obtained by the corresponding transcendental equation:

[0111]

[0112] Among them, the superscript Z represents the zero quantity in the boundary condition, N represents the corresponding non-zero quantity in the boundary condition, and S is the state vector of the tapered beam.

[0113] Furthermore, the step five is to bring the natural frequency into the original transfer matrix and solve equation (10):

[0114] The linear relationship between the non-zero state variables can be obtained, and then the expression of the deflection at any position can be obtained by using the transfer relationship, and then the formation of the tapered beam at this order natural frequency can be obtained:

[0115]

[0116] Among them, S is the state vector of the tapered beam, T is the transfer matrix, N is the corresponding non-zero quantity in the boundary condition, L is the beam length, and Z is the zero quantity in the boundary condition.

[0117] Specifically, the comparison of the first three natural frequencies is shown in the following table:

[0118]

[0119] Get the first three-stage formation comparison Figure 3 As shown, the modal confidence matrix results are as follows:

[0120]

[0121] Implementation Method 2

[0122] This embodiment provides a system for analyzing the free vibration characteristics of a chain-type beam structure containing a tapered beam. The system uses the method for analyzing the free vibration characteristics of a chain-type beam structure containing a tapered beam as described in Embodiment 1. The system includes:

[0123] The vibration equation building module, based on the transverse vibration equation of the Euler-Bernoulli beam, considers the geometric characteristics of the tapered beam, takes the cross-sectional inertia moment and area of ​​the tapered beam changing with the axial length as variables, and obtains the transverse vibration equation of the tapered beam;

[0124] Theoretical analytical solution module, which solves the lateral vibration equation of the tapered beam and obtains the theoretical analytical solution of the tapered beam deflection, rotation, bending moment and shear force containing Bessel functions;

[0125] The transfer matrix calculation module solves the transfer relationship of the tapered tube parameters according to the obtained deflection, rotation angle, bending moment and shear force of the tapered beam, and further obtains the transfer relationship of the mechanical parameters of the beginning and end of the tapered beam;

[0126] Based on the constructed transfer relationship, the transfer matrix expression of the tapered beam based on the Bessel equation is further formed. At the same time, the obtained transfer matrix expression has good ductility and can be connected with the transfer matrices of other beams with different characteristics to solve the free vibration characteristics of the chain beam structure containing tapered beams.

[0127] The inherent characteristics solving module, based on the typical boundary conditions of the chain beam structure, the transfer matrix of the tapered beam and other characteristic beams, can obtain the singular submatrix under the field transfer matrix, derive the transcendental equation, solve the equation to obtain the free vibration frequency characteristics of the chain beam structure, obtain the inherent frequency, bring the inherent frequency back into the transfer matrix, and further solve the chain beam structure formation, thereby realizing the solution of the inherent characteristics of the chain beam structure containing the tapered beam.

[0128] Implementation Method 3

[0129] An embodiment of the present invention provides an electronic device, which includes a memory, a processor, and a computer program stored in the memory and executable on the processor, wherein the memory is used to store software programs and modules, and the processor executes various functional applications and data processing by running the software programs and modules stored in the memory. The memory and the processor are connected via a bus. Specifically, the processor implements any step in the first embodiment above by running the computer program stored in the memory.

[0130] It should be understood that in the embodiments of the present invention, the processor referred to may be a central processing unit (CPU), and the processor may also be other general-purpose processors, digital signal processors (DSP), application-specific integrated circuits (ASIC), field-programmable gate arrays (FPGA) or other programmable logic devices, discrete gate or transistor logic devices, discrete hardware components, etc. A general-purpose processor may be a microprocessor or the processor may also be any conventional processor, etc.

[0131] The memory may include a read-only memory, a flash memory, and a random access memory, and provides instructions and data to the processor. A part or all of the memory may also include a nonvolatile random access memory.

[0132] It can be seen from the above that the electronic device provided in the embodiment of the present invention can implement the analytical analysis method for the free vibration characteristics of the chain-type beam structure containing tapered beams as described in the first embodiment by running a computer program, and can accurately analytically express the tapered beam part during the transmission process of the chain structure containing tapered beams, thereby providing technical support for vibration prediction and vibration optimization of tapered beams.

[0133] It should be understood that if the above-mentioned integrated module / unit is implemented in the form of a software functional unit and sold or used as an independent product, it can be stored in a computer-readable storage medium. Based on such an understanding, the present invention implements all or part of the processes in the above-mentioned implementation method, and can also be completed by instructing the relevant hardware through a computer program. The above-mentioned computer program can be stored in a computer-readable storage medium, and the computer program can implement the steps of the above-mentioned various method implementations when executed by the processor. Among them, the above-mentioned computer program includes computer program code, and the above-mentioned computer program code can be in source code form, object code form, executable file or some intermediate form. The above-mentioned computer-readable medium may include: any entity or device capable of carrying the above-mentioned computer program code, recording medium, U disk, mobile hard disk, disk, optical disk, computer memory, read-only memory (ROM), random access memory (RAM), electric carrier signal, telecommunication signal and software distribution medium. It should be noted that the content contained in the above-mentioned computer-readable storage medium can be appropriately increased or decreased according to the requirements of legislation and patent practice in the jurisdiction.

[0134] The above description of the disclosed embodiments enables one skilled in the art to implement or use the present invention. Various modifications to these embodiments will be apparent to one skilled in the art, and the general principles defined herein may be implemented in other embodiments without departing from the spirit or scope of the present invention. Therefore, the present invention will not be limited to the embodiments shown herein, but rather to the widest range consistent with the principles and novel features disclosed herein.

[0135] Those skilled in the art can clearly understand that for the convenience and simplicity of description, only the division of the above-mentioned functional units and modules is used as an example for illustration. In actual applications, the above-mentioned function allocation can be completed by different functional units and modules as needed, that is, the internal structure of the above-mentioned device can be divided into different functional units or modules to complete all or part of the functions described above. The functional units and modules in the implementation method can be integrated into a processing unit, or each unit can exist physically separately, or two or more units can be integrated into one unit. The above-mentioned integrated unit can be implemented in the form of hardware or in the form of software functional units. In addition, the specific names of the functional units and modules are only for the convenience of distinguishing each other, and are not used to limit the scope of protection of the present invention. The specific working process of the units and modules in the above-mentioned system can refer to the corresponding process in the aforementioned method implementation method, which will not be repeated here.

[0136] It should be noted that the methods and detailed examples provided in the above embodiments can be combined with the devices and equipment provided in the embodiments, and references can be made to each other, and no further details will be given.

[0137] Those skilled in the art will appreciate that the units and algorithm steps of each example described in conjunction with the embodiments disclosed herein can be implemented in electronic hardware, or a combination of computer software and electronic hardware. Whether these functions are performed in hardware or software depends on the specific application and design constraints of the technical solution. Professional and technical personnel can use different methods to implement the described functions for each specific application, but such implementation should not be considered to be beyond the scope of the present invention.

[0138] In the embodiments provided by the present invention, it should be understood that the disclosed device / terminal equipment and method can be implemented in other ways. For example, the device / equipment implementation described above is only illustrative, for example, the division of the above modules or units is only a logical function division, and in actual implementation, there may be other division methods, such as multiple units or components can be combined or integrated into another system, or some features can be ignored or not executed.

[0139] The above-mentioned embodiments are only used to illustrate the technical solutions of the present invention, rather than to limit them. Although the present invention has been described in detail with reference to the above-mentioned embodiments, those skilled in the art should understand that they can still modify the technical solutions described in the above-mentioned embodiments, or replace some of the technical features therein by equivalents. These modifications or replacements do not deviate the essence of the corresponding technical solutions from the spirit and scope of the technical solutions of the various embodiments of the present invention, and should all be included in the protection scope of the present invention.

Claims

1. A method for analyzing the free vibration characteristics of a chain beam structure containing a tapered beam, characterized in that: The method comprises the following steps, Step 1: Based on the Euler-Bernoulli beam lateral vibration equation, considering the geometric characteristics of the tapered beam, the cross-sectional inertia moment and area of ​​the tapered beam varying with the axial length are taken as variables to obtain the tapered beam lateral vibration equation; Step 2: Solve the lateral vibration equation of the tapered beam obtained in step 1 to obtain the theoretical analytical solution of the tapered beam deflection, rotation angle, bending moment and shear force containing Bessel functions; Step 3: Solve the transfer relationship of the tapered tube parameters based on the deflection, rotation angle, bending moment and shear force of the tapered beam obtained in step 2, and further obtain the transfer relationship of the mechanical parameters of the beginning and end of the tapered beam; Step 4: Based on the transfer relationship constructed in step 3, a transfer matrix expression of the tapered beam based on the Bessel equation is further formed. At the same time, the obtained transfer matrix expression has good ductility and can be connected with the transfer matrices of other beams with different characteristics to solve the free vibration characteristics of the chain beam structure containing the tapered beam; Step 5: Based on the typical boundary conditions of the chain-beam structure, the transfer matrix of the tapered beam and other characteristic beams, the singular submatrix under the field transfer matrix can be obtained, and the transcendental equation can be derived. The free vibration frequency characteristics of the chain-beam structure are obtained by solving the equation, and the natural frequency is obtained. The natural frequency is substituted into the transfer matrix, and the chain-beam structure formation is further solved, thereby realizing the solution of the natural characteristics of the chain-beam structure containing the tapered beam.

2. The method according to claim 1, characterized in that: Specifically, the vibration equation of the Euler-Bernoulli beam is: Where x represents the axial position of the cross section, E represents the elastic modulus of the beam structure, I(x) is the moment of inertia at the x position, w is the main vibration frequency of the beam, and ρ is the density of the beam structure; The length of the beam is L, the radius of the circular section on the left is r0, and the radius of the circular section on the right is r1. Therefore, the expressions of the cross-sectional area A(x) and moment of inertia I(x) of the beam are: Where x represents the axial position of the section, and the parameter Indicates the rate of change of the cross-sectional radius along the axial direction.

3. The method according to claim 2, characterized in that: Specifically, the step 2 is to first solve the variable coefficient ordinary differential vibration equation, which can be simplified to: In the formula By introducing coordinate transformation: Finally, we can get the analytical solution containing Bessel equation: Substituting equation (4) into the relationship between the rotation angle, bending moment, shear force and deflection of the Euler-Bernoulli beam model, we can obtain: Among them, B1, B2, B3, B4 are constant coefficients, J n is the nth-order Bessel function, Y n is the nth-order Neumann function, I n is the nth order imaginary Bessel function, K n is the nth-order imaginary Neumann function, θ z (β) is the rotation angle of the beam structure at β, M z (β) is the bending moment of the beam structure at β, Q y (β) is the shear force of the beam structure at position β, and I(0) is the moment of inertia at position x=0.

4. The method according to claim 1, characterized in that: Specifically, the step three is as follows: according to formula (4)-(6), the transfer relationship is as follows: where E1 = 1 / β 2 , E2 = k 2 ΔR / (2Lβ 2 ), E3 = -EI0dr 2 β 3 / (4L 2 k 2 ), E4 = EI0ΔR 3 β 2 / (8k 2 L 3 ).

5. The method according to claim 1, characterized in that: The step three is specifically to obtain the transfer matrix: T=Z(L)*Z(0) -1 (8) Among them, Z(L) is the transfer relationship matrix at the position x=L, Z(0) -1 It is the inverse matrix of the transfer relation matrix at the cross section x=0.

6. The method according to claim 1, characterized in that: The typical boundary conditions of the chain beam structure in step 5 are as follows: elastic modulus is 210, density is 7850, length is 3, left radius is 0.1, and right radius is 0.

01.

7. The method according to claim 5, characterized in that: Specifically, for a beam with a uniform cross-section, the transfer relationship can be written as: where p 4 =ρA / (EI)ω 2 , Then its transfer matrix is: T T =Z T (L)*Z T (0) -1 (10) For the chain beam structure, the transfer matrix can be written as: T L-0 =T T ·...·T (11) The order of the transfer matrices in the formula is determined by the left-to-right order in the chain beam structure.

8. The method according to claim 6, characterized in that: After bringing in typical boundary conditions, the transfer matrix can be written as follows: The final natural frequency is obtained by the corresponding transcendental equation: Among them, the superscript Z represents the zero quantity in the boundary condition, N represents the corresponding non-zero quantity in the boundary condition, and S is the natural frequency of the tapered beam at this order.

9. The method according to claim 7, characterized in that: Specifically, the step 5 is to bring the natural frequency into the original transfer matrix, solve equation (10) to obtain the linear relationship between the non-zero state variables, and then use the transfer relationship to obtain the expression of the deflection at any position, and then obtain the formation of the tapered beam at this order natural frequency: Among them, S is the natural frequency of the tapered beam, T is the transfer matrix, N is the corresponding non-zero quantity in the boundary condition, L is the beam length, and Z is the zero quantity in the boundary condition.

10. A computer-readable storage medium, characterized in that: The computer-readable storage medium stores a computer program, and when the computer program is executed by a processor, the method according to any one of claims 1 to 9 is implemented.

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