Analytical method for transverse free vibration of rectangular thin plate structure of satellite

Abstract

The application discloses an analytic method for transverse free vibration of a rectangular thin plate structure of a flat plate satellite, and belongs to the technical field of vibration analysis of a rectangular thin plate structure. The method can solve the problem that the existing analytic method for transverse free vibration of a rectangular thin plate cannot solve the problem of arbitrary combination of boundary conditions. The method comprises the following steps: establishing a differential equation of free vibration of a rectangular thin plate under a small disturbance; combining a first-order main vibration expression of the rectangular thin plate to obtain an eigen-differential equation of the rectangular thin plate; establishing a mode function in a separation variable form, and simplifying the eigen-differential equation of the rectangular thin plate to obtain a simplified eigen-differential equation; obtaining an eigen-function expression based on a method of assuming spatial eigenvalues; and establishing constraint equations based on three boundary conditions of a simply supported rectangular thin plate, a fixed supported rectangular thin plate and a free boundary rectangular thin plate, combining the eigen-function expression, and calculating mode functions corresponding to the three boundary conditions and natural vibration frequencies of the rectangular thin plate corresponding to the three boundary conditions. The method can be used for analyzing transverse free vibration of a rectangular thin plate structure under various boundary conditions.

Description

Technical Field

[0001] The invention relates to a method for analyzing the transverse free vibration of a rectangular thin plate structure of a flat-plate satellite, and belongs to the technical field of vibration analysis of rectangular thin plate structures. Background Art

[0002] With the rapid global growth in demand for low-orbit communications satellites, low-cost, efficient satellite launch methods have become crucial. Against this backdrop, the "multi-satellite launch" model has emerged. To meet the demands of this "multi-satellite launch," the physical configuration of the satellites must be optimized accordingly. Flat-panel satellites, with their compact structure and small size, are well-suited for this launch method. This configuration effectively utilizes the limited space within the launch vehicle's fairing, enabling close arrangement of multiple satellites for maximum launch efficiency. Furthermore, its design supports a high degree of modularity and mass production, further reducing production and launch costs.

[0003] The payload platform of a flat-panel satellite is primarily assembled from various rectangular thin plates under various boundary conditions, with the highest percentage of these being those with four-sided clamps. To ensure the stability and operational reliability of the flat-panel satellite structure, the payload plate prototypes must undergo ground-based vibration testing after machining to determine whether their frequency, amplitude, and other responses meet design requirements. During the payload plate design process, software is used to simulate and solve its vibrations, and the design is then updated, adjusted, and modified based on the response results. The most important aspect of this is analyzing the lateral free vibration of the rectangular thin plates.

[0004] The lateral free vibration of rectangular thin plates has been extensively studied, and through numerous research and experiments by various scholars, a systematic solution has emerged. However, it is generally accepted that an exact solution to the lateral free vibration of a rectangular thin plate can only be obtained if at least one pair of opposing edges are simply supported. This approach, often referred to as the inverse method, has certain limitations. For a rectangular thin plate with at least one pair of opposing edges being simply supported, its eigenvalues ​​and characteristic equations can be guessed, allowing the inverse method to be used to solve its frequencies and mode shapes. However, for any combination of other boundary conditions, a solution is more difficult.

[0005] Therefore, how to obtain the analytical solution of the lateral free vibration of rectangular thin plates under any combination of boundary conditions is a problem that urgently needs an innovative solution. Summary of the Invention

[0006] Aiming at the problem that the existing analytical method for the transverse free vibration of a rectangular thin plate cannot solve the situation of arbitrary combination of boundary conditions, the present invention provides an analytical method for the transverse free vibration of a rectangular thin plate structure of a flat-panel satellite.

[0007] The present invention provides a method for analyzing the lateral free vibration of a rectangular thin plate structure of a flat-panel satellite, comprising:

[0008] Based on the differential equation at the neutral surface of the rectangular thin plate under static equilibrium conditions and combined with the free vibration displacement of the rectangular thin plate under small disturbances, the free vibration differential equation of the rectangular thin plate under small disturbances is obtained; then, the free vibration differential equation is simplified by combining the expression of the first-order principal vibration of the rectangular thin plate to obtain the eigendifferential equation of the rectangular thin plate;

[0009] Considering the separation of variables method, the mode function of the separated variables form is established; based on the mode function, the eigendifferential equation of the rectangular thin plate is simplified to obtain the simplified eigendifferential equation; based on the method of assuming spatial eigenvalues, the eigenfunction expressions of the simplified eigendifferential equation in the X and Y axis directions are obtained;

[0010] Constraint equations based on three boundary conditions of rectangular thin plates, namely, simply supported, clamped, and free boundaries, are established. Combined with the eigenfunction expressions, the mode functions corresponding to the three boundary conditions and the corresponding natural vibration frequencies of the rectangular thin plates are calculated.

[0011] According to the analytical method for the lateral free vibration of the rectangular thin plate structure of the flat-panel satellite of the present invention, the differential equation at the neutral surface of the rectangular thin plate under static equilibrium conditions is obtained as follows:

[0012] Assume that the deflection of the rectangular plate under static equilibrium is w st , the lateral static load that produces the deflection is q st According to Newton's second law, the differential equation at the neutral surface of the rectangular thin plate under static equilibrium conditions is:

[0013]

[0014] in is the elastic force per unit area of ​​the rectangular thin plate during free vibration; D is the bending stiffness of the rectangular thin plate:

[0015] D=Eh 3 / 12(1-υ 2 ),

[0016] Where E is Young's modulus, h is the thickness of the rectangular plate, and υ is Poisson's ratio;

[0017] is a biharmonic operator:

[0018]

[0019] In the formula is the Laplace operator:

[0020]

[0021] Where x represents the X-axis coordinate of the rectangular thin plate in the transverse direction, and y represents the Y-axis coordinate of the rectangular thin plate in the transverse direction.

[0022] According to the analytical method for the lateral free vibration of the rectangular thin plate structure of the flat-panel satellite of the present invention, the method for obtaining the differential equation of the free vibration of the rectangular thin plate under small disturbance is:

[0023] Assume that the free vibration displacement of the rectangular thin plate under small disturbance is w free , the inertia force per unit volume of the rectangular thin plate is Where ρ is the mass per unit volume of the rectangular thin plate, and t is the time;

[0024] Under small disturbance, the elastic force per unit area of ​​the rectangular thin plate during free vibration is With the lateral static load q st and the inertia force per unit volume of the rectangular plate Phase equilibrium:

[0025]

[0026] Subtracting formula (2) from formula (1) yields:

[0027]

[0028] According to the deflection w under static equilibrium conditions st : Rewrite formula (3) as:

[0029]

[0030] Taking the static equilibrium position of the rectangular thin plate as the coordinate origin, the instantaneous deflection w of the rectangular thin plate is: w = w free -w st ; By transforming formula (4), we can obtain the differential equation of free vibration of rectangular thin plate under small disturbance:

[0031]

[0032] According to the analytical method for the lateral free vibration of the rectangular thin plate structure of the flat-panel satellite of the present invention, the method for obtaining the eigendifferential equation of the rectangular thin plate is:

[0033] The expression of the first-order main vibration of the rectangular thin plate is:

[0034] w=[A cos(ωt)+B sin(ωt)]W(x,y) (6),

[0035] Where A is the cosine coefficient, ω is the angular frequency of the rectangular thin plate, B is the sine coefficient, and W(x,y) is the vibration mode function;

[0036] Based on formula (6), formula (5) is simplified and the non-zero factor A cos(ωt)+B sin(ωt) is eliminated to obtain the eigendifferential equation of the rectangular thin plate:

[0037]

[0038] make

[0039] Where γ is the frequency parameter;

[0040] The final form of the eigendifferential equation for the rectangular thin plate is obtained:

[0041]

[0042] According to the analytical method for lateral free vibration of the rectangular thin plate structure of a flat-panel satellite of the present invention, the mode function in the form of separated variables is:

[0043] W(x,y)=e μx e λy (10),

[0044] Where μ is the spatial eigenvalue in the X-axis direction, and λ is the spatial eigenvalue in the Y-axis direction.

[0045] According to the analytical method for the lateral free vibration of the rectangular thin plate structure of the flat-panel satellite of the present invention, the mode function in the form of separated variables is substituted into formula (9) to obtain the simplified eigendifferential equation:

[0046] λ 4 +2λ 2 +μ 4 =γ 4 (11),

[0047] Right now:

[0048] (λ 2 +μ 2 ) 2 =γ 4 (12).

[0049] According to the analytical method for the transverse free vibration of the rectangular thin plate structure of the flat-panel satellite of the present invention, the method for obtaining the eigenfunction expressions of the simplified eigendifferential equation in the X and Y axis directions is:

[0050] Assume that in formula (12), the solution corresponding to μ is μ 1,2 、μ 3,4 , the solution corresponding to λ is λ 1,2 ,λ 3,4 :

[0051] μ 1,2 =±iα1,μ 3,4 = ±α2(13),

[0052] λ 1,2 = ±iβ1,λ 3,4 = ±β2(14),

[0053] in:

[0054]

[0055] Let μ = iα1 and λ = iβ1, and substitute into formulas (15) and (16), we get:

[0056]

[0057] The eigenfunction expressions in the X and Y axis directions are calculated as follows:

[0058] X1=A1 cosh(α2x)+A2 sinh(α2x)+A3 cos(α1x)+A4 sin(α1x) (19),

[0059] Y1=B1 cosh(β2y)+B2 sinh(β2y)+B3 cos(β1y)+B4 sin(β1y) (20),

[0060] Where X1 is the eigenfunction in the X-axis direction, and Y1 is the eigenfunction in the Y-axis direction;

[0061] A1 is the X-axis direction eigenfunction coefficient one, A2 is the X-axis direction eigenfunction coefficient two, A3 is the X-axis direction eigenfunction coefficient three, and A4 is the X-axis direction eigenfunction coefficient four;

[0062] B1 is the Y-axis direction eigenfunction coefficient one, B2 is the Y-axis direction eigenfunction coefficient two, B3 is the Y-axis direction eigenfunction coefficient three, and B4 is the Y-axis direction eigenfunction coefficient four.

[0063] The present invention overcomes the limitations of existing methods and analyzes the transverse free vibration of rectangular thin plates, taking into account common vibration phenomena in engineering. Starting with the transverse free vibration equation for a rectangular thin plate, the present method uses the separation of variables method to analytically derive the free vibration frequency equation, obtaining the frequency and mode shape functions of the rectangular thin plate. This method is applicable to the analysis of the transverse free vibration of rectangular thin plates under any combination of boundary conditions and possesses universal applicability. BRIEF DESCRIPTION OF THE DRAWINGS

[0064] Figure 1 is a schematic diagram of the rectangular thin plate of the present invention in the XOY plane in the XYZ coordinate system;

[0065] Figure 2 This is a schematic diagram of a rectangular thin plate with clamped boundary conditions on all four sides;

[0066] Figure 3 is a schematic diagram of a rectangular thin plate model in a simulation experiment;

[0067] Figure 4 is a projection diagram of the first nine order modal shapes of the rectangular thin plate in the XOY direction under the four-side fixed boundary condition.

[0068] Figure 5 is a projection diagram of the first nine order modal shapes of the rectangular thin plate in the XOY direction under the four-side fixed boundary condition. DETAILED DESCRIPTION

[0069] The technical solutions in the embodiments of the present application will be clearly and completely described below with reference to the drawings in the embodiments of the present application. Obviously, the described embodiments are only a part of the embodiments of the present application, rather than all the embodiments of the present application. Based on the embodiments in the present application, all other embodiments obtained by those skilled in the art without creative work fall within the scope of protection of the present application.

[0070] It should be noted that the embodiments in the present application and the features in the embodiments can be combined with each other without conflict.

[0071] The present application will be further described below with reference to the drawings and specific embodiments, but is not limited to the present application.

[0072] Specific embodiment one, in combination Figure 1 and Figure 2 As shown in the drawings, the present application provides a rectangular thin plate structure transverse free vibration analytical method of a flat plate satellite, comprising:

[0073] Based on the differential equation at the neutral surface of the rectangular thin plate under the static equilibrium condition, the free vibration differential equation of the rectangular thin plate under the small disturbance is obtained by combining the free vibration displacement of the rectangular thin plate under the small disturbance. Then, the eigen differential equation of the rectangular thin plate is obtained by simplifying the free vibration differential equation in combination with the expression of the first order main vibration of the rectangular thin plate.

[0074] The separation of variables method is considered, and the mode shape function in the form of separation of variables is established. The eigen differential equation after simplification is obtained by simplifying the eigen differential equation of the rectangular thin plate based on the mode shape function. The eigen function expression in the X and Y axis directions of the eigen differential equation after simplification is obtained based on the method of assuming spatial eigenvalues.

[0075] The constraint equations based on the three boundary conditions of simply supported, fixed and free boundary conditions of the rectangular thin plate are established, and the mode shape functions and the corresponding natural vibration frequencies of the rectangular thin plate corresponding to the three boundary conditions are calculated in combination with the eigen function expression.

[0076] Further, the method for obtaining the differential equation at the neutral surface of the rectangular thin plate under the static equilibrium condition is:

[0077] The problem of analyzing the free vibration of a rectangular thin plate is described as follows:

[0078] Given a rectangular thin plate, which satisfies the basic assumptions of thin plate theory. The initial thin plate is in an equilibrium position under the action of a given lateral load, and deviates from the equilibrium position after being disturbed, and vibrates slightly in amplitude near the equilibrium position. The main task of the analysis of the free vibration of a rectangular thin plate is to solve the natural vibration mode and the natural frequency of the vibration of the thin plate, especially to obtain the fundamental frequency of the thin plate.

[0079] Assume that the deflection of the rectangular thin plate under static equilibrium condition is w st , the lateral static load that generates the deflection is q st , and the lateral static load can be gravity; according to Newton's second law, the differential equation of the neutral surface of the rectangular thin plate under static equilibrium condition is obtained as follows:

[0080]

[0081] where is the elastic force per unit area of the rectangular thin plate when it is in free vibration; D is the bending stiffness of the rectangular thin plate:

[0082] D = Eh 3 / 12(1-υ 2 ),

[0083] where E is the Young's modulus, h is the thickness of the rectangular thin plate, and υ is the Poisson's ratio;

[0084] is a biharmonic operator:

[0085]

[0086] where is the Laplace operator:

[0087]

[0088] where x represents the X-axis coordinate in the lateral direction of the rectangular thin plate, and y represents the Y-axis coordinate in the lateral direction of the rectangular thin plate.

[0089] Equation (1) shows that when the thin plate is in a static equilibrium position, the elastic force per unit area is in equilibrium with the lateral static load q st .

[0090] In this embodiment, the method for obtaining the free vibration differential equation of the rectangular thin plate under a small disturbance is as follows:

[0091] When the thin plate is subjected to a small disturbance, it will deviate from the static equilibrium position, and the displacement of the rectangular thin plate in free vibration under a small disturbance is assumed to be wfree , the inertia force per unit volume of the rectangular thin plate is where p is the mass per unit volume of the rectangular thin plate, and t is time;

[0092] Under a small disturbance, the elastic force per unit area of the rectangular thin plate in free vibration is and the transverse static load is q st and the inertia force per unit volume of the rectangular thin plate is are in equilibrium:

[0093]

[0094] Subtracting formula (2) from formula (1), we obtain:

[0095]

[0096] According to the deflection w of the static equilibrium condition st does not change with time, we obtain: Rewriting formula (3) as:

[0097]

[0098] In vibration, the static equilibrium position of the rectangular thin plate is taken as the coordinate origin, and the instantaneous deflection w of the rectangular thin plate is: w = w free -w st ; Deforming formula (4), we obtain the differential equation of free vibration of the rectangular thin plate under a small disturbance:

[0099]

[0100] Solving the differential equation, the solution is the sum of each order of principal vibration independent of each other.

[0101] The method for obtaining the eigen-differential equation of the rectangular thin plate is:

[0102] The expression of any order of principal vibration of the rectangular thin plate is:

[0103] w = [A cos (ωt) + B sin (ωt)] W (x, y) (6),

[0104] where A is the cosine coefficient, ω is the angular frequency of the rectangular thin plate, B is the sine coefficient, W (x, y) is the vibration mode function, also known as the modal function, iω is the eigenvalue corresponding to the time coordinate, and i 2 = -1;

[0105] Based on formula (6), formula (5) is simplified, and the non-zero factor A cos (ωt) + B sin (ωt) is eliminated, to obtain the eigen-differential equation of the rectangular thin plate:

[0106]

[0107] Let

[0108] where γ is a frequency parameter;

[0109] The final form of the eigen-differential equation of the rectangular thin plate is obtained:

[0110]

[0111] In summary, only when ρh and D are constants, the exact solution of the mode shape function and the natural frequency can be obtained according to the boundary condition constraint. Then the integral coefficients corresponding to the mode shape function are determined according to the orthogonality of the mode shape function and the initial condition. For formula (9), it is equivalent to solving a differential equation group about deflection, that is, the core problem of calculating the exact solution of the transverse free vibration of the rectangular thin plate is to solve the mode shape function and the corresponding natural frequency of the system, and on this basis, the constraint equation brought by the boundary condition is considered.

[0112] The mode shape function in the form of separation of variables is:

[0113] W(x,y)=e μx e λy (10),

[0114] where μ is the spatial eigenvalue in the X-axis direction and λ is the spatial eigenvalue in the Y-axis direction.

[0115] Substitute the mode shape function in the form of separation of variables into formula (9) to obtain the simplified eigen-differential equation:

[0116] λ 4 +2λ 2 +μ 4 =γ 4 (11),

[0117] That is:

[0118] (λ 2 +μ 2 ) 2 =γ 4 (12)。

[0119] In formula (12), the two spatial eigenvalues λ and μ corresponding to the mode shape function W(x,y) and the frequency parameter γ are all unknown quantities. The two spatial eigenvalues obtained by solving formula (12) can be real numbers, pure imaginary numbers and conjugate imaginary numbers, and each pair of spatial eigenvalues (λ, μ) corresponds to a natural frequency ω.

[0120] Further, the method for obtaining the eigenfunction expressions of the X-axis and Y-axis directions of the simplified eigen-differential equation is:

[0121] Assume that in formula (12), the solution corresponding to μ is μ 1,2 、μ 3,4 , the solution corresponding to λ is λ 1,2 ,λ 3,4 :

[0122] μ 1,2 =±iα1,μ 3,4 = ±α2 (13),

[0123] λ 1,2 = ±iβ1,λ 3,4 = ±β2 (14),

[0124] in:

[0125]

[0126] Let μ = iα1 and λ = iβ1, and substitute into formulas (15) and (16), we get:

[0127]

[0128] The eigenfunction expressions in the X and Y axis directions are calculated as follows:

[0129] X1=A1 cosh(α2x)+A2 sinh(α2x)+A3 cos(α1x)+A4 sin(α1x) (19),

[0130] Y1=B1 cosh(β2y)+B2 sinh(β2y)+B3 cos(β1y)+B4 sin(β1y) (20),

[0131] Where X1 is the eigenfunction in the X-axis direction, and Y1 is the eigenfunction in the Y-axis direction;

[0132] A1 is the X-axis direction eigenfunction coefficient one, A2 is the X-axis direction eigenfunction coefficient two, A3 is the X-axis direction eigenfunction coefficient three, and A4 is the X-axis direction eigenfunction coefficient four;

[0133] B1 is the Y-axis direction eigenfunction coefficient one, B2 is the Y-axis direction eigenfunction coefficient two, B3 is the Y-axis direction eigenfunction coefficient three, and B4 is the Y-axis direction eigenfunction coefficient four.

[0134] Going further, the method for establishing the constraint equations based on the three boundary conditions of the rectangular thin plate: simply supported, clamped, and free boundary is as follows:

[0135] The boundary conditions of the simply supported edge are: w = 0, M n =0;

[0136] The boundary condition of the clamped edge is: w = 0,

[0137] The boundary conditions of the free edge are: M n =0,

[0138] Where M n is the normal bending moment per unit width on the neutral surface of the rectangular thin plate, n is the outer normal direction of the rectangular thin plate boundary, Q n is the normal shear force per unit width on the neutral surface of the rectangular thin plate, s is the tangent direction of the rectangular thin plate boundary, M ns is the torque within the boundary of the rectangular thin plate;

[0139] When solving the mode function and natural frequency from the free vibration differential equation of a rectangular thin plate, the boundary conditions need to be expressed in terms of mode function. Figure 1 For the rectangular plate shown, the mode shape function is expressed as:

[0140] W(x,y)=X1(x)Y1(y)(21);

[0141] Assuming that the side length of the rectangular plate in the X-axis direction is x=a, the constraint equation is:

[0142]

[0143]

[0144] Where SS represents the boundary condition corresponding to the simply supported edge, C represents the boundary condition corresponding to the clamped edge, and F represents the boundary condition corresponding to the free edge;

[0145] Arranging formulas (22) to (24) yields:

[0146]

[0147] For the other three sides of the rectangular thin plate, x=0, y=b, and y=0, three typical boundary conditions are considered. The constraint equations are the same as above and will not be repeated here.

[0148] For the case where the four edges of the rectangular thin plate are simply supported, clamped, or free, based on the boundary conditions, the four boundary conditions corresponding to the two edges x = 0 and x = a give a homogeneous linear equation system about A1 to A4. Similarly, the four boundary conditions corresponding to the two edges y = 0 and y = b give a homogeneous linear equation system about B1 to B4. Where b is the length of the side of the rectangular thin plate in the Y-axis direction.

[0149] Let the determinants of the two homogeneous linear equations be equal to zero, and obtain the transcendental equation about (λ,μ). Then solve the transcendental equation to obtain the mode function W(x,y) and the natural vibration frequency.

[0150] As an example, combined with Figure 2As shown in the figure, the four sides of the rectangular thin plate are clamped boundary conditions, that is, the rectangular thin plate is fixed to the base by bolts, or fixed to the base by clamped hinges. The four sides of the rectangular thin plate are expressed as:

[0151] C(x=0 side)-C(x=a side)-C(y=0 side)-C(y=b side);

[0152] The exact solution for the lateral free vibration of a rectangular thin plate under this boundary condition is derived below. First, the parameters of the rectangular thin plate are given, with the length and width being a and b respectively, and the boundary condition being that all four sides are clamped.

[0153] According to formula (23), the constraint equation of the rectangular thin plate with four sides as clamped boundary conditions is:

[0154]

[0155] Substituting formulas (19) and (20) into formula (28), we can obtain:

[0156]

[0157] Solve formulas (29) and (30) to obtain the expressions of coefficients A1~A4 and B1~B4; According to the relevant theory of linear algebra, the homogeneous equations have non-zero solutions, which requires that the corresponding coefficient matrix determinant is equal to zero;

[0158] Setting the determinant of the coefficient matrix corresponding to coefficients A1 to A4 and B1 to B4 equal to zero yields:

[0159]

[0160] According to formulas (17) and (18), we can obtain:

[0161]

[0162] Formulas (31) to (33) are combined and solved using the Newton-Raphson iterative numerical method to obtain the coefficients A1~A4 and B1~B4, determine the eigenfunction and obtain the spatial eigenvalue (λ,μ), and then obtain the mode function and the corresponding natural vibration frequency of the rectangular thin plate.

[0163] This embodiment derives a separation of variables method and uses the Newton-Raphson numerical iteration method to solve the simultaneous characteristic equations to obtain the solved frequency and spatial eigenvalues, thereby calculating the closed-loop exact analytical solution for the lateral free vibration of the rectangular thin plate.

[0164] The calculation results show that: Figure 2 The CCCC situation shown:

[0165] The eigenvalue equation is:

[0166]

[0167] The mode function is:

[0168]

[0169] From the formula of frequency parameter:

[0170]

[0171] After arranging and simplifying, the formula for solving the angular frequency of the rectangular thin plate is:

[0172]

[0173] Where f n is the natural vibration frequency of the rectangular plate.

[0174] Simulation experiment: Simulation verification of the lateral free vibration of a rectangular thin plate clamped on four sides.

[0175] Aiming at the problem of accurately solving the lateral free vibration of a rectangular thin plate under four-side clamped boundary conditions, the solution strategy and the corresponding fourth-order homogeneous partial differential equations to be solved have been derived.

[0176] Since the fourth-order homogeneous partial differential equations to be solved are transcendental equations and cannot be solved using traditional methods, they can be solved through programming using the Newton-Raphson iterative numerical method.

[0177] Physical simulation modeling process:

[0178] First, the physical parameters of the rectangular thin plate to be simulated are given as shown in Table 1. The material selected for the rectangular thin plate is A212 hard aluminum alloy.

[0179] Since the rectangular thin plate is 220 mm long and 145 mm wide, with small dimensions and large stiffness, analysis shows that the order of magnitude of the result obtained when measuring its natural frequency should be large. During program writing, it is necessary to prevent the occurrence of singular solutions or situations where the solution exceeds the set range.

[0180] Table 1 Physical parameters of rectangular thin plates

[0181]

[0182] Afterwards, based on the parameters and derived formulas of the rectangular thin plate, a lateral free vibration model of the rectangular thin plate under four-side clamped boundary conditions was compiled, and the vibration mode function and natural frequency of the rectangular thin plate were solved.

[0183] The rectangular thin plate model established by the simulation software is as follows Figure 3 shown.

[0184] Next, we write a program based on the derived formula and use the Newton-Raphson iterative numerical method to solve the eigenvalues ​​of the mode shape function. We then plot the amplitude of the lateral free vibration of the rectangular plate based on the mode shape function. Finally, we use the frequency parameter to solve for the natural frequency of the rectangular plate.

[0185] The physical simulation results and corresponding analysis are as follows:

[0186] Physical simulation results:

[0187] First, the mode shapes corresponding to the first nine orders of lateral free vibration of a rectangular thin plate under four-side clamped boundary conditions are obtained as the analysis object, as shown in Figure 4 shown.

[0188] Secondly, after solving the eigenvalue equation, the natural frequencies corresponding to the first fourteenth mode shapes of the lateral free vibration of the rectangular thin plate under the four-side clamped boundary condition can be calculated, as shown in Table 2.

[0189] Table 2 Natural frequencies corresponding to the first fourteen modes of the rectangular thin plate

[0190]

[0191] Under four-side clamped boundary conditions, the first-order natural frequency (also known as the fundamental frequency) of a rectangular thin plate is 617.1660 Hz. This fundamental frequency is the most critical natural frequency of a rectangular thin plate and is required for subsequent experimental verification and data analysis.

[0192] Conclusion analysis:

[0193] The following conclusions can be drawn from the analysis of the simulation image results:

[0194] First, each vibration mode is dimensionless, with the maximum value being 1, and the rest are distributed proportionally; and the vibration modes are both positive and negative, which are reflected in the vibration mode diagram as the vibration mode in the positive direction is green with a yellowish color, while the vibration mode in the negative direction is green with a blueish color. The direction and size of the vibration mode can be distinguished by the depth of the color.

[0195] Second, except for the image of the first-order vibration mode, the images of the other order vibration modes are basically symmetrical. On the upper and lower positive and negative sides of the XY plane, the specific projection diagrams of several orders are as follows: Figure 5 shown.

[0196] Observe the projection Figure 5 It can be found that the first-order mode only has positive vibration in the Z direction, which is peak-shaped. The amplitude is the largest in the center and decreases in circles toward the periphery until it reaches the boundary of the rectangular thin plate. Due to the four-sided clamped boundary condition, the amplitude is zero at the boundary.

[0197] Taking the fifth-order mode as an example, not only is there symmetry in the positive and negative Z directions, but by observing the XY plane projection, we can also see symmetry within the XY plane. For example, the fifth-order mode exhibits a centrosymmetrical vibration mode distribution with one diagonal angle being positive and the other diagonal angle being negative.

[0198] Third, as the modal order increases, the peak and valley values ​​of the corresponding vibration modes also increase. Figure 5 The 6th to 9th order mode shapes given prove this conclusion;

[0199] Fourth, since the lateral free vibration response of the rectangular thin plate is to superimpose all the modes together to obtain the final vibration shape, by comparing the vibration shapes of the first few modals and the subsequent modals, it can be found that the low-order modes play a major role. The high-order modes have more peaks and valleys, and the peaks and valleys corresponding to the modes with similar shapes and the previous low-order modes can be approximately considered to cancel each other out. Therefore, the main role is still played by the first few modes, especially the fundamental frequency of the rectangular thin plate, that is, the first-order mode.

[0200] Fifth, the fundamental frequency of the lateral free vibration of an A212 rectangular thin plate with a length of 220 mm, a width of 145 mm, and a thickness of 2 mm under the four-side clamped boundary conditions is calculated to be 617.1660 Hz.

[0201] Although the present invention is described herein with reference to specific embodiments, it should be understood that these embodiments are merely illustrative of the principles and applications of the invention. It should be understood that many modifications may be made to the illustrative embodiments, and that other arrangements may be devised, without departing from the spirit and scope of the invention as defined by the appended claims. It should be understood that the various dependent claims and features described herein may be combined in ways other than those described in the original claims. It should also be understood that features described in conjunction with individual embodiments may be used in conjunction with other described embodiments.

Claims

1. A method for analyzing the lateral free vibration of a rectangular thin plate structure of a flat-panel satellite, characterized in that: include: Based on the differential equation at the neutral surface of the rectangular thin plate under static equilibrium conditions and combined with the free vibration displacement of the rectangular thin plate under small disturbances, the free vibration differential equation of the rectangular thin plate under small disturbances is obtained; then, the free vibration differential equation is simplified by combining the expression of the first-order principal vibration of the rectangular thin plate to obtain the eigendifferential equation of the rectangular thin plate; Considering the separation of variables method, the mode function of the separated variables form is established; based on the mode function, the eigendifferential equation of the rectangular thin plate is simplified to obtain the simplified eigendifferential equation, (λ 2 +μ 2 ) 2 =γ 4 , where μ is the spatial eigenvalue in the X-axis direction, λ is the spatial eigenvalue in the Y-axis direction, and γ is the frequency parameter; based on the method of assuming spatial eigenvalues, the eigenfunction expressions of the simplified eigendifferential equation in the X-axis and Y-axis directions are obtained; Establish constraint equations based on three boundary conditions of the rectangular thin plate: simply supported, clamped, and free. Combined with the eigenfunction expression, calculate the mode functions corresponding to the three boundary conditions and the corresponding natural vibration frequencies of the rectangular thin plate. For the four sides of the rectangular thin plate with fixed support boundary conditions, the four sides of the rectangular thin plate are expressed as: C(x=0 side)-C(x=a side)-C(y=0 side)-C(y=b side), The constraint equations for a rectangular thin plate with four clamped boundary conditions are: Where X1 is the eigenfunction in the X-axis direction, and Y1 is the eigenfunction in the Y-axis direction; Substituting the eigenfunction expressions of the X and Y axes into formula (28), we can obtain: A1 is the X-axis direction eigenfunction coefficient one, A2 is the X-axis direction eigenfunction coefficient two, A3 is the X-axis direction eigenfunction coefficient three, and A4 is the X-axis direction eigenfunction coefficient four; B1 is the Y-axis direction eigenfunction coefficient one, B2 is the Y-axis direction eigenfunction coefficient two, B3 is the Y-axis direction eigenfunction coefficient three, and B4 is the Y-axis direction eigenfunction coefficient four; Solve formulas (29) and (30) to obtain expressions for coefficients A1~A4 and B1~B4; Setting the determinant of the coefficient matrix corresponding to coefficients A1 to A4 and B1 to B4 equal to zero yields: According to the formula get: Formulas (31) to (33) are combined and solved using the Newton-Raphson iterative numerical method to obtain the coefficients A1~A4 and B1~B4, determine the eigenfunction and obtain the spatial eigenvalue (λ,μ), and then obtain the mode function and the corresponding natural vibration frequency of the rectangular thin plate.

2. The method for analyzing the lateral free vibration of a rectangular thin plate structure of a flat-panel satellite according to claim 1, characterized in that: The differential equation at the neutral surface of a rectangular thin plate under static equilibrium conditions is obtained as follows: Assume that the deflection of the rectangular plate under static equilibrium is w st , the lateral static load that produces the deflection is q st According to Newton's second law, the differential equation at the neutral surface of the rectangular thin plate under static equilibrium conditions is: in is the elastic force per unit area of ​​the rectangular thin plate during free vibration; D is the bending stiffness of the rectangular thin plate: D=Yes 3 / 12(1−υ 2 ), Where E is Young's modulus, h is the thickness of the rectangular plate, and υ is Poisson's ratio; is a biharmonic operator: In the formula is the Laplace operator: Where x represents the X-axis coordinate of the rectangular thin plate in the transverse direction, and y represents the Y-axis coordinate of the rectangular thin plate in the transverse direction.

3. The method for analyzing the lateral free vibration of a rectangular thin plate structure of a flat-panel satellite according to claim 2, characterized in that: The method to obtain the differential equation of free vibration of a rectangular thin plate under small perturbations is: Assume that the free vibration displacement of the rectangular thin plate under small disturbance is w free , the inertia force per unit volume of the rectangular thin plate is Where ρ is the mass per unit volume of the rectangular thin plate, and t is the time; Under small disturbance, the elastic force per unit area of ​​the rectangular thin plate during free vibration is With the lateral static load q st and the inertia force per unit volume of the rectangular plate Phase equilibrium: Subtracting formula (2) from formula (1) yields: According to the deflection w under static equilibrium conditions st : Rewrite formula (3) as: Taking the static equilibrium position of the rectangular thin plate as the coordinate origin, the instantaneous deflection w of the rectangular thin plate is: w = w free -w st ; By transforming formula (4), we can obtain the differential equation of free vibration of rectangular thin plate under small disturbance:

4. The method for analyzing the lateral free vibration of a rectangular thin plate structure of a flat-panel satellite according to claim 3, characterized in that: The method to obtain the eigendifferential equation of the rectangular thin plate is: The expression of the first-order main vibration of the rectangular thin plate is: w=[Acos(ωt)+Bsin(ωt)]W(x,y)(6), Where A is the cosine coefficient, ω is the angular frequency of the rectangular thin plate, B is the sine coefficient, and W(x,y) is the vibration mode function; Based on formula (6), formula (5) is simplified and the non-zero factor Acos(ωt)+Bsin(ωt) is eliminated to obtain the eigendifferential equation of the rectangular thin plate: make The final form of the eigendifferential equation for the rectangular thin plate is obtained:

5. The method for analyzing the lateral free vibration of a rectangular thin plate structure of a flat-panel satellite according to claim 4, characterized in that: The vibration mode function in the form of separated variables is: W(x,y)=e μx e λy (10)。 6. The method for analyzing the lateral free vibration of a rectangular thin plate structure of a flat-panel satellite according to claim 5, characterized in that: Substituting the mode function in the form of separated variables into formula (9), we obtain the simplified eigendifferential equation: l 4 +2min 2 +m 4 =c 4 (11), Right now: (l 2 +m 2 ) 2 =c 4 (12)。 7. The method for analyzing the lateral free vibration of a rectangular thin plate structure of a flat-panel satellite according to claim 6, characterized in that: The method for obtaining the eigenfunction expressions of the simplified eigendifferential equation in the X and Y axis directions is: Assume that in formula (12), the solution corresponding to μ is μ 1,2 、μ 3,4 , the solution corresponding to λ is λ 1,2 ,λ 3,4 : m 1,2 =±iα1,μ 3,4 =±α2(13), l 1,2 =±iβ1,λ 3,4 =±β2(14), in: Let μ = iα1 and λ = iβ1, and substitute into formulas (15) and (16), we get: The eigenfunction expressions in the X and Y axis directions are calculated as follows: X1=A1cosh(α2x)+A2 sinh(α2x)+A3 cos(α1x)+A4sin(α1x)(19), Y1=B1cosh(β2y)+B2 sinh(β2y)+B3 cos(β1y)+B4sin(β1y) (20).

8. The method for analyzing the lateral free vibration of a rectangular thin plate structure of a flat-panel satellite according to claim 7, characterized in that: The method for establishing the constraint equations based on the three boundary conditions of simply supported, clamped, and free boundaries of rectangular thin plates is as follows: The boundary conditions of the simply supported edge are: w = 0, M n =0; The boundary condition of the clamped edge is: w = 0, The boundary conditions of the free edge are: M n =0, Where M n is the normal bending moment per unit width on the neutral surface of the rectangular thin plate, n is the outer normal direction of the rectangular thin plate boundary, Q n is the normal shear force per unit width on the neutral surface of the rectangular thin plate, s is the tangent direction of the rectangular thin plate boundary, M ns is the torque within the boundary of the rectangular thin plate; The mode function is expressed as: W(x,y)=X1(x)Y1(y)(21); Assuming that the side length of the rectangular plate in the X-axis direction is x=a, the constraint equation is obtained: Where SS represents the boundary condition corresponding to the simply supported edge, C represents the boundary condition corresponding to the clamped edge, and F represents the boundary condition corresponding to the free edge; Arranging formulas (22) to (24) yields:

9. The method for analyzing the lateral free vibration of a rectangular thin plate structure of a flat-panel satellite according to claim 8, characterized in that: For the case where the four edges of the rectangular thin plate are simply supported, clamped, or free, based on the boundary conditions, the four boundary conditions corresponding to the two edges x = 0 and x = a give a homogeneous linear equation system about A1 to A4. Similarly, the four boundary conditions corresponding to the two edges y = 0 and y = b give a homogeneous linear equation system about B1 to B4. Where b is the length of the side of the rectangular thin plate in the Y-axis direction. Let the determinants of the two homogeneous linear equations be equal to zero, and obtain the transcendental equation about (λ,μ). Then solve the transcendental equation to obtain the mode function W(x,y) and the natural vibration frequency.

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