A method for equivalent plate dynamics modeling of a periodic truss structure

Abstract

The application discloses a kind of equivalent plate dynamics modeling methods of periodic truss structure, comprising: based on Mindlin plate theory, strain high-order Taylor expansion of periodic truss unit is carried out and is modified, to obtain the strain description of periodic truss unit characterized by equivalent plate transverse vibration and in-plane vibration;According to the strain description of periodic truss unit, the strain energy and kinetic energy of periodic truss unit are calculated;According to the energy equivalence principle and the strain energy and kinetic energy of periodic truss unit, the equivalent stiffness matrix and equivalent inertia matrix of the equivalent plate of periodic truss structure are calculated;The equivalent stiffness matrix and equivalent inertia matrix of the equivalent plate are dimensionally reduced to construct the orthotropic equivalent Mindlin plate dynamics model of periodic truss structure.The application breaks through the difficulty that traditional equivalent beam model cannot depict the distributed hinge mechanical characteristics in the width direction of cross section, and solves the problem that traditional equivalent plate model does not consider describing the lateral bending vibration of periodic truss structure.

Description

Technical Field

[0001] This invention relates to the field of spacecraft technology, and in particular to an equivalent plate dynamics modeling method for periodic truss structures. Background Technology

[0002] Due to their advantages such as lightweight, high stiffness-to-density ratio, and high packing capacity, space deployable structures are increasingly widely used in the aerospace field. To meet the demands of larger spacecraft payloads, space periodic antenna structures (such as...) are also increasingly being developed. Figure 1 As shown, a typical space-deployable structure is widely used, and its periodic antenna elements (such as...) Figure 2 As shown, the periodic truss and the probe load are combined by hinges and extend along the length direction. The hinges connecting the periodic truss and the probe load are distributed at intervals along the width direction of the periodic truss section. Therefore, the periodic truss is subjected to nonlinear concentrated forces transmitted by the hinges, which makes the periodic truss structure exhibit strong connection nonlinearity at the hinge connection.

[0003] However, the dynamic models of periodic truss structures established using the traditional finite element method typically have a high number of degrees of freedom, resulting in large computational loads and long processing times, making them unsuitable for the design of vibration control systems. Conversely, the continuum equivalent modeling method, as an analytical or approximate analytical approach, can guarantee both solution efficiency and accuracy requirements, providing convenience for the dynamic modeling of periodic truss structures. Existing methods for equivalent continuum modeling of truss structures mainly fall into two categories: the first category, for truss structures extending in one dimension, uses points to describe the cross-section, equating the truss structure to a continuous beam model; the second category, for planar truss structures extending in two dimensions, uses lines to describe the cross-section, equating the truss structure to a continuous plate model with a consistent geometric shape.

[0004] For a spatial periodic antenna structure, there are two connecting hinges along the width of its cross-section. Due to factors such as the non-smoothness of these hinges, the periodic truss structure exhibits strong local nonlinearity. Existing equivalent beam modeling theories, while capable of describing three-dimensional spatial vibration characteristics, rely on point descriptions of the cross-section, such as... Figure 3 As shown, the installation position characteristics of the two hinges in the width direction of the cross-section cannot be characterized. Therefore, equating the periodic truss structure to a continuous beam model presents the problem of difficulty in representing the connection nonlinearity caused by the hinges. Existing equivalent plate theories use lines to describe the cross-section, such as... Figure 4 As shown, the above-mentioned nonlinear characterization problem of hinge connections can be solved. However, current equivalent plate models are all for planar truss structures extending in two dimensions, which fails to consider the lateral bending vibration of truss structures with spatial three-dimensional vibration characteristics, and still faces difficulties in strain description. Therefore, for such periodic truss structures with nonlinear connection characteristics, current equivalent plate modeling methods cannot solve this problem well. Summary of the Invention

[0005] The purpose of this invention is to provide an equivalent plate dynamic modeling method for periodic truss structures. This method uses the in-plane vibration of an orthogonal anisotropic equivalent plate to simulate the lateral bending vibration of periodic truss structures, overcoming the problem that existing equivalent plate modeling methods do not consider describing the lateral bending vibration of periodic truss structures with spatial three-dimensional vibration characteristics.

[0006] To achieve the above objectives, the present invention is implemented through the following technical solution:

[0007] A method for equivalent plate dynamics modeling of a periodic truss structure, wherein the periodic truss structure comprises a plurality of periodic truss elements, including:

[0008] Step S110: Based on Mindlin plate theory, the strain of the periodic truss element is expanded and corrected to obtain the strain description of the periodic truss element that characterizes the transverse vibration and in-plane vibration of the equivalent plate.

[0009] Step S120: Calculate the strain energy and kinetic energy of the periodic truss unit based on the strain description of the periodic truss unit;

[0010] Step S130: Based on the principle of energy equivalence and the strain energy and kinetic energy of the periodic truss unit, calculate the equivalent stiffness matrix and equivalent inertia matrix of the equivalent plate of the periodic truss structure.

[0011] Step S140: Reduce the dimensionality of the equivalent stiffness matrix and equivalent inertia matrix of the equivalent plate of the periodic truss structure to construct the orthogonal anisotropic equivalent Mindlin plate dynamic model of the periodic truss structure, and perform inherent characteristic analysis.

[0012] Optionally, each of the periodic truss units is a regular triangular prism, and a plurality of the periodic truss units extend in the longitudinal direction to form the periodic truss structure, and each of the periodic truss units consists of 3 longitudinal beams, 6 transverse beams and 6 stay cables.

[0013] Optionally, step S110 includes:

[0014] Based on Mindlin plate theory and the displacement of the periodic truss element, the strain components of the periodic truss element along the equivalent plate thickness direction are obtained, and their expressions are as follows:

[0015]

[0016] Where, ε x ε y and ε z γ represents the tensile and compressive strain of the periodic truss element along the equivalent plate thickness direction;xy γ xz and γ yz The shear strain of the periodic truss unit along the equivalent plate thickness direction; and This represents the tensile and compressive strain on the equivalent plate surface. and This represents the shear strain on the equivalent plate surface. and Let be the curvature on the surface of the equivalent plate. Torque; and for The partial derivative; and Both are functions of coordinates x and y, and their specific expressions are as follows:

[0017]

[0018]

[0019] Among them, u 0 v 0 and w 0 This represents the displacement of the mid-surface of the equivalent plate; and This represents the rotation angle of the cross section at the mid-surface of the equivalent plate;

[0020] To characterize the transverse bending-torsional coupled vibration and transverse local deformation characteristics of the periodic truss unit, the strain components of the periodic truss unit along the equivalent plate thickness direction are subjected to a two-dimensional higher-order Taylor expansion at the origin of the coordinate system to obtain the strain description of the periodic truss unit characterizing the transverse vibration of the equivalent plate; and the origin of the coordinate system is the center of the left cross section of the periodic truss unit.

[0021] Specifically, shear strain γ xy The torsional deformation of the equivalent plate is characterized by the shear strain γ in the strain components of the periodic truss element. xy The Taylor expansion is extended to the third order to more accurately describe the torsional deformation characteristics of the truss using the equivalent plate model; the strain expression of the truss element characterizing the lateral vibration of the equivalent plate is as follows:

[0022]

[0023] in, and It represents a partial differential.

[0024] Optionally, in step S110, the in-plane vibration of the equivalent plate of the periodic truss structure is used to simulate the lateral bending vibration of the periodic truss structure, and the in-plane vibration of the equivalent plate is decoupled from the lateral vibration. To accurately characterize the lateral bending vibration of the periodic truss structure, the strain components of the periodic truss element need to be redefined, and ε needs to be deleted. x ε y and γ xy In the expression and The relevant coupling terms decouple the in-plane vibration from the transverse vibration. Furthermore, removing the coupling terms will cause changes in the in-plane tensile and compressive stiffness. To ensure that the equivalent plate's in-plane vibration mode frequency is equal to the transverse bending vibration mode frequency, ε must be adjusted. x and ε y Make corrections;

[0025] For the strain component of the periodic truss element along the equivalent plate thickness direction, a correction factor for the normal strain is introduced to obtain a strain description of the periodic truss element characterizing the in-plane vibration of the equivalent plate.

[0026] The strain expression for the periodic truss element characterizing the equivalent in-plane vibration is as follows:

[0027]

[0028] Where k1 and k2 are correction coefficients for normal strain.

[0029] Optionally, in step S120, the strain energy of the periodic truss element is calculated using the following formula:

[0030]

[0031]

[0032]

[0033] Among them, U e ε is the strain energy of the periodic truss element; s The strain parameter vector of the equivalent plate of the periodic truss structure; N is the number of components in the periodic truss element, and k is the component number; E (k) A (k) L represents the tensile and compressive stiffness of the k-th member; (k) Indicates the length of the k-th component; F is the strain transformation matrix of the k-th member; (k) Let be the coordinate transformation matrix of the k-th component; This represents the contribution matrix of the bending strain energy and torsional strain energy of the k-th member to the overall stiffness of the periodic truss element; (l( k ),m(k ),n( k )) represents the direction cosine of the k-th component;

[0034] When ε s When representing the lateral vibration strain parameters of the equivalent plate of the periodic truss structure,

[0035]

[0036] When ε s When representing the in-plane vibration strain parameters of the equivalent plate of the periodic truss structure,

[0037]

[0038] The kinetic energy of the periodic truss unit is calculated using the following formula:

[0039]

[0040]

[0041]

[0042]

[0043] Among them, T e The kinetic energy of the periodic truss unit; u s The displacement parameter vector of the equivalent plate of the periodic truss structure; Let be the displacement transformation matrix of the k-th component.

[0044] Optionally, step S130 includes:

[0045] Based on the principle of energy equivalence, the strain energy of the periodic truss unit is made equal to the strain energy of the equivalent plate of the periodic truss structure, so as to obtain the equivalent stiffness matrix of the equivalent plate of the periodic truss structure.

[0046] The expression for the strain energy of the equivalent plate of the periodic truss structure is as follows:

[0047]

[0048] Among them, U p L1 represents the strain energy of the equivalent plate of the periodic truss structure; L2 represents the length of the longitudinal beam and L1 represents the length of the transverse beam. Let be the equivalent stiffness matrix of the equivalent plate of the periodic truss structure;

[0049] byU e =U p The expression for the equivalent stiffness matrix of the equivalent plate of the periodic truss structure is as follows:

[0050]

[0051] Where Δ is the area of ​​the mid-surface of the equivalent plate, and Δ=L1L2;

[0052] Based on the principle of energy equivalence, the kinetic energy of the periodic truss unit is made equal to the kinetic energy of the equivalent plate of the periodic truss structure, so as to obtain the equivalent inertia matrix of the equivalent plate of the periodic truss structure.

[0053] The expression for the kinetic energy of the equivalent plate of the periodic truss is as follows:

[0054]

[0055] Among them, T p Let be the kinetic energy of the equivalent plate of the periodic truss structure; Let be the equivalent inertia matrix of the equivalent plate of the periodic truss structure;

[0056] By T e =T p The expression for the equivalent plate of the periodic truss structure is obtained as follows:

[0057]

[0058] Optionally, in step S140, the strain parameter vector ε of the equivalent plate of the periodic truss structure is... s and displacement parameter vector u s (Used in steps S120 and S130) contains multiple redundant variables, which do not match the variables of the classic Mindlin board model;

[0059] The strain parameter vector ε of the equivalent plate of the periodic truss structure is eliminated by the static condensation method. s Redundant variables in the equation are removed to reduce the dimensionality of the equivalent stiffness matrix of the equivalent plate of the periodic truss structure; the displacement parameter vector u in the equivalent plate of the periodic truss structure is eliminated using a static condensation method. s Redundant variables in the data are used to reduce the dimensionality of the equivalent inertia matrix of the equivalent plate of the periodic truss structure.

[0060] The strain parameter vector ε of the equivalent plate of the periodic truss structure is... s Separate the variables in the data:

[0061] ε s =[ε s1 ε s2 ε s3 ] T

[0062] When ε s When representing the strain parameter of transverse vibration, i.e., out-of-plane vibration, εs1 ε represents the strain variable that is identical to the transverse vibration strain variable of the equivalent Mindlin plate; s2 ε represents the strain variable related to the neglected stress. s3 This represents the strain variable that needs to be deleted. The specific expression is as follows:

[0063]

[0064]

[0065]

[0066] When ε s When representing the strain parameter of in-plane vibration, ε s1 This represents the strain variable that is identical to the in-plane vibration strain variable of the equivalent Mindlin plate. The specific expression is as follows:

[0067]

[0068]

[0069] The displacement parameter vector u of the equivalent plate of the periodic truss structure is... s Separate the variables in the data:

[0070] u s =[u s1 u s2 ] T

[0071] Where u s1 u represents the same displacement variable as the equivalent Mindlin plate. s2 The displacement variable to be deleted is expressed as follows:

[0072]

[0073] The formulas for the equivalent stiffness matrix and equivalent inertia matrix of the equivalent plate of the periodic truss structure can be expressed as follows:

[0074]

[0075] in and All are symmetric matrices. and respectively with ε s1 ε s2 and ε s3 They have the same dimension; and respectively with u s1 and u s2 They have the same dimension;

[0076] Static condensation method is used to condense the equivalent stiffness matrix and inertia matrix By performing dimensionality reduction, we can obtain the equivalent stiffness matrix and equivalent inertia matrix of the simplified equivalent plate of the periodic truss structure, as shown below:

[0077]

[0078]

[0079] Where D is the equivalent stiffness matrix of the equivalent plate of the periodic truss structure after dimensionality reduction; G is the equivalent inertia matrix of the equivalent plate of the periodic truss structure after dimensionality reduction.

[0080] Based on the equivalent stiffness matrix and equivalent inertia matrix of the equivalent plate of the reduced periodic truss structure, an orthogonal anisotropic equivalent Mindlin plate dynamic model of the periodic truss structure is constructed.

[0081] The dynamic model of the equivalent plate is subjected to inherent characteristic analysis to obtain the natural frequencies and mode shapes of the equivalent plate of the periodic truss, and the effectiveness of the model is verified by comparing it with the finite element calculation results.

[0082] Compared with the prior art, the present invention has at least one of the following advantages:

[0083] This invention provides a method for modeling the equivalent plate dynamics of a periodic truss structure. Based on Mindlin plate theory, the strain of the periodic truss element is expanded and corrected using a higher-order Taylor expansion to obtain a strain description of the periodic truss element characterizing the transverse and in-plane vibrations of the equivalent plate. The strain energy and kinetic energy of the periodic truss element can be calculated based on this strain description. According to the energy equivalence principle and the strain energy and kinetic energy of the periodic truss element, the equivalent stiffness matrix and equivalent inertia matrix of the equivalent plate of the periodic truss structure can be calculated. The equivalent stiffness matrix and equivalent inertia matrix of the periodic truss equivalent plate are then dimensionally reduced to construct an orthogonal anisotropic equivalent Mindlin plate dynamic model of the periodic truss structure.

[0084] Compared with the traditional truss structure equivalent beam modeling method based on point section assumption, the present invention adopts the equivalent plate modeling method based on line section assumption, which can be applied to the nonlinear dynamic modeling of space antenna structures with distributed hinges installed in the cross-sectional width direction, and is more engineering applicable.

[0085] Compared with the traditional planar equivalent plate modeling method, this invention uses the in-plane vibration of orthogonal anisotropic equivalent plates to simulate the lateral bending vibration of periodic truss structures, overcoming the problem that existing equivalent plate modeling methods do not consider describing the lateral bending vibration of periodic truss structures with spatial three-dimensional vibration characteristics.

[0086] Compared with traditional numerical calculation methods such as finite element method, the equivalent plate dynamic modeling method adopted in this invention has higher calculation efficiency while ensuring the same accuracy;

[0087] The present invention adopts the obtained orthogonal anisotropic equivalent plate dynamic model, which can more comprehensively consider the characteristics of multiple hinge connections in the cross-sectional width direction, providing important support for subsequent research on the dynamics and vibration control of spatial periodic antenna structures with nonlinear connections, and even studying their complex nonlinear dynamic behavior.

[0088] This invention overcomes the problem that traditional equivalent beam models cannot characterize the mechanical features of hinges distributed along the width of the cross section; it solves the problem that traditional equivalent plate models do not consider the lateral bending vibration of periodic truss structures; and it adopts a semi-analytical method, which is more efficient than numerical methods such as the finite element method, providing important support for the study of the dynamics and vibration control of spatial periodic antenna structures with nonlinear connections. Attached Figure Description

[0089] Figure 1 This is a schematic diagram of a spatial periodic antenna structure.

[0090] Figure 2 This is a schematic diagram of the structure of a periodic antenna element in a spatial periodic antenna structure;

[0091] Figure 3 This is a schematic diagram of the point cross-section assumption of a periodic antenna element;

[0092] Figure 4 This is a schematic diagram of the assumed line cross-section of a periodic antenna element;

[0093] Figure 5 This is a flowchart of an equivalent plate dynamic modeling method for a periodic truss structure provided in an embodiment of the present invention;

[0094] Figure 6 This is a schematic diagram of a periodic truss structure provided in an embodiment of the present invention;

[0095] Figure 7 This is a schematic diagram of the structure of a periodic truss unit in a periodic truss structure according to an embodiment of the present invention;

[0096] Figure 8 This is a schematic diagram of the equivalent plate geometric model of a periodic truss unit provided in an embodiment of the present invention. Detailed Implementation

[0097] The following detailed description, in conjunction with the accompanying drawings and specific embodiments, provides a further detailed explanation of the equivalent plate dynamics modeling method for a periodic truss structure proposed in this invention. The advantages and features of this invention will become clearer from the following description. It should be noted that the accompanying drawings are in a very simplified form and use non-precise scales, used only to facilitate and clearly illustrate the embodiments of this invention. Please refer to the accompanying drawings to make the objectives, features, and advantages of this invention more apparent and understandable. It should be understood that the structures, scales, sizes, etc., depicted in the accompanying drawings are only for illustrative purposes to aid those skilled in the art and are not intended to limit the implementation conditions of this invention. Therefore, they have no substantial technical significance. Any modifications to the structure, changes in scale, or adjustments to size, without affecting the effects and objectives achieved by this invention, should still fall within the scope of the technical content disclosed in this invention.

[0098] It should be noted that, in this document, relational terms such as "first" and "second" are used only to distinguish one entity or operation from another, and do not necessarily require or imply any such actual relationship or order between these entities or operations. Furthermore, the terms "comprising," "including," or any other variations thereof are intended to cover non-exclusive inclusion, such that a process, method, article, or apparatus that comprises a list of elements includes not only those elements but also other elements not expressly listed, or elements inherent to such a process, method, article, or apparatus. Without further limitations, an element defined by the phrase "comprising one..." does not exclude the presence of other identical elements in the process, method, article, or apparatus that includes said element.

[0099] Combined with appendix Figures 5-8 As shown, this embodiment provides a method for modeling the equivalent plate dynamics of a periodic truss structure. The periodic truss structure includes several periodic truss elements, including: Step S110, based on Mindlin plate theory, performing a higher-order Taylor expansion of the strain of the periodic truss elements and making corrections to obtain the strain description of the periodic truss elements characterizing the transverse vibration and in-plane vibration of the equivalent plate; Step S120, calculating the strain energy and kinetic energy of the periodic truss elements according to the strain description of the periodic truss elements; Step S130, calculating the equivalent stiffness matrix and equivalent inertia matrix of the equivalent plate of the periodic truss structure according to the energy equivalence principle and the strain energy and kinetic energy of the periodic truss elements; Step S140, reducing the dimensionality of the equivalent stiffness matrix and equivalent inertia matrix of the equivalent plate of the periodic truss to construct an orthogonal anisotropic equivalent Mindlin plate dynamic model of the periodic truss structure, and performing inherent characteristic analysis.

[0100] Specifically, each of the periodic truss units is a regular triangular prism structure, and several of the periodic truss units extend longitudinally along the longitudinal beam direction to form the periodic truss structure, such as... Figure 6 As shown. More specifically, the periodic truss unit includes three types of components: longitudinal beams, transverse beams, and stay cables, such as... Figure 7 As shown, the structure consists of 15 components. Components 11, 12, and 13 are longitudinal beams; components 14, 15, 16, 17, 18, and 19 are stay cables; and components 20, 21, 22, 23, 24, and 25 are crossbeams. In other words, in the periodic truss unit, the components parallel to the longitudinal extension direction are longitudinal beams, the components parallel to the cross-section are crossbeams, and the diagonal components of the quadrilateral formed by adjacent longitudinal and crossbeams are stay cables. Simultaneously, an inertial coordinate system oxyz is constructed with the centroid of the left cross-section of the periodic truss unit (i.e., the centroid of the truss cross-section) as the origin o. The x-axis extends along the longitudinal beam direction of the periodic truss unit, the y-axis extends along the direction of a crossbeam (e.g., component 20) in the periodic truss unit, and the z-axis extends along a direction perpendicular to the plane xoy. More specifically, as... Figure 8 As shown, a structure with the same length as the periodic truss unit can be selected as the equivalent plate of the periodic truss unit, and the surface ABCD located in the plane xoy is the mid-surface of the equivalent plate of the periodic truss unit, but the present invention is not limited thereto.

[0101] Please also refer to Figure 5 , Figure 7 and Figure 8 Step S110 includes: obtaining the strain components of the periodic truss unit along the equivalent plate thickness direction based on Mindlin plate theory and the displacement of the periodic truss unit; performing a higher-order Taylor expansion of the strain components of the periodic truss unit along the equivalent plate thickness direction at the origin of the coordinate system to obtain a strain description of the periodic truss unit characterizing the transverse vibration of the equivalent plate; and the origin of the coordinate system is the centroid of the left cross section of the periodic truss unit; correcting the normal strain in the strain components of the periodic truss unit along the equivalent plate thickness direction to obtain a strain description of the periodic truss unit characterizing the in-plane vibration of the equivalent plate, and decoupling the in-plane vibration and transverse vibration of the equivalent plate of the periodic truss structure.

[0102] It is understandable that the mapping relationship between the displacement of the periodic truss element and its equivalent plate displacement is as follows:

[0103]

[0104] Where u, v, and w are the displacements of the periodic truss unit; u 0 (x,y), v 0 (x,y) and w 0(x,y) represents the displacement on the equivalent plate mid-surface ABCD of the periodic truss unit. and Let be the cross-sectional rotation angle of the equivalent plate mid-surface ABCD of the periodic truss unit; Let be the principal strain along the z-direction of the equivalent plate mid-surface ABCD of the periodic truss unit.

[0105] Based on the Mindlin plate theory, from the displacement expression (1) of the periodic truss element, the strain component expression of the periodic truss element along the equivalent plate thickness direction can be obtained as follows:

[0106]

[0107] Where, ε x ε y and ε z γ represents the tensile and compressive strain of the periodic truss element along the equivalent plate thickness direction; xy γ xz and γ yz The shear strain of the periodic truss unit along the equivalent plate thickness direction; and This represents the tensile and compressive strain on the equivalent plate surface. and This represents the shear strain on the equivalent plate surface. and Let be the curvature on the surface of the equivalent plate. Torque; and for The partial derivative; and Both are functions of coordinates x and y, and their specific expressions are as follows:

[0108]

[0109] Among them, u 0 v 0 and w 0 This represents the displacement of the mid-surface of the equivalent plate; and This represents the rotation angle of the cross section of the mid-surface of the equivalent plate.

[0110] Specifically, in this embodiment, when using an equivalent plate to describe the lateral vibration of the periodic truss unit, in order to characterize the lateral bending-torsional coupled vibration and lateral local deformation characteristics of the periodic truss unit, the strain component formula (2) is expanded in two variables at the center of the left end section of the periodic truss unit (i.e., at the origin of the coordinate system). In particular, the shear strain γ xyThe torsional deformation of the equivalent plate is characterized. In Mindlin's medium-thick plate theory, torsional deformation and bending deformation are coupled, which differs from the decoupled torsional deformation and bending deformation when Timoshenko beam theory is used for the symmetrical structure of the periodic truss element. Therefore, in order to make the equivalent plate model more accurately describe the torsional deformation characteristics of the periodic truss structure, the shear strain γ needs to be... xy The Taylor expansion is extended to the third order. More specifically, the strain expression for the periodic truss element characterizing the equivalent plate's lateral vibration is as follows:

[0111]

[0112] in, and It represents a partial differential.

[0113] Specifically, in this embodiment, the in-plane vibration of the equivalent plate of the periodic truss structure is used to simulate the lateral bending vibration of the periodic truss structure; therefore, the in-plane vibration and lateral vibration of the equivalent plate are decoupled. In strain formula (2), the strain describing the in-plane vibration... and It includes strain terms that reflect out-of-plane vibration (i.e., transverse vibration): and This causes in-plane and out-of-plane vibrations to couple, which is inconsistent with the vibration characteristics of the periodic truss element, resulting in its inability to accurately characterize the lateral bending vibration of the periodic truss structure. Therefore, it is necessary to re-evaluate ε. x ε y and γ xy Define, i.e., delete ε x ε y and γ xy In the expression and The relevant coupling terms decouple the in-plane vibration from the transverse vibration. Furthermore, removing the coupling terms will cause changes in the in-plane tensile and compressive stiffness. To ensure that the equivalent plate's in-plane vibration mode frequency is equal to the transverse bending vibration mode frequency, ε must be adjusted. x and ε y Corrections are made. Therefore, correction factors k1 and k2 for the normal strain are introduced, and ε is assumed to be... z γ xz and γ yz All are zero. More specifically, the expression for the strain of the periodic truss element characterizing the equivalent in-plane vibration is as follows:

[0114]

[0115] Wherein, k1 and k2 are correction coefficients for normal strain, which can be determined according to the geometric configuration and material parameters of the periodic truss unit.

[0116] Specifically, in this embodiment, in step S120, the strain energy of the longitudinal and transverse beams in the periodic truss unit includes tensile and compressive strain energy, bending strain energy, and torsional strain energy, while the strain energy of the stay cables only includes tensile and compressive strain energy. More specifically, the axial strain of the k-th (k=1,...,N) member in the periodic truss unit... It can be represented as:

[0117]

[0118] in

[0119]

[0120]

[0121] Where N represents the number of components in the periodic truss unit, k represents the component number, and ε (k) F represents the strain component vector of the k-th member. (k) Let (l, m, n) represent the coordinate transformation matrix of the k-th component. (k) Let represent the direction cosine of the k-th component.

[0122] The strain energy of the periodic truss element is:

[0123]

[0124] Among them, L (k) E represents the length of the k-th component; (k) A (k) , and G (k) J (k) These are the tensile, compressive, bending, and torsional stiffnesses of the k-th member, respectively. Local coordinate system Local coordinates along the axial direction of the lower edge component. and Local coordinate systems The displacement and rotation angle.

[0125] The strain formula (4) characterizing transverse vibration and the strain formula (5) characterizing in-plane vibration can be uniformly transformed into:

[0126]

[0127] Where, ε s Let be the strain parameter vector of the equivalent plate of the periodic truss structure. Let be the strain transformation matrix of the k-th component.

[0128] When ε sWhen representing the lateral vibration strain parameters of the equivalent plate of the periodic truss structure,

[0129]

[0130] When ε s When representing the in-plane vibration strain parameters of the equivalent plate of the periodic truss structure,

[0131]

[0132] Substituting equations (10)-(12) into equation (9), the strain energy U of the periodic truss unit is... e It can be represented as the strain parameter vector ε s Functions:

[0133]

[0134]

[0135] in, This represents the contribution matrix of the bending strain energy and torsional strain energy of the k-th member to the overall stiffness of the periodic truss unit.

[0136] Specifically, in this embodiment, the kinetic energy of the periodic truss unit can be expressed as:

[0137]

[0138] Among them, T e The kinetic energy of the periodic truss unit;

[0139] and It is the velocity component at the endpoint k of component. and The velocity component at the other end of component k can be obtained by differentiating formula (1) with respect to time t; ρ (k) and A (k) Let K represent the density and cross-sectional area of ​​component K, respectively.

[0140] Transform formula (1) into:

[0141]

[0142] in, Let u be the displacement transformation matrix of the k-th component; s The displacement parameter vector of the equivalent plate of the periodic truss structure can be expressed as:

[0143]

[0144] Substituting equations (16)-(17) into equation (15), the kinetic energy of the periodic truss unit can be expressed as the displacement parameter vector u. s Functions:

[0145]

[0146] in

[0147]

[0148]

[0149] Please also refer to Figure 5 , Figure 7 and Figure 8 Step S130 includes: based on the principle of energy equivalence, making the strain energy of the periodic truss unit equal to the strain energy of the equivalent plate of the periodic truss structure, so as to obtain the equivalent stiffness matrix of the equivalent plate of the periodic truss structure; based on the principle of energy equivalence, making the kinetic energy of the periodic truss unit equal to the kinetic energy of the equivalent plate of the periodic truss structure, so as to obtain the equivalent inertia matrix of the equivalent plate of the periodic truss structure.

[0150] It is understood that the expression for the strain energy of the equivalent plate of the periodic truss structure is as follows:

[0151]

[0152] Among them, U p L1 represents the strain energy of the equivalent plate of the periodic truss structure; L2 represents the length of the longitudinal beam and L1 represents the length of the transverse beam. Let be the equivalent stiffness matrix of the equivalent plate of the periodic truss structure;

[0153] byU e =U p The expression for the equivalent stiffness matrix of the equivalent plate of the periodic truss structure is as follows:

[0154]

[0155] Where Δ is the area of ​​the mid-surface of the equivalent plate, and Δ=L1L2;

[0156] The expression for the kinetic energy of the equivalent plate of the periodic truss structure is as follows:

[0157]

[0158] in Let be the equivalent inertia matrix of the continuous plate.

[0159] By T e =T pThe expression for the equivalent plate of the periodic truss structure is obtained as follows:

[0160]

[0161] Please also refer to Figure 5 , Figure 7 and Figure 8 Step S140 includes: eliminating redundant variables in the strain parameter vector of the equivalent stiffness matrix of the equivalent plate of the periodic truss structure using a static condensation method to reduce the dimensionality of the equivalent stiffness matrix of the equivalent plate of the periodic truss structure; eliminating redundant variables in the displacement parameter vector of the equivalent plate of the periodic truss structure using a static condensation method to reduce the dimensionality of the equivalent inertia matrix of the equivalent plate of the periodic truss structure; constructing an orthogonal anisotropic equivalent Mindlin plate dynamic model of the periodic truss structure based on the reduced equivalent stiffness matrix and the reduced equivalent inertia matrix of the periodic truss structure; performing inherent characteristic analysis on the equivalent plate dynamic model to obtain the natural frequencies and mode shapes of the equivalent plate of the periodic truss structure, and verifying the effectiveness of the model by comparing it with the finite element calculation results.

[0162] Specifically, in this embodiment, it can be seen from formulas (11), (12), and (17) that the strain parameter vector ε in the equivalent stiffness matrix and equivalent inertia matrix of the equivalent plate of the periodic truss structure is... s and displacement parameter vector u s It contains numerous redundant variables, which do not conform to the variables in the classic Mindlin plate model. To establish an orthogonal anisotropic equivalent Mindlin plate model of the periodic truss structure, a static condensation method is used to eliminate the strain parameter vector ε. s and displacement parameter vector u s Redundant variables in the model are used to reduce the dimensionality of the equivalent plate model. More specifically, the strain parameter vector ε of the equivalent plate of the periodic truss structure is... s Separate the variables in the data:

[0163] ε s =[ε s1 ε s2 ε s3 ] T (25)

[0164] When ε s When representing the transverse vibration strain parameter, ε in formula (25) s1 ε represents the strain variable that is identical to the transverse vibration strain variable of the equivalent Mindlin plate; s2 ε represents the strain variable related to the neglected stress. s3 This represents the strain variable that needs to be deleted. The specific expression is as follows:

[0165]

[0166] When ε s When expressing the in-plane vibration strain parameter, ε in formula (25) s1 This represents the strain variable that is identical to the in-plane vibration strain variable of the equivalent Mindlin plate. The specific expression is as follows:

[0167]

[0168] The displacement parameter vector u of the equivalent plate of the periodic truss structure is... s Separate the variables in the data:

[0169] u s =[u s1 u s2 ] T (28)

[0170] Among them, u s1 u represents the same displacement variable as the equivalent Mindlin plate. s2 The displacement variable to be deleted is expressed as follows:

[0171]

[0172] The equivalent stiffness matrix formula (22) and equivalent inertia matrix formula (24) of the periodic truss equivalent plate can be expressed as follows:

[0173]

[0174]

[0175] in, and All are symmetric matrices. and respectively with ε s1 ε s2 and ε s3 They have the same dimension; and respectively with u s1 and u s2 They have the same dimension.

[0176] The equivalent stiffness matrix in formula (30) is obtained by using the static condensation method. and the inertia matrix in formula (31) Dimensionality reduction yields the simplified equivalent stiffness matrix and equivalent inertia matrix of the equivalent plate model, as shown below:

[0177]

[0178]

[0179] Where D is the equivalent stiffness matrix of the equivalent plate of the periodic truss structure after dimensionality reduction; G is the equivalent inertia matrix of the equivalent plate of the periodic truss structure after dimensionality reduction.

[0180] The dimension of the equivalent stiffness matrix D of the equivalent plate of the periodic truss structure after dimension reduction is the same as the dimension of the strain variables of the Mindlin plate. Similarly, the dimension of the equivalent inertia matrix G of the equivalent plate of the periodic truss structure after dimension reduction is the same as the dimension of the displacement variables of the Mindlin plate. Therefore, an orthogonal anisotropic equivalent Mindlin plate model of the periodic truss structure is established. Then, a MATLAB program is used to analyze the inherent characteristics of the equivalent plate dynamic model, thereby solving for the natural frequencies and mode shapes of the equivalent plate. To further verify the correctness of the equivalent plate model, a finite element model of the periodic truss structure under cantilever boundary conditions is established using the finite element simulation software MSC.NASTRAN, and the calculation results are compared with those of the equivalent plate model. However, this invention is not limited to this.

[0181] In summary, this embodiment provides a method for modeling the equivalent plate dynamics of a periodic truss structure. Based on Mindlin plate theory, the strain of the periodic truss element is expanded and corrected using a higher-order Taylor expansion to obtain a strain description of the periodic truss element characterizing the transverse and in-plane vibrations of the equivalent plate. The strain energy and kinetic energy of the periodic truss element can be calculated based on this strain description. The equivalent stiffness matrix and equivalent inertia matrix of the equivalent plate of the periodic truss structure can be calculated based on the energy equivalence principle and the strain energy and kinetic energy of the periodic truss element. The equivalent stiffness matrix and equivalent inertia matrix of the equivalent plate of the periodic truss structure are then dimension-reduced to construct an orthogonal anisotropic equivalent Mindlin plate dynamic model of the periodic truss structure. Compared to the traditional method for modeling equivalent beams of truss structures based on point section assumptions, this embodiment uses an equivalent plate modeling method based on line section assumptions. This method can be applied to the nonlinear dynamics modeling of space antenna structures with distributed hinges installed in the width direction of the cross-section, making it more suitable for engineering applications. Compared to traditional planar equivalent plate modeling methods, this embodiment uses the in-plane vibration of an orthotropic equivalent plate to simulate the lateral bending vibration of a periodic truss structure. This overcomes the problem that existing equivalent plate modeling methods do not consider describing the lateral bending vibration of periodic truss structures with spatial three-dimensional vibration characteristics. The orthotropic equivalent plate dynamic model obtained in this embodiment can more comprehensively consider characteristics such as multiple hinge connections along the cross-sectional width, providing important support for subsequent research on the dynamics and vibration control of spatial periodic antenna structures with nonlinear connections, and even for studying their complex nonlinear dynamic behavior.

[0182] Although the present invention has been described in detail through the preferred embodiments above, it should be understood that the above description should not be considered as a limitation of the present invention. Various modifications and substitutions to the present invention will be apparent to those skilled in the art after reading the above description. Therefore, the scope of protection of the present invention should be defined by the appended claims.

Claims

1. A method for modeling the equivalent plate dynamics of a periodic truss structure, wherein the periodic truss structure comprises a plurality of periodic truss elements, characterized in that, include: Step S110: Based on Mindlin plate theory, the strain of the periodic truss element is expanded and corrected to obtain the strain description of the periodic truss element that characterizes the transverse vibration and in-plane vibration of the equivalent plate. Step S120: Calculate the strain energy and kinetic energy of the periodic truss unit based on the strain description of the periodic truss unit; Step S130: Based on the principle of energy equivalence and the strain energy and kinetic energy of the periodic truss unit, calculate the equivalent stiffness matrix and equivalent inertia matrix of the equivalent plate of the periodic truss structure. Step S140: Reduce the dimensionality of the equivalent stiffness matrix and equivalent inertia matrix of the equivalent plate of the periodic truss structure to construct the orthogonal anisotropic equivalent Mindlin plate dynamic model of the periodic truss structure and perform inherent characteristic analysis. Step S110 includes: Based on Mindlin plate theory and the displacement of the periodic truss element, the strain components of the periodic truss element along the equivalent plate thickness direction are obtained, and their expressions are as follows: in, , and The tensile and compressive strain of the periodic truss unit along the equivalent plate thickness direction; , and The shear strain of the periodic truss unit along the equivalent plate thickness direction; , and This represents the tensile and compressive strain on the equivalent plate surface. , and This represents the shear strain on the equivalent plate surface. and Let be the curvature of the surface in the equivalent plate. Torque; and for The partial derivative; , , , , , , and All are about coordinates x and y The function, specifically the expression, is as follows: in, , and This represents the displacement of the mid-surface of the equivalent plate; and This represents the rotation angle of the cross section at the mid-surface of the equivalent plate; To characterize the transverse bending-torsional coupled vibration and transverse local deformation characteristics of the periodic truss unit, the strain components of the periodic truss unit along the equivalent plate thickness direction are subjected to a two-dimensional higher-order Taylor expansion at the origin of the coordinate system to obtain the strain description of the periodic truss unit characterizing the transverse vibration of the equivalent plate; and the origin of the coordinate system is the center of the left cross-section of the periodic truss unit. Shear strain Characterizing the degree of torsional deformation of the equivalent plate, the shear strain in the strain components of the periodic truss element is... The Taylor expansion is extended to the third order to allow an equivalent plate model to describe the torsional deformation characteristics of the periodic truss structure; the strain expression of the periodic truss element characterizing the lateral vibration of the equivalent plate is as follows: in, , , , , , and It represents a partial differential.

2. The equivalent plate dynamics modeling method for periodic truss structures as described in claim 1, characterized in that, Each of the periodic truss units is a regular triangular prism, and several of the periodic truss units extend in the longitudinal direction to form the periodic truss structure. Each of the periodic truss units consists of 3 longitudinal beams, 6 transverse beams, and 6 stay cables.

3. The equivalent plate dynamics modeling method for periodic truss structures as described in claim 1, characterized in that, In step S110, the in-plane vibration of the equivalent plate of the periodic truss structure is used to simulate the lateral bending vibration of the periodic truss structure, and the in-plane vibration of the equivalent plate is decoupled from the lateral vibration; to characterize the lateral bending vibration of the periodic truss structure, the strain components of the periodic truss element are redefined and deleted. , and In the expression , and The relevant coupling terms decouple the in-plane vibration from the transverse vibration; however, removing the coupling terms will cause changes in the in-plane tensile and compressive stiffness. To ensure that the equivalent plate's in-plane vibration modal frequency is equal to the transverse bending vibration modal frequency, the following is necessary: and Make corrections; For the strain component of the periodic truss element along the equivalent plate thickness direction, a correction factor for the normal strain is introduced to obtain a strain description of the periodic truss element characterizing the in-plane vibration of the equivalent plate. The strain expression for the periodic truss element characterizing the equivalent in-plane vibration is as follows: in, and This is the correction factor for the normal strain.

4. The equivalent plate dynamics modeling method for a periodic truss structure as described in claim 3, characterized in that, In step S120, the strain energy of the periodic truss unit is calculated using the following formula: in, The strain energy of the periodic truss unit; The strain parameter vector of the equivalent plate of the periodic truss structure; N The number of components in the periodic truss unit. k Number the components; For the first k Tensile and compressive stiffness of the root member; Indicates the first k The length of the root component; For the first k The strain transformation matrix of the root component; For the first k The coordinate transformation matrix of the root component; Indicates the first k The contribution matrix of the bending strain energy and torsional strain energy of the root member to the overall stiffness of the periodic truss unit; Indicates the first k The direction cosine of the root component; when When representing the lateral vibration strain parameters of the equivalent plate of the periodic truss structure, ； when When representing the in-plane vibration strain parameters of the equivalent plate of the periodic truss structure, The kinetic energy of the periodic truss unit is calculated using the following formula: in, The kinetic energy of the periodic truss unit; The displacement parameter vector of the equivalent plate of the periodic truss structure; For the first k The displacement transformation matrix of the root component.

5. The equivalent plate dynamics modeling method for a periodic truss structure as described in claim 4, characterized in that, Step S130 includes: Based on the principle of energy equivalence, the strain energy of the periodic truss unit is made equal to the strain energy of the equivalent plate of the periodic truss structure, so as to obtain the equivalent stiffness matrix of the equivalent plate of the periodic truss structure. The expression for the strain energy of the equivalent plate of the periodic truss structure is as follows: in, The strain energy of the equivalent plate of the periodic truss structure is given. The length of the longitudinal beam. The length of the beam. Let be the equivalent stiffness matrix of the equivalent plate of the periodic truss structure; Depend on The expression for the equivalent stiffness matrix of the equivalent plate of the periodic truss structure is as follows: in, Let be the area of ​​the equivalent plate's mid-surface, and ; Based on the principle of energy equivalence, the kinetic energy of the periodic truss unit is made equal to the kinetic energy of the equivalent plate of the periodic truss structure, so as to obtain the equivalent inertia matrix of the equivalent plate of the periodic truss structure. The expression for the kinetic energy of the equivalent plate of the periodic truss structure is as follows: in, The kinetic energy of the equivalent plate of the periodic truss structure; Let be the equivalent inertia matrix of the equivalent plate of the periodic truss structure; Depend on The expression for the equivalent inertia matrix of the equivalent plate of the periodic truss structure is as follows: 。 6. The equivalent plate dynamics modeling method for a periodic truss structure as described in claim 5, characterized in that, The strain parameter vector of the equivalent plate of the periodic truss structure and displacement parameter vector It contains multiple redundant variables, which do not conform to the variables of the classic Mindlin board model; step S140 includes: The strain parameter vector of the equivalent plate of the periodic truss structure is eliminated by the static condensation method. Redundant variables in the equation are removed to reduce the dimensionality of the equivalent stiffness matrix of the equivalent plate of the periodic truss structure; a static condensation method is used to eliminate the displacement parameter vectors in the equivalent plate of the periodic truss structure. Redundant variables in the data are used to reduce the dimensionality of the equivalent inertia matrix of the equivalent plate of the periodic truss structure. The strain parameter vector of the equivalent plate of the periodic truss structure Separate the variables in the data: when When representing transverse vibration strain parameters, This represents the strain variable that is identical to the transverse vibration strain variable of the equivalent Mindlin plate; This represents the strain variable related to the neglected stress; This represents the strain variable that needs to be deleted; the specific expression is as follows: when When representing in-plane vibration strain parameters, This represents the strain variable that is the same as the in-plane vibration strain variable of the equivalent Mindlin plate; the specific expression is as follows: The displacement parameter vector of the equivalent plate of the periodic truss structure Separate the variables in the data: in This represents the same displacement variables as the equivalent Mindlin plate. The displacement variable to be deleted is expressed as follows: , The formulas for the equivalent stiffness matrix and equivalent inertia matrix of the equivalent plate of the periodic truss structure can be expressed as follows: , in and All are symmetric matrices. , and respectively with , and They have the same dimension; and respectively with and They have the same dimension; Static condensation method is used to condense the equivalent stiffness matrix and inertia matrix By performing dimensionality reduction, we can obtain the equivalent stiffness matrix and equivalent inertia matrix of the simplified equivalent plate of the periodic truss structure, as shown below: in, This is the equivalent stiffness matrix of the equivalent plate of the reduced-dimensional periodic truss structure; This is the equivalent inertia matrix of the equivalent plate of the reduced-dimensional periodic truss structure; Based on the equivalent stiffness matrix and equivalent inertia matrix of the equivalent plate of the reduced periodic truss structure, an orthogonal anisotropic equivalent Mindlin plate dynamic model of the periodic truss structure is constructed. The dynamic model of the equivalent plate is subjected to inherent characteristic analysis to obtain the natural frequencies and mode shapes of the periodic truss equivalent plate, and the effectiveness of the model is verified by comparing it with the finite element calculation results.

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