Dynamic behavior analysis of tensegrity module based on energy preserving matrix perturbation method

Abstract

The application provides a kind of based on energy-preserving matrix perturbation method's tension integral module dynamic behavior analysis, and the existing modeling method for tension integral structure is different from the method that rope and bar are simulated as not bending stiffness bar or both are regarded as beam, the method is based on energy-preserving matrix perturbation method, string element is used to model rope, and the actual vibration condition of structure can be reflected.Taking three-bar tension integral structure module unit as an example, first, the dynamic stiffness matrix with prestress is established for the rope and bar in the structure respectively;Then the W-W algorithm is used to solve the established dynamic stiffness matrix, and the structure frequency is obtained;Finally, the structure response is obtained under the existing frequency and verified, and from the verification result, it can be seen that the method has good calculation efficiency under the premise of ensuring accuracy.

Description

Technical Field

[0001] This invention belongs to the field of aerospace, and specifically relates to a dynamic behavior analysis of a tensioned integral module based on the energy preservation matrix perturbation method. Background Technology

[0002] Tensile monolithic structures were first proposed and named by Buckmister Fuller based on the artistic forms of Ioganson and Snelson. Although they were initially used primarily in the field of art, tensile monolithic structures have gained widespread recognition from researchers in many fields such as architecture, aerospace, biology, and robotics due to their superior overall uniform load-bearing capacity and significant advantages in lightweighting and controllability. In particular, the importance of tensile monolithic structures is becoming increasingly prominent in aerospace applications.

[0003] Existing modeling methods for tensioned monolithic structures can be essentially divided into two categories: one simulates the ropes and members as rods with negligible bending stiffness, and the other treats both as beams. In fact, the bending stiffness of the rope is a small quantity, but due to the stress stiffening effect caused by pretension, the influence of the rope's geometric stiffness on the system's modes cannot be ignored. However, these key factors have not been fully considered in existing research.

[0004] Currently, most scholars use the finite element method for modeling and often simplify each component into a single element in the calculation, without considering the flexible vibration of the component. However, the finite element method has its inherent limitations. When using the finite element method to calculate a structure, the results are highly dependent on the number of elements. If only one element is used, it is difficult to guarantee the accuracy of the results and it is impossible to represent the vibration of the flexible body. On the other hand, if there are too many elements, it will affect the calculation efficiency.

[0005] Therefore, for tensioned integral structures, a dynamic behavior analysis method that is more realistic and has a shorter computation time is needed. Summary of the Invention

[0006] The purpose of this invention is to overcome the shortcomings of existing technologies and provide a more realistic and computationally efficient method for analyzing the dynamic behavior of tensioned monolithic structures. Based on the energy-preserving matrix perturbation method, taking a three-bar tensioned monolithic structure module as an example, this invention first establishes a dynamic stiffness matrix with prestress for the ropes in the structure; then, it establishes a dynamic stiffness matrix with prestress for the members in the structure; next, it solves the established dynamic stiffness matrices using the WW algorithm to obtain the structural frequencies; finally, it calculates the structural response given the established frequencies. This method can reflect the actual vibration of the structure and has good computational efficiency while ensuring accuracy.

[0007] The technical solution of this invention is as follows:

[0008] Dynamic behavior analysis of a tensioned overall module based on the energy-preserving matrix perturbation method, characterized by the following specific steps:

[0009] Step 1: Model the rope in the structure using string elements, establish the element stiffness matrix with prestress, and then assemble the obtained element stiffness matrix to obtain the overall stiffness matrix of the rope.

[0010] Step 2: Model the members in the structure using the Euler-Bernoulli beam model, establish the element stiffness matrix with prestress, and then assemble the obtained element stiffness matrix to obtain the overall stiffness matrix of the members.

[0011] Step 3: Summate the overall stiffness matrix of the rope obtained in Step 1 and Step 2 with the overall stiffness matrix of the bar to obtain the overall stiffness matrix of the structure. Then, use the WW algorithm to solve the obtained overall stiffness matrix of the structure to obtain the structural frequency.

[0012] Step 4: After obtaining the structural frequency, the response is calculated using the precise integration method based on the energy-preserving matrix perturbation method.

[0013] Furthermore, the element stiffness matrix of the rope established in step 1 is as follows:

[0014]

[0015]

[0016] in,

[0017] The dynamic stiffness matrix of the rope element is represented by the superscript e, which indicates the element and the subscript a, which indicates the rope. Indicates rope prestress; Indicates the length of the rope; Indicates the structural frequency; Indicates the mass density of the rope; This represents the cross-sectional area of ​​the rope.

[0018] Furthermore, the overall stiffness matrix of the rope obtained in step 1 is:

[0019]

[0020] in,

[0021] Represents the overall stiffness matrix of the rope; superscript Represents the transpose of a matrix; The transformation matrix between the local coordinate system and the global coordinate system is expressed as follows:

[0022]

[0023]

[0024] In the formula, Representing the local coordinate system axis; Representing the global coordinate system axis; Representing the local coordinate system axis; Representing the global coordinate system axis; Representing the local coordinate system axis; Representing the global coordinate system axis.

[0025] Furthermore, the element stiffness matrix of the rod established in step 2 is as follows:

[0026]

[0027] in,

[0028] The element dynamic stiffness matrix represents the member element, with the superscript 'e' indicating an element and the subscript 'b' indicating a member element. Indicates the prestress of the member; The stiffness matrix of the rod is represented. The geometric matrix representing the bar. The dynamic mass matrix of the member is given by the following formulas:

[0029]

[0030]

[0031]

[0032] In the formula, Indicates the length of the rod; Indicates the mass density of the rod; Indicates the cross-sectional area of ​​the member; Indicates the moment of inertia of the cross section of the member; Indicates the elastic modulus of a rod; Represents the shape function of the rod; superscript The superscript ' indicates the transpose of the matrix; the superscript ' indicates the first derivative; the superscript " indicates the second derivative.

[0033] Furthermore, the overall stiffness matrix of the member obtained in step 2 is:

[0034]

[0035] in,

[0036] This represents the overall stiffness matrix of the member.

[0037] Furthermore, the overall structural stiffness matrix obtained in step 3 is:

[0038]

[0039] in,

[0040] This represents the overall stiffness matrix of the structure.

[0041] Furthermore, the specific steps in step 3 of solving the overall structural stiffness matrix using the WW algorithm are as follows:

[0042] First, give the desired i-th order frequency. The upper realm and the lower realm ;

[0043] Next, the upper and lower bounds are adjusted using the bisection method so that the number of frequencies at the upper and lower bounds meets the condition.

[0044] Finally, assuming the number of upper and lower bound frequencies meets the condition, it is determined whether the upper and lower bound frequencies are within the set error limit: if they are within the set error limit, then at this time... This is the exact solution for the i-th order frequency; otherwise, readjust the upper and lower bounds until the number of upper and lower bound frequencies and the upper and lower bound frequencies both satisfy the conditions.

[0045] Further:

[0046] The condition that the number of upper and lower bound frequencies must satisfy is:

[0047]

[0048]

[0049] in,

[0050] It is a constant; The number of frequencies is expressed as:

[0051]

[0052] In the formula, This represents the fixed-end frequency number of the structure, taking the axial direction as an example. ,in Indicates the structural frequency. Indicates the length of the structure. Indicates the structural mass density. This represents the structural elastic modulus, and ent() indicates rounding down; This indicates that Gaussian elimination is used to eliminate the elements. The number of negative elements on the main diagonal after reducing to an upper triangular matrix.

[0053] Furthermore, step 4 specifically includes:

[0054] First, the structural modal equations are established as follows:

[0055]

[0056] in,

[0057] Indicates modal coordinates; subscript Indicates time; superscript Indicates the first First mode; Represents modal generalized force;

[0058] Next, dual variables are introduced. The modal equations are transformed into:

[0059]

[0060] in,

[0061] In the formula Represents the identity matrix; ;

[0062] Find:

[0063]

[0064] in,

[0065] In the formula , Generally, N=20; , Given load Written to satisfy ;

[0066] Finally, the i-th order frequency Substitution Find the first Mode shape Then, the structural response was calculated using frequency and mode shape. :

[0067]

[0068] in,

[0069] for A matrix formed by combining elements; for A vector formed by combining elements.

[0070] The beneficial effects of this invention are as follows:

[0071] There are two existing modeling methods for tensioned monolithic structures: one simulates the rope and members as rods neglecting bending stiffness, and the other treats both as beams. Neither method reflects the original physical properties of the rope, thus compromising the accuracy of the calculation results. Compared to existing techniques, this method, based on the energy-preserving matrix perturbation method, uses chord elements to model the rope and Euler-Bernoulli beam models to model the members. This hybrid element modeling approach accurately characterizes the relatively low stiffness of the rope but its stress-stiffening effect on the overall structure, while also fully considering the bending deformation and buckling phenomena that may occur in the members of the tensioned monolithic structure. This physics-preserving hybrid modeling method enables precise simulation and analysis of the mechanical behavior of the tensioned monolithic structure, while also maintaining high computational efficiency. Attached Figure Description

[0072] The above or additional aspects and advantages of the present invention will become apparent and readily understood from the description of the embodiments taken in conjunction with the following drawings, in which:

[0073] Figure 1 This is a flowchart illustrating the dynamic behavior analysis of the tensioned integral module based on the energy-preserving matrix perturbation method of this invention.

[0074] Figure 2 This is a numerical example of the overall structure of the three-bar tensioning system of the present invention, where the thicker parts are the bars and the thinner parts are the ropes;

[0075] Figure 3 The diagram shows the structural response curves of the numerical example of this invention. Detailed Implementation

[0076] The embodiments of the present invention will be further described below with reference to the accompanying drawings.

[0077] This invention provides a dynamic behavior analysis of a tensioned integral module based on an energy-preserving matrix perturbation method. Compared with existing technologies, it has a shorter computation time and better reflects actual conditions, as shown in the attached figure. Figure 1 As shown, the specific steps include:

[0078] Step 1: Establish a dynamic stiffness matrix with prestress for the ropes in the structure;

[0079] First, the ropes in the structure are modeled, and the element stiffness matrix with prestress is established:

[0080]

[0081]

[0082] in,

[0083] The dynamic stiffness matrix of the rope element is represented by the superscript e, which indicates the element and the subscript a, which indicates the rope. Indicates rope prestress; Indicates the length of the rope; Indicates the structural frequency; Indicates the mass density of the rope; Indicates the cross-sectional area of ​​the rope;

[0084] Next, the obtained element stiffness matrices are assembled to obtain the overall stiffness matrix of the rope:

[0085]

[0086] in,

[0087] Represents the overall stiffness matrix of the rope; superscript Represents the transpose of a matrix; The transformation matrix between the local coordinate system and the global coordinate system (independent of ropes or rods) is expressed as follows:

[0088]

[0089]

[0090] In the formula, Representing the local coordinate system axis; Representing the global coordinate system axis; Representing the local coordinate system axis; Representing the global coordinate system axis; Representing the local coordinate system axis; Representing the global coordinate system axis.

[0091] Step 2: Establish a dynamic stiffness matrix with prestress for the members in the structure;

[0092] First, the members in the structure are modeled, and the element stiffness matrix with prestress is established:

[0093]

[0094] in,

[0095] The element dynamic stiffness matrix represents the member element, with the superscript 'e' indicating an element and the subscript 'b' indicating a member element. Indicates the prestress of the member; The stiffness matrix of the rod is represented. The geometric matrix representing the bar. The dynamic mass matrix of the member is given by the following formulas:

[0096]

[0097]

[0098]

[0099] In the formula, Indicates the length of the rod; Indicates the mass density of the rod; Indicates the cross-sectional area of ​​the member; Indicates the moment of inertia of the cross section of the member; Indicates the elastic modulus of a rod; Represents the shape function of the rod; superscript The superscript ' indicates the transpose of a matrix; the superscript ' indicates the first derivative; the superscript " indicates the second derivative.

[0100] Next, the obtained element stiffness matrices are assembled to obtain the overall stiffness matrix of the members:

[0101]

[0102] in,

[0103] This represents the overall stiffness matrix of the member.

[0104] Step 3: Solve the established dynamic stiffness matrix using the WW algorithm to obtain the structural frequencies;

[0105] First, sum the overall stiffness matrix of the rope obtained in steps 1 and 2 with the overall stiffness matrix of the member to obtain the overall stiffness matrix of the structure:

[0106]

[0107] in,

[0108] Represents the overall stiffness matrix of the structure;

[0109] Next, the overall structural stiffness matrix is ​​solved using the WW algorithm to obtain the structural frequencies. Specifically:

[0110] First, give the desired i-th order frequency. The upper realm and the lower realm ;

[0111] Next, the upper and lower bounds are adjusted using a binary search method so that the number of frequencies at the upper and lower bounds satisfies the following condition:

[0112]

[0113]

[0114] in,

[0115] It is a constant; The number of frequencies is expressed as:

[0116]

[0117] In the formula, This represents the fixed-end frequency number of the structure, taking the axial direction as an example. ,in Indicates the structural frequency. Indicates the length of the structure. Indicates the structural mass density. This represents the structural elastic modulus, and ent() indicates rounding down; This indicates that Gaussian elimination is used to eliminate the elements. The number of negative elements on the main diagonal after reducing to an upper triangular matrix;

[0118] Finally, after the number of upper and lower bound frequencies meets the condition, it is then determined whether the upper and lower bound frequencies are within the set error limit. Inside, that is: when At that time, the result obtained This is the exact solution for the i-th order frequency; otherwise, readjust the upper and lower bounds until the number of upper and lower bound frequencies and the upper and lower bound frequencies both satisfy the conditions.

[0119] Step 4: After obtaining the structural frequency, the response is calculated using the precise integration method based on the energy-preserving matrix perturbation method.

[0120] First, the structural modal equations are established as follows:

[0121]

[0122] in,

[0123] Indicates modal coordinates; subscript Indicates time; superscript Indicates the first First mode; Represents modal generalized force;

[0124] Next, dual variables are introduced. The modal equations are transformed into:

[0125]

[0126] in,

[0127] In the formula Represents the identity matrix; ;

[0128] We can obtain:

[0129]

[0130] in,

[0131] In the formula , Generally, N=20; , Given load Written to satisfy ;

[0132] Finally, the i-th order frequency Substitution Find the first Mode shape Then, the structural response was calculated using frequency and mode shape. :

[0133]

[0134] in,

[0135] for A matrix formed by combining elements; for A vector formed by combining elements.

[0136] Numerical Examples

[0137] Taking a three-bar tensioned integral structure as an example, as shown in the attached diagram. Figure 2 As shown, the thicker parts are rods, and the thinner parts are ropes. The structural parameters are shown in the table below:

[0138] Table 1 Overall structural parameters of the three-bar tensioning system

[0139]

[0140] The results calculated using this method are compared with those calculated using traditional methods, as shown in the table below:

[0141] Table 2 Comparison of results and experimental values ​​between this method and the traditional method.

[0142]

[0143] As can be seen from Table 2, the calculation accuracy of this method with only one unit exceeds that of the traditional method with 30 units, which shows that this method has high accuracy.

[0144] The computational efficiency of this method is compared with that of traditional methods, as shown in the table below:

[0145] Table 3 Comparison of computational efficiency between this method and traditional methods

[0146]

[0147] As can be seen from Table 3, when the proposed method and the traditional method are divided into only one unit, the computational efficiency is comparable. However, the accuracy of the traditional method is completely unreliable at this time. When the unit is divided into multiple units, the efficiency of the proposed method will far exceed that of the traditional method.

[0148] Considering the response calculation, the response curve is shown in the attached figure. Figure 3 As shown in (a), the calculated responses of nodes 1, 2, and 3 in the x, y, and z directions are compared with the results from the finite element method, respectively. Figure 3 (b), 3(c), and 3(d) show that the results are relatively consistent, which demonstrates the accuracy of this method in response calculation.

Claims

1. Dynamic behavior analysis of a tensioned overall module based on the energy-preserving matrix perturbation method, characterized in that, The specific steps include: Step 1: Model the rope in the structure using string elements, establish the element stiffness matrix with prestress, and then assemble the obtained element stiffness matrix to obtain the overall stiffness matrix of the rope. Step 2: Model the members in the structure using the Euler-Bernoulli beam model, establish the element stiffness matrix with prestress, and then assemble the obtained element stiffness matrices to obtain the overall stiffness matrix of the members. Step 3: Summate the overall stiffness matrix of the rope obtained in Step 1 and Step 2 with the overall stiffness matrix of the bar to obtain the overall stiffness matrix of the structure. Then, use the WW algorithm to solve the obtained overall stiffness matrix of the structure to obtain the structural frequency. The specific steps for solving the overall stiffness matrix of the structure using the WW algorithm are as follows: First, give the desired i-th order frequency. The upper realm and the lower realm ; Next, the upper and lower bounds are adjusted using a binary search method so that the number of frequencies at the upper and lower bounds satisfies the condition: in, It is a constant; The number of frequencies is expressed as: In the formula, This represents the fixed-end frequency number of the structure, taking the axial direction as an example. ,in Indicates the structural frequency. Indicates the length of the structure. Indicates the structural mass density. This represents the structural elastic modulus, and ent() indicates rounding down; This indicates that Gaussian elimination is used to eliminate the elements. The number of negative elements on the main diagonal after reducing to an upper triangular matrix; Finally, assuming the number of upper and lower bound frequencies meets the condition, it is determined whether the upper and lower bound frequencies are within the set error limit: if they are within the set error limit, then at this time... This is the exact solution for the i-th frequency; otherwise, readjust the upper and lower bounds until the number of upper and lower bound frequencies and the upper and lower bound frequencies both satisfy the conditions. Step 4: After obtaining the structural frequency, the response is calculated using the refined integration method based on the energy-preserving matrix perturbation method. This specifically includes: First, the structural modal equations are established as follows: in, Indicates modal coordinates; subscript Indicates time; superscript Indicates the first First mode; Represents modal generalized force; Next, dual variables are introduced. The modal equations are transformed into: in, In the formula Represents the identity matrix; ; Find: in, In the formula , N=20; , Given load Written to satisfy ; Finally, the i-th order frequency Substitution Find the first Mode shape Then, the structural response was calculated using frequency and mode shape. : in, for A matrix formed by combining elements; for A vector formed by combining elements.

2. The dynamic behavior analysis of the tensioned integral module as described in claim 1, characterized in that, The element stiffness matrix of the rope established in step 1 is as follows: in, The dynamic stiffness matrix of the rope element is represented by the superscript e, which indicates the element and the subscript a, which indicates the rope. Indicates rope prestress; Indicates the length of the rope; Indicates the structural frequency; Indicates the mass density of the rope; This represents the cross-sectional area of ​​the rope.

3. The dynamic behavior analysis of the tensioned integral module as described in claim 1, characterized in that, The overall stiffness matrix of the rope obtained in step 1 is: in, Represents the overall stiffness matrix of the rope; superscript Represents the transpose of a matrix; The transformation matrix between the local coordinate system and the global coordinate system is expressed as follows: In the formula, Representing the local coordinate system axis; Representing the global coordinate system axis; Representing the local coordinate system axis; Representing the global coordinate system axis; Representing the local coordinate system axis; Representing the global coordinate system axis.

4. The dynamic behavior analysis of the tensioned integral module as described in claim 1, characterized in that, The element stiffness matrix of the rod established in step 2 is as follows: in, The element dynamic stiffness matrix represents the member element, with the superscript 'e' indicating an element and the subscript 'b' indicating a member element. Indicates prestress in the member; The stiffness matrix of the rod is represented. The geometric matrix representing the bar. The dynamic mass matrix of the member is given by the following formulas: In the formula, Indicates the length of the rod; Indicates the mass density of the rod; Indicates the cross-sectional area of ​​the member; Indicates the moment of inertia of the cross section of the member; Indicates the elastic modulus of a rod; Represents the shape function of the rod; superscript The superscript ' indicates the transpose of the matrix; the superscript ' indicates the first derivative; the superscript " indicates the second derivative.

5. The dynamic behavior analysis of the tensioned integral module as described in claim 1, characterized in that, The overall stiffness matrix of the rod obtained in step 2 is: in, Represents the overall stiffness matrix of the members; superscript Represents the transpose of a matrix; The transformation matrix between the local coordinate system and the global coordinate system is expressed as follows: In the formula, Representing the local coordinate system axis; Representing the global coordinate system axis; Representing the local coordinate system axis; Representing the global coordinate system axis; Representing the local coordinate system axis; Representing the global coordinate system axis.

6. The dynamic behavior analysis of the tensioned integral module as described in claim 3 or 5, characterized in that, The overall structural stiffness matrix obtained in step 3 is: in, This represents the overall stiffness matrix of the structure.

Images