Complex damping beam structure vibration complex mode analysis method suitable for any boundary condition

Abstract

The invention discloses a complex damping beam structure vibration complex modal analysis method suitable for any boundary condition, and relates to the technical field of ship and ocean engineering, and the method comprises the steps: building a unit complex dynamic stiffness matrix based on a frequency domain oscillatory differential equation of a damping beam unit with a uniformly distributed elastic layer; the double dynamic stiffness matrixes of all the units are combined, and a beam structure overall double dynamic stiffness matrix considering damping is obtained; any given force and displacement boundary conditions are applied to the overall double dynamic stiffness matrix, and a transcendental function equation used for conducting double modal analysis on the beam structure is obtained; performing triangular mesh division in a solution region in the complex plane, calculating function values of an equation at all mesh nodes, and determining candidate regions based on quadrant changes of the function values; and function zero points or poles existing in the candidate areas are calculated based on the Cauchy argument principle, so that the complex eigenvalues and eigenvectors of the beam structure are obtained, and the method has important significance on rapid and efficient expansion of complex modal analysis.

Description

Technical Field

[0001] The invention relates to the technical field of ship and ocean engineering, and in particular to a complex modal analysis method for vibration of a complex damping beam structure under arbitrary boundary conditions. Background Art

[0002] Complex beam composite structures are widely used in aviation, shipbuilding, civil engineering and other engineering fields because of their good mechanical properties and ability to withstand large loads. Their dynamic performance is greatly affected by the physical properties of the structure, such as material properties and boundary conditions. In traditional engineering structure analysis and design, structural dynamics analysis is generally a real modal analysis, without considering the material damping and viscous damping of the structure; and the boundary conditions of the structure are generally traditional boundary conditions such as fixed support, simple support, and sliding support. As for the vibration modal analysis of beam structures, the damping, elastic boundary and support conditions inside them have a crucial influence on the vibration mode. Therefore, studying the eigenvalue problem of complex beam structures under the influence of damping and elastic boundary conditions is of great significance to the dynamic characteristics analysis of the structure.

[0003] Regarding the complex eigenvalue analysis of damped beam structures, many scholars have proposed different forms of numerical methods and analytical methods and continuously optimized and developed them. Numerical calculation methods represented by finite elements can usually meet engineering requirements, and the error is within the acceptable range, but their calculation accuracy is heavily dependent on the fineness of the grid division, especially for high-frequency structural analysis, the grid needs to be further refined, and the grid needs to be re-divided for structural optimization design. It often requires high performance of the computing platform, low computational efficiency, and substandard accuracy, which is not conducive to accurate prediction of structural vibration characteristics and rapid optimization of parameters. Therefore, accurate and efficient analytical solutions are essential. Although analytical methods are more accurate and efficient than numerical methods. However, for analytical methods of complex damped beam structure problems, different formulas need to be used for different boundary conditions, and then the eigenvalues ​​and corresponding vibration modes of the structure are obtained by solving the control differential equations, which affects the calculation efficiency to a certain extent. And there are still few efficient complex modal analysis analytical methods that can meet arbitrary boundary conditions.

[0004] As a semi-analytical method based on accurate shape functions, the dynamic stiffness method has attracted widespread attention in recent years. Compared with finite element or other approximate methods, this method provides analysts with better model accuracy. However, the current application of dynamic stiffness is mainly for real modal analysis of beam structures and complex modal analysis of simple beam structures under traditional boundary conditions, and there is no complex modal analysis of complex damped beam structures under arbitrary boundary conditions. In addition, the complex eigenvalue problem that needs to be solved for the complex modal analysis of complex damped beam structures is to solve a transcendental function in a complex plane, so the traditional Wittrick-Williams algorithm, which is widely used for solving dynamic stiffness matrices without complex numbers, is no longer applicable.

[0005] In summary, among the many methods for analyzing and studying the vibration modal analysis of beam structures, there are few methods for the vibration modal analysis of complex damped beam structures. Methods with high computational efficiency and high accuracy are often only applicable to specific boundary conditions or traditional boundary conditions, while methods that meet arbitrary boundary conditions often face problems of computational efficiency and numerical accuracy stability. Therefore, in the engineering field, the accurate and efficient analysis of the complex modal vibration of complex damped beam structures under arbitrary boundary conditions faces a huge challenge. Summary of the invention

[0006] In view of the above problems and technical requirements, the inventors have proposed a complex modal analysis method for vibration of complex damping beam structures under arbitrary boundary conditions. The method adopts the complex dynamic stiffness method of the damping beam structure, and also provides an effective, accurate and widely applicable complex eigenvalue analysis method for the complex damping beam structure, providing a powerful means for complex modal analysis and optimization design of the damping beam structure. The technical solution of the present invention is as follows:

[0007] A complex modal analysis method for vibration of a complex damping beam structure under arbitrary boundary conditions includes the following steps:

[0008] Step B: Based on the frequency domain vibration differential equation of the damping beam unit with a uniformly distributed elastic layer, a unit complex dynamic stiffness matrix mapping the boundary generalized displacement coefficient and the boundary generalized force coefficient of the damping beam unit is established;

[0009] Step D: Combine the complex dynamic stiffness matrices of each unit to obtain the overall complex dynamic stiffness matrix of the beam structure considering damping;

[0010] Step F: applying given arbitrary force and displacement boundary conditions to the overall complex dynamic stiffness matrix to obtain a transcendental function equation for complex modal analysis of the beam structure;

[0011] Step H: Perform triangular meshing on the solution area in the complex plane, calculate the function values ​​of the transcendental function equation at all mesh nodes, and determine the candidate area based on the quadrant change of the function value;

[0012] Step J: Based on the Cauchy argument principle, the function zeros or poles existing in the candidate region are calculated to obtain the complex eigenvalues ​​and eigenvectors of the beam structure as the complex modal analysis results of the beam structure with damping.

[0013] A further technical solution is that the damping beam unit with a uniformly distributed elastic layer is as follows: Figure 2 As shown, the elastic layer k s Evenly distributed on the symmetry plane of the beam element along the x and z directions, step B specifically includes:

[0014] Step B1: The frequency domain vibration differential equations of the damping beam unit with uniformly distributed elastic layer are expressed as follows:

[0015] The differential equation for longitudinal vibration control of beam element based on Rayleigh-Love theory is:

[0016]

[0017] Where E is the Young's modulus of the material, ρ is the material density, υ is the Poisson's ratio of the material, A is the cross-sectional area, I p is the cross-sectional polar inertia moment, u is the axial vibration displacement, ζ k is the internal viscoelastic damping coefficient, c a is the external viscous damping coefficient of the system's axial vibration. If the material loss factor η is considered s , then the Young's modulus of the material is E * =E(1+iη s ).

[0018] The governing differential equations of torsional vibration of beam element are:

[0019]

[0020] Where G is the shear modulus, θ x is the torsional vibration displacement around the x-axis.

[0021] The differential equations governing the bending vibration of beam structures based on Timoshenko theory are:

[0022]

[0023] Where, v,θ z are the vibration displacement in the y direction and the vibration displacement around the z axis, I is the moment of inertia of the cross section, κ is the cross section correction coefficient, c s and c b is the external viscous damping coefficient of the system bending vibration, k s is the stiffness of the uniformly distributed springs around the beam.

[0024] Step B2: Solve the longitudinal vibration control differential equation of the beam unit based on Rayleigh-Love theory, the torsional vibration control differential equation of the beam unit, and the bending vibration control differential equation of the beam structure based on Timoshenko theory, respectively, obtain the general solution of each differential equation and substitute it into the natural boundary conditions of the force.

[0025] Assume that the general solution of equations (1)-(3) is H=e λx , where H represents u, v, θ x or θ z , the characteristic equations of the frequency domain vibration control differential equations are obtained, which are expressed as follows:

[0026] The characteristic equation of longitudinal vibration of beam element based on Rayleigh-Love theory is:

[0027]

[0028] in

[0029] The characteristic equation of torsional vibration of beam element is:

[0030]

[0031] in

[0032] The characteristic equation of bending vibration of beam structure based on Timoshenko theory is:

[0033]

[0034] in

[0035] Solving equations (4)-(6), we can obtain the general solutions of the frequency domain vibration control differential equations (1)-(3), which are expressed as follows:

[0036] General solution for longitudinal vibration of beam element based on Rayleigh-Love theory:

[0037]

[0038] General solution for torsional vibration of beam element:

[0039]

[0040] General solution of bending vibration of beam structure based on Timoshenko theory:

[0041]

[0042] Where C = cos(λ b1 x),S=sin(λ b1 x), C a1 , C a2 , C t1 , C t2 , A1-A4 are all arbitrary unknown constants;

[0043]

[0044]

[0045]

[0046]

[0047] Substituting the general solution of the beam element vibration control differential equation into the natural boundary conditions of the force, they are expressed as follows:

[0048] Longitudinal vibration of beam element based on Rayleigh-Love theory:

[0049]

[0050] Torsional vibration of beam element:

[0051]

[0052] Bending vibration of beam structures based on Timoshenko theory:

[0053]

[0054] where q y 、m z represent shear force and bending moment respectively.

[0055] Step B3: Substitute the boundary conditions of the force and displacement at both ends of the beam unit with a length of L into the general solution of the differential equation and the natural boundary conditions of the force to obtain the relationship between the force and displacement at both ends of the beam unit and the unknown constant vector.

[0056] Substituting 0 and L into equations (10)-(12), the boundary conditions of the forces and displacements at both ends of the beam element are:

[0057] Longitudinal vibration:

[0058]

[0059] Torsional vibration:

[0060]

[0061] Bending vibration:

[0062]

[0063] The combined equations (7)-(15) can be obtained respectively,

[0064] Longitudinal vibration:

[0065]

[0066] Torsional vibration:

[0067]

[0068] Bending vibration:

[0069]

[0070] Step B4: Perform matrix inversion operations on equations (16)-(18) to eliminate the unknown constant vectors, and obtain the unit complex dynamic stiffness matrix that maps the displacement and force boundary conditions, which are expressed as follows:

[0071] Longitudinal vibration:

[0072]

[0073] Torsional vibration:

[0074]

[0075] Bending vibration:

[0076]

[0077] in:

[0078]

[0079]

[0080]

[0081]

[0082]

[0083]

[0084]

[0085]

[0086]

[0087] Finally, by combining equations (19)-(21), the unit complex dynamic stiffness matrix K that maps the entire beam unit displacement δ and force boundary conditions f is obtained:

[0088] f=Kδ (22)

[0089] in:

[0090]

[0091]

[0092] Its further technical solution is that step D specifically includes:

[0093] Step D1: Build Figure 3The three coordinate systems shown are the global coordinate system (XYZ) of the beam structure, the local coordinate system (xyz) of the beam element, and the auxiliary coordinate system (x'y'z'). Element local coordinate system: The axis of the beam element is taken as the x-axis, and the left end of the element is the coordinate origin. At this time, there are two cases for the x-axis of the coordinate system: (1) The x-axis of the local coordinate system is parallel to the Z-axis of the global coordinate system. In this case, the y-axis of the local coordinate system can be set to be parallel to the Y-axis of the global coordinate system, and then the direction of the z-axis of the local coordinate system can be determined by the right-hand screw rule; (2) The x-axis of the local coordinate system is not parallel to the Z-axis of the global coordinate system. In this case, any plane can be made through the x-axis of the local coordinate system of the element perpendicular to the XOY plane of the global coordinate system, and the z-axis is defined in this plane. It is also stipulated that the angle between the positive direction of the z-axis and the positive direction of the Z-axis is less than 90°, and then the direction of the y-axis of the local coordinate system can be determined by the right-hand screw rule. Auxiliary coordinate system: The auxiliary coordinate system is also called the principal inertial coordinate system, which is a coordinate system related to the principal inertial plane of the beam element. Its specific definition is as follows: the x' axis coincides with the x axis of the local coordinate system, and the auxiliary coordinate system can be obtained by rotating the yoz plane of the local coordinate system around the x axis by a certain angle. It is stipulated that the right rotation around the x axis is positive, and the rotation angle is β (it can be understood that β is the angle between the y axis of the local coordinate system and the y' axis of the auxiliary coordinate system), and -90°<β<90°.

[0094] The unit complex dynamic stiffness matrix K of the damping beam unit in the local coordinate system is transformed to the global coordinate system through the coordinate transformation matrix T, which is expressed as:

[0095] K g =T T KT (25)

[0096] in

[0097]

[0098]

[0099]

[0100] (x i ,y i ,z i )、(x j ,y j ,z j ) are the coordinate values ​​of the two nodes of the unit in the global coordinate system.

[0101] Step D2: Reference Figure 4 As shown in the figure, the unit complex stiffness matrix K of all damping beam units in the global coordinate system is calculated by the common node method. g After assembly, the overall complex dynamic stiffness matrix K(ω) of the beam structure considering damping is obtained. In the figure, 1, 2, and 3 represent nodes, and ①, ②, and ③ represent damped beam units.

[0102] A further technical solution is that in step F, the method of applying arbitrary boundary conditions includes: when the force and displacement boundary conditions applied to the beam structure node are fixed constraints (such as traditional fixed support, simple support, sliding support, etc.), the corresponding rows and columns of the boundary constraint nodes in the overall complex dynamic stiffness matrix are deleted, and reference is made to Figure 5 When the force and displacement boundary conditions applied to the beam structure nodes are elastic constraints (such as spring damping, etc.), the corresponding degrees of freedom in the overall complex stiffness matrix are added with the parameters of the spring and damping, refer to Figure 6 As shown. x / y / z , k θx / θy / θz 、c x / y / z 、c θx / θy / θz Represent the parameter values ​​of spring stiffness and damping on each degree of freedom respectively.

[0103] Through the above operations, the final overall complex dynamic stiffness matrix of the damping beam structure can be obtained. If its determinant is equal to 0, it can be used to solve the eigenvalue of the beam structure. The transcendental function equation is expressed as:

[0104] f(ω)=|K(ω)|=0 (28)

[0105] A further technical solution is that step G: obtaining a solution region in a complex plane comprises:

[0106] Step G1: Given a rectangular solution region in the complex plane: the real part range is given on the real x-axis [a, b], and the imaginary part range is given on the imaginary y-axis [c, d]; give the initial grid division accuracy Δ x and Δ y ;

[0107] Step G2: using a two-dimensional bisection method to divide the given rectangular solution area into a certain number of small rectangular areas that meet the initial meshing accuracy;

[0108] Step G3: Use the Cauchy argument principle to determine the number of zeros of the transcendental function in a small rectangular area (contour integration domain), and find the small rectangular area where the zeros exist as the solution area in the complex plane.

[0109] Its further technical solution is that step G3 specifically includes:

[0110] The Cauchy argument theorem is expressed as follows:

[0111]

[0112] The geometric meaning of this formula is that when z goes around the closed curve C in the positive direction, the number of revolutions of f(z) around the origin is q, and q is always an integer, whose value is equal to the number of zeros N of f(z) in C minus the number of poles P in C; the closed curve C is discretized to obtain the discrete Cauchy argument principle:

[0113]

[0114] where z1,z2,z3,...,z p is a series of discrete points on the closed curve C, z p+1 = z1, the discrete points sequentially form a discrete curve C on the closed curve C p ,Right now The conditions satisfied by a series of discrete points are:

[0115]

[0116] According to equations (30) and (31), a small rectangular area with the number of zero points N greater than 0 is determined.

[0117] Its further technical solution is that step H specifically includes:

[0118] Step H1: Solve region Ω in the complex plane i In the example, given the target solution accuracy δ and the number of nodes in the initial mesh division, the solution area Ω is divided according to the number of nodes in the initial mesh division. i Use Delaunay triangles for meshing;

[0119] Step H2: Calculate the function value of the transcendental function equation (28) at all grid nodes and obtain the quadrant Q where the function value is located n , there are only four cases:

[0120]

[0121] where f(z n ) represents the transcendental function value corresponding to the nth grid node;

[0122] Step H3: In the Delaunay triangular mesh, since the position of the function zero or pole must be the intersection point of the four quadrants (the origin of the coordinate system), the quadrant change value |ΔQ that satisfies the function value at the two end points p1 and p2 can be P |=2 is considered as a candidate edge CE and extracted, where ΔQ P =Q p1 -Q p2 , the set of candidate edges is

[0123] Step H4: Get the candidate triangles connected to the candidate edges, and then get the candidate regions where the function zeros or poles may exist. In addition to the candidate edges, the conditions for forming the candidate triangle edges are to satisfy the quadrant change value |ΔQ of the function value at the two end points p1 and p2 P |≤1.

[0124] A further technical solution is that in step J, the function zeros or poles existing in the candidate region are calculated based on the Cauchy argument principle, including:

[0125] For the candidate region in step H4, all edges except the candidate edge satisfy |ΔQ P |≤1, then the edges of the candidate region satisfy equation (29) at the same time, and equation (29) can be replaced by equation (33):

[0126]

[0127] Where ΔQ p Indicates the quadrant change value of the function value at two adjacent nodes in the candidate area;

[0128] Then, according to the above discrete Cauchy argument principle, the number of function zeros or poles in the candidate region is calculated, and the positions of the function zeros or poles are preliminarily determined.

[0129] Its further technical solution is that the above are all calculations under the initial grid, and sometimes there is a problem that the grid accuracy does not meet the required solution accuracy. Therefore, the method also includes designing a solution area grid division strategy to refine the grid, specifically including:

[0130] After the candidate region is determined, new discrete points are added to the midpoints of the candidate edges in the candidate region to further refine the initial triangular mesh; the newly generated triangular mesh is analyzed. If the ratio of the longest side to the shortest side among the three sides is not less than or equal to 3, new discrete points are added to the centroid of the new triangle, and the new triangular mesh is refined again, forming a solution region mesh division strategy in the complex plane; this strategy is continuously iterated in the process of calculating the function zeros or poles in the candidate region based on the Cauchy argument principle until the given target solution accuracy δ is met.

[0131] The beneficial technical effects of the present invention are:

[0132] The exact solution of the vibration control differential equation of beam unit based on Rayleigh-Love and Timoshenko theory given by this method can accurately describe the vibration of the short and thick beam structure in the full frequency domain. For the overall beam structure, the unit is divided by its "discontinuity point" (the discontinuity point is all factors that cause structural discontinuity, such as external force, structural angle change, etc.), which greatly reduces the scale and degree of freedom of the problem to be solved, thereby reducing the amount of calculation and improving the calculation efficiency. It has obvious calculation advantages in the field of structural optimization and medium and high frequency analysis. The dynamic stiffness method is used to derive the overall complex dynamic stiffness matrix of the damped beam structure, which can be applied to any complex beam structure model. The spring and damping are used to simulate any boundary conditions of the structure, which can be used to simulate various boundary conditions. This method also uses Delaunay triangles to mesh the solution area in the complex plane, providing a basis for the subsequent self-updating mesh strategy. The discrete Cauchy angle principle is used to determine the number of zeros or poles in the solution area to ensure that no roots are missed. A set of automatically updated meshing strategies is used to solve the eigenvalues ​​of a given target solution accuracy δ, ensuring the accuracy of the results with extremely high robustness. BRIEF DESCRIPTION OF THE DRAWINGS

[0133] Figure 1 This is a flow chart of the complex modal analysis method for vibration of a complex damping beam structure under arbitrary boundary conditions provided by the present application.

[0134] Figure 2 It is a schematic diagram of the beam unit with elastic layer damping used in this application.

[0135] Figure 3 It is a schematic diagram of the transformation relationship of the spatial beam unit provided in this application from the local coordinate system to the global coordinate system.

[0136] Figure 4 It is a process diagram provided by the present application for assembling unit complex dynamic stiffness matrices to form an overall complex dynamic stiffness matrix.

[0137] Figure 5 This is a schematic diagram of the overall complex dynamic stiffness matrix after a fixed constraint is imposed on a certain node of the beam structure, as provided in this application.

[0138] Figure 6 This is a schematic diagram of the overall complex dynamic stiffness matrix after elastic constraint is applied to a certain node of the beam structure provided by the present application.

[0139] Figure 7 It is a schematic diagram of a planar damping beam structure under complex boundary conditions provided in this application.

[0140] Figure 8(1)-(5) are schematic diagrams comparing the complex modal analysis results of the structure in the first five modes obtained by the method of the present application and the traditional method, wherein the left side is the analysis method provided by the present application and the right side is the traditional finite element method. DETAILED DESCRIPTION

[0141] The specific implementation of the present invention will be further described below in conjunction with the accompanying drawings.

[0142] Please refer to Figure 1 As shown, the present application provides a complex modal analysis method for vibration of complex damped beam structures under arbitrary boundary conditions, which can be implemented by programming in Python or MATLAB. Specifically, the following steps are included:

[0143] Step B: Based on the frequency domain vibration differential equation of the damping beam unit with a uniformly distributed elastic layer, a unit complex dynamic stiffness matrix mapping the boundary generalized displacement coefficients and boundary generalized force coefficients of the damping beam unit is established.

[0144] For the damping beam unit with uniformly distributed elastic layer, this embodiment uses Figure 7 Taking the plane damping beam structure composed of 8 nodes and 13 beam elements as an example, the beam elements are divided according to their "discontinuity points". The discontinuity points are all factors that cause structural discontinuity, such as external forces, structural angle changes, etc. The geometric and material parameters of the beam elements in the structure are as follows: the unit radius is 0.1m, the density ρ is 7850kg / m 3 , Young's modulus E is 2×10 11 Pa, Poisson's ratio υ is 0.3, and the structural damping coefficient is c a =c b =c s =1,ζ k =0.001,k s =0.

[0145] for Figure 7 For the complex beam structure shown in the figure, starting from the control differential equation of each beam element and the boundary conditions corresponding to the corresponding force and displacement, the unit complex dynamic stiffness matrix that maps the boundary generalized displacement coefficient and boundary generalized force coefficient of the beam element can be established, refer to steps B1 to B4. The geometric parameters and material parameters of each beam element are brought into the program to calculate the unit complex dynamic stiffness matrix, which is expressed in matrix form as K by referring to formula (24). i (ω):

[0146]

[0147] Step D: Combine the complex dynamic stiffness matrices of each unit to obtain the overall complex dynamic stiffness matrix of the beam structure considering damping.

[0148] for Figure 7For the complex beam structure shown in FIG. 1 , referring to step D1, the angle β between the local coordinate system of each beam element and the global coordinate system of the beam structure can be obtained. i , by substituting the analytical expression of the coordinate transformation matrix, we can obtain the complex dynamic stiffness matrix of the beam element in the global coordinate system

[0149]

[0150] The analytical expression of the matrix T is shown in equations (26) and (27).

[0151] Then refer to step D2 for all beam element complex stiffness matrices After assembly, the overall complex dynamic stiffness matrix K(ω) of the beam structure considering damping is obtained.

[0152] Step F: Apply the given arbitrary force and displacement boundary conditions to the overall complex dynamic stiffness matrix to obtain the transcendental function equation for complex modal analysis of the beam structure.

[0153] for Figure 7 The complex beam structure shown in the figure is a plane frame beam with only two degrees of freedom constraints. Therefore, the degrees of freedom of all nodes of all beam elements in the z direction must be constrained by Figure 5 The process of applying constraints is to delete the rows and columns corresponding to the z direction in the overall complex dynamic stiffness matrix. Then, a fixed constraint is applied to node 1, and spring damping boundary conditions are applied to node 4 in the x and y directions. The parameters of the spring and damping are: k a =k b =1000N / m, k r =1000N·s / m,c ba =c bb =300N·s / m,c br =300N·m·s / rad, the overall complex dynamic stiffness matrix update process is as follows Figure 5 and Figure 6 As shown, the transcendental function equation that can be used for complex modal analysis of the structure is obtained, which is shown in formula (29).

[0154] Step G: As the complex dynamic stiffness matrix of the beam structure is a complex transcendental function of the natural frequency and is difficult to solve, a two-dimensional bisection method is used to obtain the solution region in the complex plane.

[0155] In step G1, a rectangular solution region is given in the complex plane: the real part range is given on the real x-axis [0, 450], and the imaginary part range is given on the imaginary y-axis [0, 100]; the initial grid division accuracy Δ is given x =1 and Δ y = 1. Then, according to steps G2 and G3, the set of small rectangular areas with zero points is used as the solution area Ω in the complex plane.i .

[0156] Step H: Perform triangular meshing on the solution area in the complex plane, calculate the function values ​​of the transcendental function equation at all mesh nodes, and determine the candidate area based on the quadrant change of the function value.

[0157] Step J: Based on the Cauchy argument principle, the function zeros or poles existing in the candidate region are calculated to obtain the complex eigenvalues ​​and eigenvectors of the beam structure as the complex modal analysis results of the beam structure with damping.

[0158] According to the program compiled in steps H1 to H4 and step J, the Figure 7 The complex eigenvalues ​​of the complex beam structure shown in FIG. 1 are used to obtain the corresponding eigenvectors, that is, the complex modal analysis results of the beam structure with damping are obtained.

[0159] In order to verify the accuracy of this method, this embodiment also provides the complex eigenvalues ​​and complex modal vibration shapes of the complex modes of the beam structure obtained by this method and the comparison results with the traditional finite element method as shown in Tables 1 and Figure 8 As shown, Figure 8 (1)-(5) correspond to the modal orders 1 to 5 in the table. The results show that this method is in good agreement with the finite element method, which verifies that the calculation method provided by this application is highly accurate and has great potential for engineering application.

[0160] Table 1 The first 10 modal frequencies of the damping beam structure in the embodiment obtained by the algorithm of the present invention and the finite element method

[0161]

[0162] The above is only a preferred embodiment of the present application, and the present invention is not limited to the above embodiments. It is understood that other improvements and changes directly derived or associated by those skilled in the art without departing from the spirit and concept of the present invention should be considered to be included in the protection scope of the present invention.

Claims

1. A complex modal analysis method for vibration of complex damped beam structures under arbitrary boundary conditions, characterized in that: The method comprises: Based on the frequency domain vibration differential equation of the damping beam element with uniformly distributed elastic layer, the element complex dynamic stiffness matrix mapping the boundary generalized displacement coefficient and boundary generalized force coefficient of the damping beam element is established; The complex dynamic stiffness matrix of each unit is combined to obtain the overall complex dynamic stiffness matrix of the beam structure considering damping; The given arbitrary force and displacement boundary conditions are applied to the overall complex dynamic stiffness matrix to obtain the transcendental function equations for complex modal analysis of the beam structure. Perform triangular mesh division on the solution area in the complex plane, calculate the function value of the transcendental function equation at all mesh nodes, and determine the candidate area based on the quadrant change of the function value; Based on the Cauchy argument principle, the function zeros or poles existing in the candidate region are calculated to obtain the complex eigenvalues ​​and eigenvectors of the beam structure as the complex modal analysis results of the beam structure with damping.

2. The complex modal analysis method for vibration of complex damped beam structures under arbitrary boundary conditions according to claim 1 is characterized in that: Obtaining a solution region in the complex plane includes: Given a rectangular solution region and a preliminary meshing accuracy in a complex plane, the given rectangular solution region is divided into a certain number of small rectangular regions that meet the preliminary meshing accuracy; The number of zero points of the transcendental function in the small rectangular area is determined by using the Cauchy argument principle, and the small rectangular area with zero points is found as the solution area in the complex plane.

3. The complex modal analysis method for vibration of complex damped beam structures under arbitrary boundary conditions according to claim 2 is characterized in that: The method of determining the number of zeros of the transcendental function in the small rectangular area by using the Cauchy argument principle includes: The Cauchy argument theorem is expressed as follows: The geometric meaning of this formula is that when z goes around the closed curve C in the positive direction, the number of revolutions of f(z) around the origin is q, and q is always an integer, whose value is equal to the number of zeros N of f(z) in C minus the number of poles P in C; the closed curve C is discretized to obtain the discrete Cauchy argument principle: where z1,z2,z3,...,z p is a series of discrete points on the closed curve C, z p+1 = z1, the discrete points sequentially constitute a discrete curve C on the closed curve C p ,Right now The conditions satisfied by a series of discrete points are: According to equations (2) and (3), the small rectangular area where the number of zero points N is greater than 0 is determined.

4. The complex modal analysis method for vibration of a complex damped beam structure under arbitrary boundary conditions according to claim 3 is characterized in that: The calculation of the function zeros or poles existing in the candidate region based on the Cauchy argument principle includes: Since the edges of the candidate region satisfy equation (1) at the same time, equation (1) is replaced by equation (4): Where ΔQ p Indicates the quadrant change value of the function value at two adjacent nodes in the candidate area; The number of function zeros or poles existing in the candidate region is calculated according to the discrete Cauchy argument principle, and the positions of the function zeros or poles are preliminarily determined.

5. The complex modal analysis method for vibration of complex damped beam structures under arbitrary boundary conditions according to claim 1, characterized in that: The frequency domain vibration differential equation of the damping beam unit with a uniformly distributed elastic layer is used to establish a unit complex dynamic stiffness matrix that maps the boundary generalized displacement coefficient and the boundary generalized force coefficient of the damping beam unit, including: Solve the longitudinal vibration control differential equation of beam element based on Rayleigh-Love theory, the torsional vibration control differential equation of beam element, and the bending vibration control differential equation of beam structure based on Timoshenko theory, obtain the general solution of each differential equation and substitute it into the natural boundary conditions of the force; Substituting the boundary conditions of the force and displacement at both ends of the beam unit with a length of L into the general solution of the differential equation and the natural boundary conditions of the force, the relationship between the force and displacement at both ends of the beam unit and the unknown constant vector is obtained; The matrix inversion operation is performed on the relationship to eliminate the unknown constant vectors therein, and the unit complex dynamic stiffness matrix K mapping the entire beam unit displacement δ and the force boundary condition f is obtained by combination: f=Kδ.

6. The complex modal analysis method for vibration of complex damped beam structures under arbitrary boundary conditions according to claim 1, characterized in that: The complex dynamic stiffness matrix of each unit is combined to obtain the overall complex dynamic stiffness matrix of the beam structure considering damping, including: The unit complex dynamic stiffness matrix K of the damping beam unit in the local coordinate system is transformed to the global coordinate system through the coordinate transformation matrix T to obtain K g =T T KT; The unit complex dynamic stiffness matrix K of all damping beam units in the global coordinate system is calculated by the common node method. g After assembly, the overall complex dynamic stiffness matrix of the beam structure considering damping is obtained.

7. The complex modal analysis method for vibration of complex damped beam structures under arbitrary boundary conditions according to claim 1, characterized in that: By applying the given arbitrary force and displacement boundary conditions to the overall complex dynamic stiffness matrix, the transcendental function equations used for complex modal analysis of the beam structure are obtained, including: When the force and displacement boundary conditions applied to the beam structure nodes are fixed constraints, the rows and columns corresponding to the boundary constraint nodes in the overall complex dynamic stiffness matrix are deleted; When the force and displacement boundary conditions applied to the beam structure nodes are elastic constraints, the parameters of springs and damping are added to the corresponding degrees of freedom in the overall complex dynamic stiffness matrix; The determinant of the updated overall complex dynamic stiffness matrix K(ω) is set equal to 0 and used to solve the eigenvalue of the beam structure. The transcendental function equation is expressed as: f(ω)=|K(ω)|=0.

8. The complex modal analysis method for vibration of complex damped beam structures under arbitrary boundary conditions according to claim 1, characterized in that: Perform triangular meshing on the solution area in the complex plane, calculate the function values ​​of the transcendental function equation at all mesh nodes, and determine the candidate area based on the quadrant change of the function value, including: In the solution region in the complex plane, given a target solution accuracy and an initial number of meshing nodes, the solution region is meshed using Delaunay triangles according to the initial number of meshing nodes; Calculate the function value of the transcendental function equation at all grid nodes and obtain the quadrant Q where the function value is located n , there are only four cases: where f(z n ) represents the transcendental function value corresponding to the nth grid node; In the Delaunay triangle mesh, the quadrant change value |ΔQ of the function value at the two end points p1 and p2 will be satisfied P |=2 is considered as a candidate edge and extracted, where ΔQ P =Q p1 -Q p2 ; A candidate triangle connected to the candidate edge is obtained, and then a candidate region where a function zero or pole may exist is obtained.

9. The complex modal analysis method for vibration of a complex damped beam structure under arbitrary boundary conditions according to claim 8, characterized in that: Obtain candidate triangles connected to the candidate edge, including: In addition to the candidate edges, the conditions for forming candidate triangle edges are to satisfy the quadrant change value |ΔQ of the function value at the two end points p1 and p2 P |≤1.

10. The complex modal analysis method for vibration of a complex damped beam structure under arbitrary boundary conditions according to claim 8, characterized in that: The method further comprises: After determining the candidate region, adding new discrete points at the midpoints of the candidate edges in the candidate region to further refine the initial triangular mesh; The newly generated triangular mesh is analyzed. If the ratio of the longest side to the shortest side among the three sides is not less than or equal to 3, a new discrete point is added at the centroid of the new triangle, and the new triangular mesh is refined again, forming a solution area mesh division strategy in the complex plane. This strategy is continuously iterated during the process of calculating the function zeros or poles existing in the candidate region based on the Cauchy argument principle until the given target solution accuracy is met.

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