A method for dynamic analysis of in-plane arbitrary curvature variable thickness beam structure

Abstract

This invention discloses a dynamic analysis method for beam structures with arbitrary curvature and varying thickness in a plane, belonging to the field of aerospace structure and structural dynamics modeling technology. The method first inputs the upper and lower surface functions of the curved beam to solve for its neutral surface function; secondly, it constructs a family of orthogonal displacement functions specific to this beam structure with arbitrary curvature and varying thickness, and establishes an integral mapping relationship between the global coordinate system and the dynamic sub-coordinate system; subsequently, based on the displacement function family and the integral mapping relationship, it derives and solves the energy equation of the beam structure; finally, it obtains the mode shapes and natural frequencies based on the Rayleigh-Ritz method, and calculates the dynamic response and dynamic stress of the structure using the modal superposition method. This invention overcomes the shortcomings of existing semi-analytical modeling methods for the dynamics of complex curved beam structures with varying thickness, and can efficiently and accurately predict and analyze the dynamic characteristics of beams with arbitrary curvature and varying thickness in a plane.

Description

Technical Field

[0001] This application belongs to the field of aerospace structure and structural dynamics modeling technology, specifically relating to a dynamic analysis method for a beam structure with arbitrary curvature and variable thickness in a plane. Background Technology

[0002] In the aerospace field, beam structures are fundamental building blocks for critical load-bearing components such as aircraft wing frames, engine casing support rings, and satellite deployment arms. To meet the demands of aerodynamic optimization, lightweight design, and complex load transfer paths, curved beam structures with arbitrary in-plane curvature and non-uniformly varying thickness along the axial direction are increasingly widely used. Accurately solving for the dynamic characteristics (such as mode shapes and natural frequencies) of these complex curved beam structures, as well as their dynamic response and dynamic stress under external excitation, is of significant engineering importance for evaluating the safety margin and functional reliability of aircraft structures and for conducting vibration reduction and noise reduction optimization designs.

[0003] Currently, research on dynamic modeling and modal analysis of beam structures mainly focuses on geometrically regular objects, such as straight beams with uniform cross-sections, standard circular arc beams, or simple parabolic beams. Traditional analytical or semi-analytical dynamic modeling methods are relatively convenient for handling these beam structures with specific curvature and uniform thickness. However, for beam structures with arbitrary curvature and varying thickness where both curvature and thickness exhibit complex nonlinear variations, the classical orthogonal displacement function family struggles to accurately match the geometric characteristics, and conventional integration schemes cannot be effectively implemented due to the lack of coordinate mapping relationships. This results in an extremely cumbersome modeling process, and existing semi-analytical techniques lack universal theoretical support for dealing with geometric arbitrariness.

[0004] On the other hand, the traditional finite element method is often used in engineering practice for the dynamic analysis of complex structures. However, this method faces significant limitations when dealing with curved beams of varying thickness: geometric heterogeneity requires extremely fine meshing in regions of abrupt curvature changes and thickness transitions, resulting in a large amount of manual mesh adjustment and verification during the preprocessing stage; high-density elements must be deployed to meet accuracy requirements, causing a sharp increase in computational resources and a significant decrease in solution efficiency; the mesh discretization process itself introduces geometric approximation errors, especially in regions of continuously changing curvature, where element boundaries are difficult to accurately fit the actual curved surface, ultimately limiting the reliability of dynamic response prediction.

[0005] In summary, existing dynamic analysis methods suffer from two major bottlenecks when dealing with complex curved beam structures with varying thickness: difficulties in semi-analytical modeling and cumbersome and inefficient preprocessing for numerical calculations. Engineering practice urgently needs to develop a novel dynamic analysis method that can balance geometric accuracy with computational efficiency. Summary of the Invention

[0006] The purpose of this invention is to provide a dynamic analysis method for beam structures with arbitrary curvature and varying thickness in a plane, so as to solve the technical problems of difficulty in semi-analytical modeling, low computational efficiency and insufficient accuracy of complex curved beam structures in the prior art.

[0007] This invention provides a dynamic analysis method for beam structures with arbitrary curvature and variable thickness in a plane, comprising the following steps: Step 1: Obtain the geometric parameters of a beam structure with arbitrary curvature and varying thickness in a plane, input the upper and lower surface functions of the beam structure, and solve for the high-precision approximate function of the neutral surface and the varying thickness function of the beam structure; Step 2: Establish the global coordinate system and dynamic sub-coordinate system of the beam structure, and derive the integral mapping relationship between the global coordinate system and the dynamic sub-coordinate system; Step 3: Based on the neutral surface function, construct a family of orthogonal displacement functions applicable to beam structures with arbitrary curvature and varying thickness; Step 4: Based on the family of orthogonal displacement functions and integral mapping relationships, derive and solve the kinetic energy equation, potential energy equation and boundary constraint equation for beam structures with arbitrary curvature and varying thickness. Step 5: Construct the total energy functional of the system by combining the kinetic energy equation, potential energy equation and boundary constraint equation, solve the total energy functional based on the Rayleigh-Ritz method, and obtain the mode shapes and natural frequencies of the beam structure with arbitrary curvature and variable thickness. Step 6: Based on the mode shape and natural frequency, the dynamic response and dynamic stress of the beam structure with arbitrary curvature and variable thickness under external excitation are calculated using the modal superposition method, thus completing the dynamic analysis.

[0008] Furthermore, step 1 specifically includes: Step 1.1: Top surface function based on input arbitrary curvature and variable thickness beam structure f 1( x ) and lower surface function f 2( x Assume the neutral surface function is g ( x ); after neutral surface function g ( x any point on ) P i normal and upper surface function f 1( x Intersect at A i Point, and the function of the lower surface f 2( x Intersect at B i Points, according to Timoshenko beam theory, line segments P iA i and P i B i The length satisfies the following relationship: ; In the formula, i Indicates the number starting from one end of the beam structure. i The discrete point labels on the neutral surface function N This represents the total number of points discretized on the neutral surface function. Step 1.2: Set up points A i and points B i Their coordinates in the plane coordinate system are respectively and Based on Timoshenko's beam theory, and combined with the line segment in step 1.1 P i A i and P i B i Length relationship and point A i and points B i From the coordinate parameters, the neutral surface function is derived. g ( x Any point on the equation satisfies the following equality constraint: ; Step 1.3: Transform the equality constraint relationship into an unconstrained nonlinear least squares optimization model, using the discrete ordinate vector of the neutral surface. g As decision variables, minimizing the sum of squared residuals of distance differences at all discrete sampling points, the optimization model is expressed as: ; ; In the formula, S ( g The expression represents the objective function, which is the cumulative residual of the distance differences between all sampling points. PA i ( g )and PB i ( g All of these are decision variables. g The implicit function, the value of which depends on the neutral plane in P i The direction of the normal to the position; Step 1.4: Solve the optimization model using a nonlinear least squares optimization algorithm to obtain discrete neutral surface data points, and then perform polynomial fitting on the discrete neutral surface data points to obtain a high-precision approximation function of the neutral surface. g 1( x Using approximate functions g 1( x and upper and lower surface functions f 1( x )and f 2( x The thickness function of the beam structure is calculated. h ( x ).

[0009] Furthermore, step 2 specifically includes: Step 2.1: Construct a dynamic sub-coordinate system at the end of the beam structure with arbitrary curvature and variable thickness. o 0 x 0 y 0 z 0, where x 0 direction and approximation function g 1( x The tangential direction is consistent with that of the other two. z 0 direction and approximation function g 1( x The normal direction is consistent with that of the other two directions. Step 2.2: To perform integral calculations of total kinetic and potential energy, a mapping transformation is used to change the integration domain from the dynamic sub-coordinate system. o 0 x 0 y 0 z Convert 0 to global coordinate system oxyz The integral mapping relationship is established as follows: ; In the formula, [ x a , x b ] represents an approximate function in the global coordinate system. g 1( x The domain of ) D Let the energy function be the function to be integrated. G ( x ) represents the mapping weight function after coordinate mapping. L ( x ) represents an approximate function g 1( x ) in the interval [ x a , x The arc length on [].

[0010] Furthermore, step 3 specifically includes: Step 3.1: Based on Timoshenko beam theory, establish the displacement field of a beam structure with arbitrary curvature and varying thickness in a dynamic sub-coordinate system, and determine the displacement of any point on the planar beam in the tangential and normal directions. and It can be represented as: ; in, t Indicates time; u Indicates along the dynamic sub-coordinate system x Linear displacement in the 0 direction; w Indicates along the dynamic sub-coordinate system z Linear displacement in the 0 direction; Indicates the movement around the dynamic sub-coordinate system y Angular displacement along the 0-axis; linear displacement u , w and angular displacement The expression is: ; in, ξ The normalized variable representing the length of the curved beam along the tangential direction on the neutral surface function satisfies... ξ = L ( x ) / L ( x b ); ω Angular frequency; j The imaginary unit; U ( ξ ), W ( ξ )and Φ ( ξ ) respectively represent the dynamic sub-coordinate system x 0 direction, z 0 direction around y Modal displacement function of a beam structure with arbitrary curvature and variable thickness in the 0-axis rotation direction; m For order labels; NT The order of operations for the family of orthogonal displacement functions; a m , b m and c m Representing modal systems number Represents the dynamic sub-coordinate system x 0 direction, z 0 direction around y A family of orthogonal displacement functions in the direction of rotation around the zero axis; Step 3.2: Construct a family of characteristic functions. Set the normalized first-order orthogonal displacement function satisfying the classical boundary conditions as the initial basis function, and construct higher-order orthogonal displacement functions through the following recursive relationship: ;

[0011] The formula for calculating the intermediate variable is as follows: ; Among the symbols P ( ξ ) represents the weight function, defined as P ( ξ ) = h ( ξ ) × G ( ξ ), h ( ξ () is an axially normalized variable thickness function. G ( ξ ) is the coordinate mapping weight function with axial normalization; Step 3.3: Calculate the 2-norm of the higher-order orthogonal displacement function, its expression is: ; By normalizing the orthogonal polynomial series, we can construct the standardized orthogonal basis functions: ; The standardized orthogonal basis functions satisfy the following Kronecker orthogonality relation: ; In the formula, The Kronecker function is used to derive a family of orthogonal displacement functions applicable to beam structures with arbitrary curvature and varying thickness.

[0012] Furthermore, step 4 specifically includes: Step 4.1: Based on Timoshenko beam theory and the strain-displacement relationship, combined with the integral mapping relationship in Step 2, derive the potential energy equation of an arbitrary curvature variable thickness beam structure in the global coordinate system. Its expression is: ; In the formula, U ε Represents potential energy. A 11 , B 11 , D 11 and A 55 These represent the tensile stiffness, tensile coupling stiffness, bending coupling stiffness, and shear stiffness coefficients, respectively.G ( x ) is the coordinate mapping weight function. R It is a function of radius of curvature; Step 4.2: Derive the kinetic energy equation for a beam structure with arbitrary curvature and variable thickness. Its expression is: ; In the formula, T Indicates kinetic energy. I 0、 I 1 and I 2 are the moment of inertia coefficients obtained by integrating along the thickness direction; Step 4.3: Under the condition of multi-point arbitrary stiffness constraint, establish the boundary spring potential energy equation for the beam structure with arbitrary curvature and variable thickness. Its expression is: ; In the formula, U b This represents the total boundary spring potential energy. Indicates that it is located at The position at the first One constraint , and These are the corresponding constraint stiffnesses.

[0013] Furthermore, step 5 specifically includes: Step 5.1: Substitute the family of orthogonal displacement functions into the potential energy equation, kinetic energy equation, and boundary spring potential energy equation respectively to construct the total energy functional of the system; Step 5.2: Based on the Rayleigh-Ritz method, calculate the partial derivatives of each Ritz parameter in the total energy functional and set them to zero to derive the eigenvalue equations of the system: ; In the formula, K is the total stiffness matrix, satisfying K=K ε +K b K ε Let K be the elastic potential energy matrix. b M is the elastic boundary matrix; M is the mass matrix; ω denoted as the natural circular frequency; X is the Ritz vector containing unknown modal coefficients; Step 5.3: Solve the eigenvalue equations using the matrix decomposition algorithm to obtain the natural frequency vector and modal coefficient matrix of the beam structure with arbitrary curvature and variable thickness, and then combine them with the orthogonal displacement function family to obtain the mode shape functions of each order.

[0014] Furthermore, step 6 specifically includes: Step 6.1: Based on the obtained mode shape functions and the Lagrange equations, establish the dynamic differential equations for a beam structure with arbitrary curvature and variable thickness: ; In the formula, , and Let F represent the mass matrix, damping matrix, and stiffness matrix, respectively, and let F represent the excitation force vector applied at any position in the beam structure. Step 6.2: By combining coordinate transformation with modal orthogonality, the dynamic differential equations are decoupled into the modal coordinate system, so that the mass matrix and stiffness matrix are converted into diagonal matrix form; Step 6.3: Solve the decoupled dynamic differential equations using a numerical solution algorithm to obtain the dynamic response of the beam structure with arbitrary curvature and variable thickness; Step 6.4: Substitute the dynamic response into the strain-displacement relationship and constitutive equation to calculate and derive the dynamic stress of an arbitrary curvature variable thickness beam structure under forced vibration.

[0015] The beneficial effects of this invention are as follows: (1) This invention establishes an integral mapping relationship between the global coordinate system and the dynamic sub-coordinate system, and constructs a fusion variable thickness function. h ( ξ and mapping weight function G ( ξ By combining the exclusive family of orthogonal displacement functions with coordinate mapping transformation relationship integral calculation, a unified semi-analytical modeling of beam structures with arbitrary curvature and varying thickness in the plane is realized, overcoming the technical bottleneck of traditional semi-analytical methods that cannot handle complex curved beams due to insufficient geometric adaptability. (2) The present invention uses the Rayleigh-Ritz method based on the energy principle combined with the modal superposition method for solving the problem. It does not require complex mesh generation and preprocessing operations. Compared with the traditional finite element method, it significantly reduces the preprocessing workload and computational resource consumption, and greatly improves the computational efficiency. (3) This invention uses a nonlinear least squares optimization algorithm to accurately solve the neutral surface function and considers the effects of shear deformation and rotational inertia based on Timoshenko beam theory, which makes the prediction accuracy of the mode shape, natural frequency, dynamic response and dynamic stress of complex variable thickness curved beam structures higher and can effectively replace some expensive physical test methods. Attached Figure Description

[0016] Figure 1 A schematic diagram of a beam structure with arbitrary curvature and variable thickness in a plane in the aerospace field, provided by the present invention; Figure 2 A flowchart illustrating the dynamic analysis method for beam structures with arbitrary curvature and variable thickness in a plane provided by this invention; Figure 3 A schematic diagram illustrating the solution process and coordinate mapping of the neutral surface in the geometric model of an arbitrary curvature variable thickness beam structure provided by this invention. Figure 4 This is a schematic diagram of the modal shape results provided by the present invention; Figure 5 This is a schematic diagram of harmonic external excitation provided by the present invention; Figure 6 A schematic diagram comparing the dynamic response and finite element simulation results provided by this invention; Figure 7 This is a schematic diagram of the dynamic stress amplitude results provided by the present invention. Detailed Implementation

[0017] The technical solutions of this invention will now be clearly and completely described with reference to the accompanying drawings. Obviously, the described embodiments are merely some embodiments of this invention, and not all embodiments. All other embodiments obtained by those skilled in the art based on the embodiments of this invention without inventive effort are within the scope of protection of this invention.

[0018] like Figure 1 , Figure 2 As shown, this invention provides a dynamic analysis method for a planar beam structure with arbitrary curvature and variable thickness, using a curved beam frame structure of a certain type of aircraft wing as an example. To meet aerodynamic requirements, the curvature of this curved beam frame changes continuously along the axial direction, and its thickness is non-uniformly distributed along the axial direction, making it a typical planar beam structure with arbitrary curvature and variable thickness.

[0019] Step 1: Solve for the neutral surface approximation function and the variable thickness function. like Figure 3 As shown, the geometric parameters of the curved beam skeleton are obtained. In this embodiment, the upper surface function of the curved beam is extracted from the CAD model. f 1( x ) and lower surface function f 2( x ), is defined as follows: ; In the formula, f 01 ( x )and f 02 ( x It is defined as follows: ; Specifically, in step 1.1, it is assumed that the neutral surface function is... g ( x After passing through the neutral surface functiong ( x any point on ) P i On the normal line, let it intersect the surface function. f 1( x )At A i Point, intersecting the surface function f 2( x )At B i Points, according to Timoshenko beam theory, line segments P i A i and P i B i The length satisfies the following relationship: ; In the formula, i Indicates the number starting from one end of the curved beam. i The discrete point labels on the neutral surface function N This represents the total number of points discretized on the neutral surface function. In this embodiment, points are selected uniformly along the length of the curved beam. N = 3000 discrete sampling points.

[0020] In step 1.2, set point A i and points B i Their coordinates in the plane coordinate system are respectively and Based on Timoshenko's beam theory, and combined with the line segment in step 1.1 P i A i and P i B i Length relationship and point A i and points B i From the coordinate parameters, the neutral surface function is derived. g ( x Any point on the equation satisfies the following equality constraint: ; In step 1.3, the equality constraint relationship is transformed into an unconstrained nonlinear least squares optimization model, using the discrete ordinate vector of the neutral surface. gAs decision variables, minimizing the sum of squared residuals of distance differences at all discrete sampling points, the optimization model is expressed as: ; ; In the formula, S ( g The expression represents the objective function, which is the cumulative residual of the distance differences between all sampling points. PA i ( g )and PB i ( g All of these are decision variables. g The implicit function, whose value depends on the neutral plane in P i The direction of the normal to the position.

[0021] In step 1.4, the trust-region-reflective algorithm is used to solve the above optimization model, obtaining discrete neutral surface data points. Polynomial fitting is then performed on the discrete neutral surface data points to obtain a high-precision approximation function of the neutral surface. g 1( x Using approximate functions g 1( x and upper and lower surface functions f 1( x )and f 2( x The thickness function of the curved beam is calculated. h ( x ).

[0022] Approximate function g 1( x The expression for ) is as follows: ; In the formula, a i , b i The specific values ​​of the coefficients w and w are shown in the table below: Table 1: Parameter Items a i , b i and w

[0023] Step 2: Establish integral mapping relationship like Figure 3 As shown, in step 2.1, a dynamic sub-coordinate system is constructed at the left end of the curved beam skeleton structure. o 0x 0 y 0 z 0, where x 0 direction and approximation function g 1( x The tangential direction is consistent with that of the other two. z 0 direction and approximation function g 1( x The normal direction is consistent with that of the ).

[0024] In step 2.2, the integration domain is transformed from the dynamic sub-coordinate system using a mapping transformation. o 0 x 0 y 0 z Convert 0 to global coordinate system oxyz The integral mapping relationship is established as follows:

[0025] In the formula, [ x a , x b ] represents an approximate function in the global coordinate system. g 1( x The domain of ) D Let the energy function be the function to be integrated. G ( x ) represents the mapping weight function after coordinate mapping. L ( x ) represents an approximate function g 1( x ) in the interval [ x a , x The arc length on the [ ]. Through the above mapping integral, a unified mathematical integral for arbitrary curved beam structures can be effectively achieved.

[0026] Step 3: Construct a family of dedicated orthogonal displacement functions In step 3.1, based on Timoshenko beam theory, the displacement field of the curved beam skeleton in the dynamic sub-coordinate system is established. The displacements of any point on the planar beam in the tangential and normal directions are calculated. and It can be represented as: ; in, t Indicates time. u Indicates along the dynamic sub-coordinate system x Linear displacement in the 0 direction, w Indicates along the dynamic sub-coordinate system z Linear displacement in the 0 direction, Indicates the movement around the dynamic sub-coordinate system yAngular displacement along the 0-axis. Linear displacement. u , w and angular displacement The expression is: ; in, ξ The normalized variable representing the length of the curved beam along the tangential direction on the neutral surface function satisfies... ξ = L ( x ) / L ( x b ), ω Angular frequency, j It is the imaginary unit. U ( ξ ), W ( ξ )and Φ ( ξ ) respectively represent the dynamic sub-coordinate system x 0 direction, z 0 direction around y Modal displacement function of a beam structure with arbitrary curvature and variable thickness in the 0-axis rotation direction. m For order labeling, NT is the operational order of the family of orthogonal displacement functions. a m , b m and c m Representing modal systems number , Represents the dynamic sub-coordinate system x 0 direction, z 0 direction around y A family of orthogonal displacement functions in the direction of rotation along the 0-axis.

[0027] In step 3.2, a family of characteristic functions is constructed. A normalized first-order orthogonal displacement function satisfying classical boundary conditions is set as the initial basis function. Higher-order orthogonal displacement functions are then constructed through the following recursive relationship: ;

[0028] The formula for calculating the intermediate variable is as follows: ; Among them, symbols P ( ξ ) represents the weight function, defined as P ( ξ ) = h ( ξ ) × G ( ξ ),h ( ξ () is an axially normalized variable thickness function. G ( ξ ) is the axially normalized coordinate mapping weight function. This weight function integrates the variable thickness and curvature mapping characteristics of the beam, enabling the constructed family of orthogonal displacement functions to adaptively reflect the actual geometric characteristics of beam structures with arbitrary curvature and variable thickness.

[0029] In step 3.3, the 2-norm of the higher-order orthogonal displacement function is calculated: ; By normalizing the orthogonal polynomial series, we can construct the standardized orthogonal basis functions: ; The standardized orthogonal basis functions satisfy the following Kronecker orthogonality relation: ; In the formula, This is the Kronecker function. From this, we obtain a family of orthogonal displacement functions applicable to beam structures with arbitrary curvature and varying thickness.

[0030] Step 4: Derive the energy equation In step 4.1, based on Timoshenko beam theory and the strain-displacement relationship, combined with the integral mapping relationship in step 2, the potential energy equation of the curved beam skeleton in the global coordinate system is derived: ; In the formula, U ε Represents potential energy. A 11 , B 11 , D 11 and A 55 These represent the tensile stiffness, tensile coupling stiffness, bending coupling stiffness, and shear stiffness coefficients, respectively. G ( x ) is the coordinate mapping weight function. R This is a function of the radius of curvature. In this embodiment, the curved beam material is aerospace structural steel, and the Young's modulus is... E = 210 GPa, Poisson's ratio ν = 0.3, density ρ = 7850 kg / m 3 Damping ratio ξ c = 0.02.

[0031] In step 4.2, the kinetic energy equation is derived: ; In the formula, T Indicates kinetic energy. I 0、 I 1 and I 2 are the moment of inertia coefficients obtained by integrating along the thickness direction.

[0032] In step 4.3, under the condition of multi-point arbitrary stiffness constraint, the boundary spring potential energy equation is established: ; In the formula, U b This represents the total boundary spring potential energy. Indicates that it is located at The position at the first One constraint , and These represent the corresponding constraint stiffnesses. In this embodiment, the left end of the curved beam is fixedly constrained. The right end adopts a free boundary ( ).

[0033] Step 5: Modal Solution In step 5.1, the family of orthogonal displacement functions is substituted into the potential energy equation, kinetic energy equation and boundary spring potential energy equation respectively to construct the total energy functional of the system.

[0034] In step 5.2, based on the Rayleigh-Ritz method, the partial derivatives of each Ritz parameter in the total energy functional are calculated and set to zero to derive the eigenvalue equations of the system: ; In the formula, K is the total stiffness matrix, satisfying K=K ε +K b K ε Let K be the elastic potential energy matrix. b Let M be the elastic boundary matrix and M be the mass matrix. ω Let X be the natural circular frequency, and let X be the Ritz vector containing unknown modal coefficients.

[0035] In step 5.3, construct matrix S=M -1 (K ε +K b By solving the eigenvalue equations through QR decomposition, the natural frequency vector and modal coefficient matrix of the curved beam skeleton are obtained. Then, combined with a family of orthogonal displacement functions, the mode shape functions of each order are obtained, such as... Figure 4 As shown.

[0036] Step 6: Solving for dynamic response and dynamic stress In step 6.1, based on the obtained mode shape functions and the Lagrange equations, the dynamic differential equations of the curved beam skeleton are established: ; In the formula, , and Let F represent the mass matrix, damping matrix, and stiffness matrix, respectively, and let F represent the excitation force vector applied at any location on the beam structure. In this embodiment, F is a 1000 Pa harmonic uniformly distributed excitation, such as... Figure 5 As shown. The damping matrix adopts a proportional damping model, and the damping ratio is taken as... ξ c = 0.02.

[0037] In step 6.2, the dynamic differential equations are decoupled to the modal coordinate system by coordinate transformation and modal orthogonality, so that the mass matrix and stiffness matrix are converted into diagonal matrix form.

[0038] In step 6.3, the transfer function method is used to solve the decoupled dynamic differential equations to obtain the amplitude-frequency response curve of the curved beam frame under harmonic excitation, as shown below. Figure 6 As shown. For transient excitation conditions, the Runge-Kutta method is used for numerical solution to obtain the time-domain dynamic response of the beam structure with arbitrary curvature and varying thickness.

[0039] In step 6.4, the dynamic response is substituted into the strain-displacement relationship and constitutive equations to calculate and derive the dynamic stress distribution of the curved beam skeleton under forced vibration. Specifically, according to Timoshenko beam theory, the normal strain and shear strain can be obtained from the derivative of the displacement field, combined with the elastic modulus of the material. E and shear modulus G This allows us to calculate the variation of normal and shear stresses on the beam cross-section over time, such as... Figure 7 The figure shows the dynamic stress amplitude distribution of the entire beam structure under different resonance orders under harmonic excitation, which provides data support for fatigue life assessment and safety verification of the structure.

[0040] The above description is merely an embodiment of the present invention and is not intended to limit the scope of protection of the present invention. For those skilled in the art, the present invention can have various modifications and variations. Any modifications, equivalent substitutions, improvements, etc., made within the spirit and principles of the present invention should be included within the scope of protection of the present invention.

Claims

1. A dynamic analysis method for a beam structure with arbitrary curvature and variable thickness in a plane, characterized in that, Includes the following steps: Step 1: Obtain the geometric parameters of a beam structure with arbitrary curvature and varying thickness in a plane, input the upper and lower surface functions of the beam structure, and solve for the high-precision approximate function of the neutral surface and the varying thickness function of the beam structure; Step 2: Establish the global coordinate system and dynamic sub-coordinate system of the beam structure, and derive the integral mapping relationship between the global coordinate system and the dynamic sub-coordinate system; Step 3: Based on the neutral surface function, construct a family of orthogonal displacement functions applicable to beam structures with arbitrary curvature and varying thickness; Step 4: Based on the orthogonal displacement function family and the integral mapping relationship, derive and solve the kinetic energy equation, potential energy equation and boundary constraint equation of the beam structure with arbitrary curvature and variable thickness; Step 5: Construct the total energy functional of the system by combining the kinetic energy equation, potential energy equation and boundary constraint equation, solve the total energy functional based on the Rayleigh-Ritz method, and obtain the mode shapes and natural frequencies of the beam structure with arbitrary curvature and variable thickness. Step 6: Based on the modal shapes and natural frequencies, the dynamic response and dynamic stress of the beam structure with arbitrary curvature and variable thickness under external excitation are calculated using the modal superposition method, thus completing the dynamic analysis.

2. The dynamic analysis method for a beam structure with arbitrary curvature and variable thickness in a plane according to claim 1, characterized in that, Step 1 specifically includes: Step 1.1: Top surface function based on input of arbitrary curvature and variable thickness beam structure f 1( x ) and lower surface function f 2( x Assume the neutral surface function is g ( x ); after the neutral surface function g ( x any point on ) P i normal and upper surface function f 1( x Intersect at A i Point, and the function of the lower surface f 2( x Intersect at B i Points, according to Timoshenko beam theory, line segments P i A i and P i B i The length satisfies the following relationship: ； In the formula, i Indicates the number starting from one end of the beam structure. i The discrete point labels on the neutral surface function N This represents the total number of points discretized on the neutral surface function. Step 1.2: Let the point be... A i and points B i Their coordinates in the plane coordinate system are respectively and Based on Timoshenko's beam theory, and combined with the line segment described in step 1.1 P i A i and P i B i The length relationship and the point A i and points B i The coordinate parameters are used to derive the neutral surface function. g ( x Any point on the equation satisfies the following equality constraint: ； Step 1.3: Transform the aforementioned equality constraint relationship into an unconstrained nonlinear least squares optimization model, using the discrete ordinate vector of the neutral surface. g As decision variables, minimizing the sum of squared residuals of distance differences at all discrete sampling points, the optimization model is expressed as: ； ； In the formula, S ( g The expression represents the objective function, which is the cumulative residual of the distance differences between all sampling points. PA i ( g )and PB i ( g All of these are decision variables. g The implicit function, the value of which depends on the neutral plane in P i The direction of the normal to the position; Step 1.4: Solve the optimization model using a nonlinear least squares optimization algorithm to obtain discrete neutral surface data points, and perform polynomial fitting on the discrete neutral surface data points to obtain a high-precision approximation function of the neutral surface. g 1( x Using the approximate function g 1( x and upper and lower surface functions f 1( x )and f 2( x The thickness function of the beam structure is calculated. h ( x ).

3. The dynamic analysis method for a beam structure with arbitrary curvature and variable thickness in a plane according to claim 2, characterized in that, Step 2 specifically includes: Step 2.1: Construct a dynamic sub-coordinate system at the end of the beam structure. o 0 x 0 y 0 z 0, where x 0 direction and the approximate function g 1( x The tangential direction is consistent with that of the other two. z 0 direction and the approximate function g 1( x The normal direction is consistent with that of the other two directions. Step 2.2: To perform integral calculations of total kinetic and potential energy, a mapping transformation is used to change the integration domain from the dynamic sub-coordinate system. o 0 x 0 y 0 z Convert 0 to global coordinate system oxyz The integral mapping relationship is established as follows: ； In the formula, [ x a , x b ] represents the approximate function in the global coordinate system. g 1( x The domain of ) D Let the energy function be the function to be integrated. G ( x ) represents the mapping weight function after coordinate mapping. L ( x ) represents the approximate function g 1( x ) in the interval [ x a , x The arc length on [].

4. The dynamic analysis method for a beam structure with arbitrary curvature and variable thickness in a plane according to claim 3, characterized in that, Step 3 specifically includes: Step 3.1: Based on Timoshenko beam theory, establish the displacement field of the beam structure in the dynamic sub-coordinate system, and the displacement of any point on the planar beam in the tangential and normal directions. and It can be represented as: ； in, t Indicates time; u Indicates along the dynamic sub-coordinate system x Linear displacement in the 0 direction; w Indicates along the dynamic sub-coordinate system z Linear displacement in the 0 direction; ϕ Indicates the orbit around the dynamic sub-coordinate system y Angular displacement along the 0 axis; linear displacement u , w and angular displacement ϕ The expression is: ； in, ξ The normalized variable representing the length of the curved beam along the tangential direction on the neutral surface function satisfies... ξ = L ( x ) / L ( x b ); ω Angular frequency; j The imaginary unit; U ( ξ ), W ( ξ )and Φ ( ξ ) respectively represent the dynamic sub-coordinate system x 0 direction z 0 direction around y Modal displacement function of a beam structure with arbitrary curvature and variable thickness in the 0-axis rotation direction; m For order labels; NT The order of operation for the orthogonal displacement function family; a m , b m and c m Representing modal systems number; Represents the dynamic sub-coordinate system x 0 direction z 0 direction around y A family of orthogonal displacement functions in the direction of rotation along the 0-axis; Step 3.2: Construct a family of characteristic functions. Set the normalized first-order orthogonal displacement function satisfying the classical boundary conditions as the initial basis function, and construct higher-order orthogonal displacement functions through the following recursive relationship: ； The formula for calculating the intermediate variable is as follows: ； Among the symbols P ( ξ ) represents the weight function, defined as P ( ξ ) = h ( ξ ) × G ( ξ ), h ( ξ () is an axially normalized variable thickness function. G ( ξ ) is the coordinate mapping weight function with axial normalization; Step 3.3: Calculate the 2-norm of the higher-order orthogonal displacement function, its expression is: ； By normalizing the orthogonal polynomial series, we can construct the standardized orthogonal basis functions: ； The standardized orthogonal basis functions satisfy the following Kronecker orthogonality relation: ； In the formula, The Kronecker function is used to obtain a family of orthogonal displacement functions applicable to the beam structure with arbitrary curvature and varying thickness.

5. The dynamic analysis method for a beam structure with arbitrary curvature and variable thickness in a plane according to claim 4, characterized in that, Step 4 specifically includes: Step 4.1: Based on Timoshenko beam theory and the strain-displacement relationship, combined with the integral mapping relationship in Step 2, derive the potential energy equation of the beam structure in the global coordinate system. Its expression is: ； In the formula, U ε Represents potential energy. A 11 , B 11 , D 11 and A 55 These represent the tensile stiffness, tensile coupling stiffness, bending coupling stiffness, and shear stiffness coefficients, respectively. G ( x ) is the coordinate mapping weight function. R It is a function of radius of curvature; Step 4.2: Derive the kinetic energy equation of the beam structure, its expression is: ； In the formula, T Indicates kinetic energy. I 0、 I 1 and I 2 are the moment of inertia coefficients obtained by integrating along the thickness direction; Step 4.3: Under the condition of multi-point arbitrary stiffness constraint, establish the boundary spring potential energy equation of the beam structure, the expression of which is: ； In the formula, U b This represents the total boundary spring potential energy. Indicates that it is located at The position at the first One constraint , and These are the corresponding constraint stiffnesses.

6. The dynamic analysis method for a beam structure with arbitrary curvature and variable thickness in a plane according to claim 5, characterized in that, Step 5 specifically includes: Step 5.1: Substitute the family of orthogonal displacement functions into the potential energy equation, kinetic energy equation, and boundary spring potential energy equation respectively to construct the total energy functional of the system; Step 5.2: Based on the Rayleigh-Ritz method, calculate the partial derivatives of each Ritz parameter in the total energy functional and set them to zero to derive the eigenvalue equations of the system: ； In the formula, K is the overall stiffness matrix, satisfying K = K ε + K b K ε Let K be the elastic potential energy matrix. b M is the elastic boundary matrix; M is the mass matrix; ω is the natural circular frequency; X is the Ritz vector containing unknown modal coefficients; Step 5.3: Solve the eigenvalue equations using a matrix decomposition algorithm to obtain the natural frequency vector and modal coefficient matrix of the beam structure, and then combine them with the orthogonal displacement function family to obtain the mode shape functions of each order.

7. The dynamic analysis method for a beam structure with arbitrary curvature and variable thickness in a plane according to claim 6, characterized in that, Step 6 specifically includes: Step 6.1: Based on the obtained mode shape functions and the Lagrange equations, establish the dynamic differential equations of the arbitrary curvature variable thickness beam structure: ； In the formula, , and Let F represent the mass matrix, damping matrix, and stiffness matrix, respectively, and let F represent the excitation force vector applied at any position in the beam structure. Step 6.2: By combining coordinate transformation with modal orthogonality, the dynamic differential equation is decoupled to the modal coordinate system, so that the mass matrix and stiffness matrix are converted into diagonal matrix form; Step 6.3: Solve the decoupled dynamic differential equations using a numerical solution algorithm to obtain the dynamic response of the arbitrary curvature variable thickness beam structure; Step 6.4: Substitute the dynamic response into the strain-displacement relationship and constitutive equation to calculate and derive the dynamic stress of the arbitrary curvature variable thickness beam structure under forced vibration.

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