Transverse free vibration analysis method for rectangular thin plate structure of flat plate type satellite

Abstract

The invention discloses a rectangular thin plate structure transverse free vibration analysis method of a flat plate type satellite, and belongs to the technical field of rectangular thin plate structure vibration analysis. The method aims at solving the problem that an existing rectangular thin plate transverse free vibration analysis method cannot solve the arbitrary combination condition of boundary conditions. Comprising the steps of establishing a rectangular thin plate free vibration differential equation under tiny disturbance; performing simplification in combination with a first-order main vibration expression of the rectangular thin plate to obtain an intrinsic differential equation of the rectangular thin plate; establishing a vibration mode function in a variable separation form, and simplifying the intrinsic differential equation of the rectangular thin plate to obtain a simplified intrinsic differential equation; obtaining an eigenfunction expression based on a method of assuming a spatial eigenvalue; and establishing a constraint equation based on three boundary conditions of simple support, fixed support and free boundary of the rectangular thin plate, and calculating to obtain vibration mode functions corresponding to the three boundary conditions and corresponding natural vibration frequencies of the rectangular thin plate in combination with an intrinsic function expression. The method is used for analyzing the transverse free vibration of the rectangular thin plate structure under various boundary conditions.

Description

Technical Field

[0001] The invention relates to a method for analyzing the lateral 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 growth of global demand for low-orbit communication satellites, low-cost and high-efficiency satellite launch methods have become crucial. Against this background, the "multiple satellites in one rocket" launch method came into being. In order to meet the launch requirements of "multiple satellites in one rocket", the physical configuration of the satellite must be optimized accordingly. Flat-panel satellites are very suitable for this type of launch due to their compact structure and small size. Satellites of this configuration can effectively utilize the limited space in the fairing of the launch vehicle to achieve close arrangement of multiple satellites, thereby maximizing the carrying efficiency; at the same time, their design supports a high degree of modularization and mass production, further reducing production and launch costs.

[0003] The payload platform of a flat-panel satellite is mainly assembled from various rectangular thin plates under different boundary conditions as satellite payload plates, among which the boundary conditions with four sides fixed account for the highest proportion. In order to ensure the stability of the flat-panel satellite structure and the reliability of its operation, the payload plate needs to be subjected to a vibration test on the ground after the sample processing is completed to determine whether its frequency, amplitude and other responses meet the design requirements. In the design process of the payload plate, it is necessary to simulate and solve its vibration through software, and update, adjust and correct the design based on the response results. The most important of these is to analyze the lateral free vibration problem of the rectangular thin plate.

[0004] The problem of lateral free vibration of rectangular thin plates has been widely studied. Through a large number of research and experiments by different scholars, a systematic solution has been developed. It is generally believed that only when there is at least one set of simply supported boundary conditions on the opposite sides of a rectangular thin plate can the exact solution of its lateral free vibration be obtained, and the method used is the inverse method, which has certain limitations. When facing a rectangular thin plate with at least one set of simply supported boundary conditions on the opposite sides, its eigenvalues ​​and characteristic equations can be guessed, and its frequency and mode function can be solved using the inverse method; however, when facing any combination of the remaining boundary conditions, it is more difficult to solve.

[0005] Therefore, how to obtain the analytical solution of the lateral free vibration of a rectangular thin plate 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 transverse free vibration of rectangular thin plates cannot solve the situation of arbitrary combination of boundary conditions, the present invention provides an analytical method for transverse free vibration of a rectangular thin plate structure of a flat-plate satellite.

[0007] A method for analyzing the lateral free vibration of a rectangular thin plate structure of a flat-plate satellite of the present invention comprises:

[0008] Based on the differential equation at the neutral surface of the rectangular thin plate under static equilibrium conditions, 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 eigenvalue differential equation of the rectangular thin plate;

[0009] Considering the separation of variables method, the vibration mode function in the form of separation of variables is established; based on the vibration 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, fixedly supported and free boundaries, are established. Combined with the eigenfunction expression, 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 lateral free vibration of the rectangular thin plate structure of the flat-plate satellite of the present invention, the method for obtaining the differential equation at the neutral surface of the rectangular thin plate under static equilibrium conditions is:

[0012] Assume that the deflection of the rectangular thin plate under static equilibrium conditions 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] Wherein 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 lateral free vibration of the rectangular thin plate structure of the flat-plate satellite of the present invention, the method for obtaining the differential equation of 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 a small disturbance, the elastic force per unit area of ​​a rectangular thin plate when it vibrates freely is With 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), we get:

[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 lateral free vibration of the rectangular thin plate structure of the flat-plate 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 plate is obtained:

[0041]

[0042] According to the analytical method for lateral free vibration of the rectangular thin plate structure of a flat-plate satellite of the present invention, the vibration 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 lateral free vibration of the rectangular thin plate structure of a flat-panel satellite of the present invention, the vibration 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 lateral 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 , 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 formula (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 eigenfunction coefficient one in the Y-axis direction, B2 is the eigenfunction coefficient two in the Y-axis direction, B3 is the eigenfunction coefficient three in the Y-axis direction, and B4 is the eigenfunction coefficient four in the Y-axis direction.

[0063] Beneficial effects of the present invention: The method of the present invention overcomes the limitations of the existing methods, considers the common vibration phenomena in engineering, and analyzes the transverse free vibration of a rectangular thin plate. The method of the present invention takes the transverse free vibration equation of a rectangular thin plate as a starting point, combines the separation of variables method to perform analytical derivation of the free vibration frequency equation, and obtains the frequency and vibration mode function of the rectangular thin plate. The method is applicable to the analysis of the transverse free vibration of a rectangular thin plate under any combination of boundary conditions, and has universality. 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 It is a schematic diagram of the rectangular thin plate with fixed boundary conditions on all four sides;

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

[0067] Figure 4 These are the first nine modal vibration diagrams of the lateral free vibration of a rectangular thin plate under the four-side clamped boundary conditions;

[0068] Figure 5 It is the projection diagram of the first nine modal vibration shapes of the lateral free vibration of the rectangular thin plate in the XOY direction under the four-side clamped boundary conditions. DETAILED DESCRIPTION

[0069] The following will be combined with the drawings in the embodiments of the present invention to clearly and completely describe the technical solutions in the embodiments of the present invention. Obviously, the described embodiments are only part of the embodiments of the present invention, not all of the embodiments. Based on the embodiments of the present invention, all other embodiments obtained by ordinary technicians in this field without creative work are within the scope of protection of the present invention.

[0070] It should be noted that, in the absence of conflict, the embodiments of the present invention and the features in the embodiments may be combined with each other.

[0071] The present invention will be further described below with reference to the accompanying drawings and specific embodiments, but they are not intended to limit the present invention.

[0072] Specific implementation method 1. Combination Figure 1 and Figure 2 As shown, the present invention provides a method for analyzing the lateral free vibration of a rectangular thin plate structure of a flat-plate satellite, comprising:

[0073] Based on the differential equation at the neutral surface of the rectangular thin plate under static equilibrium conditions, 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 eigenvalue differential equation of the rectangular thin plate;

[0074] Considering the separation of variables method, the vibration mode function in the form of separation of variables is established; based on the vibration 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;

[0075] Constraint equations based on three boundary conditions of rectangular thin plates, namely, simply supported, fixedly supported and free boundaries, are established. Combined with the eigenfunction expression, the mode functions corresponding to the three boundary conditions and the corresponding natural vibration frequencies of the rectangular thin plates are calculated.

[0076] Furthermore, the differential equation at the neutral surface of the rectangular thin plate under static equilibrium conditions is obtained as follows:

[0077] The problem of lateral free vibration analysis of a rectangular thin plate is described as:

[0078] Given a rectangular thin plate, the basic assumptions of thin plate theory are met. Initially, the thin plate is in equilibrium under the action of a given lateral load. After the initial disturbance, it deviates from the equilibrium position and vibrates slightly near the equilibrium position. The main purpose of the lateral free vibration analysis of a rectangular thin plate is to solve the natural vibration mode and natural frequency of the thin plate vibration, especially to solve the fundamental frequency of the thin plate.

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

[0080]

[0081] 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:

[0082] D=Eh 3 / 12(1-υ 2 ),

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

[0084] is a biharmonic operator:

[0085]

[0086] In the formula is the Laplace operator:

[0087]

[0088] Wherein 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.

[0089] Formula (1) shows that when the thin plate is in static equilibrium, the elastic force per unit area is With the lateral static load q st Phase balance.

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

[0091] When the thin plate is subjected to a small disturbance, it will deviate from the static equilibrium position. Assume that the free vibration displacement of the rectangular thin plate under a small disturbance is wfree , 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;

[0092] Under a small disturbance, the elastic force per unit area of ​​a rectangular thin plate when it vibrates freely is With lateral static load q st and the inertia force per unit volume of the rectangular plate Phase equilibrium:

[0093]

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

[0095]

[0096] According to the deflection w under static equilibrium conditions st Does not change with time, we get: Rewrite formula (3) as:

[0097]

[0098] In vibration theory, the static equilibrium position of a 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 ; By transforming formula (4), we can obtain the differential equation of free vibration of rectangular thin plate under small disturbance:

[0099]

[0100] Solving this differential equation, the solution obtained is the sum of the independent main vibrations of each order.

[0101] The method to obtain the eigendifferential equation of the rectangular thin plate is:

[0102] The expression of any one-order main vibration of a 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 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, 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 eigendifferential equation of the rectangular thin plate:

[0106]

[0107] make

[0108] Where γ is the frequency parameter;

[0109] The final form of the eigendifferential equation for the rectangular plate is obtained:

[0110]

[0111] In summary, only when ρh and D are constants, according to the boundary conditions, can the exact solutions of the mode functions and natural frequencies be obtained. Then the corresponding integral coefficients are determined according to the orthogonality of the mode functions and the initial conditions. For formula (9), it is equivalent to solving a set of differential equations about deflection, that is, when calculating the exact solution of the lateral free vibration of a rectangular thin plate, the core problem is to solve the mode functions and corresponding natural frequencies of the system, and on this basis consider the constraint equations brought by the boundary conditions.

[0112] The vibration mode function in the form of separated 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] Substituting the vibration mode function in the form of separated variables into formula (9), we can obtain the simplified eigendifferential equation:

[0116] λ 4 +2λ 2 +μ 4 =γ 4 (11)

[0117] Right now:

[0118] (λ 2 +μ 2 ) 2 =γ 4 (12).

[0119] In formula (12), the two spatial eigenvalues ​​λ and μ corresponding to the vibration mode function W(x, y), as well as 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] Furthermore, the method for obtaining the eigenfunction expressions of the simplified eigendifferential equation in the X and Y axis directions 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 , 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 formula (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 eigenfunction coefficient one in the Y-axis direction, B2 is the eigenfunction coefficient two in the Y-axis direction, B3 is the eigenfunction coefficient three in the Y-axis direction, and B4 is the eigenfunction coefficient four in the Y-axis direction.

[0134] Furthermore, the method for establishing constraint equations based on three boundary conditions of rectangular thin plates: simply supported, fixed, and free boundaries is as follows:

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

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

[0137] The boundary condition of the free edge is: 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 vibration mode function and natural frequency from the differential equation of free vibration of a rectangular thin plate, it is necessary to express the boundary conditions in terms of vibration mode function. Figure 1 For the rectangular plate shown, the vibration mode function is expressed as:

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

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

[0142]

[0143]

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

[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, and the constraint equations are the same as above and will not be repeated.

[0148] For the cases where the four sides of the rectangular thin plate are simply supported, fixedly supported, and free, based on the boundary conditions, the four boundary conditions corresponding to the two sides x=0 and x=a give the homogeneous linear equations about A1~A4; similarly, the four boundary conditions corresponding to the two sides y=0 and y=b give the homogeneous linear equations about B1~B4; where b is the length of the side of the rectangular thin plate in the Y-axis direction;

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

[0150] As an example, combining Figure 2As shown, for the four sides of the rectangular thin plate, the fixed boundary conditions are that the rectangular thin plate is fixed to the base by bolts, or fixed to the base by fixed 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 of 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, the length and width are a and b respectively, and the boundary condition is that all four sides are fixed.

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

[0154]

[0155] Substituting formula (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] Let the determinant of the coefficient matrix corresponding to coefficients A1~A4 and B1~B4 be equal to zero, and we get:

[0159]

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

[0161]

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

[0163] This embodiment derives a separation of variables method, uses the Newton-Raphson numerical iteration method to solve the characteristic equations obtained by the combination, obtains the solved frequency and spatial eigenvalues, and thus calculates the closed and accurate analytical solution of the lateral free vibration of the rectangular thin plate.

[0164] After calculation, we get: Figure 2 CCCC situation shown:

[0165] The eigenvalue equation is:

[0166]

[0167] The vibration mode function is:

[0168]

[0169] From the frequency parameter formula:

[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 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 220mm long and 145mm wide, with small size and large stiffness, analysis shows that the order of magnitude of the result obtained when measuring its natural frequency should be large. In the process of writing the program, it is necessary to prevent the occurrence of singular solutions or situations that exceed the set range.

[0180] Table 1 Physical parameters of rectangular thin plates

[0181]

[0182] Afterwards, according to the parameters and derived formulas of the rectangular thin plate, a lateral free vibration model of the rectangular thin plate under the 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 for the derived formula, use the Newton-Raphson iterative numerical method to solve the eigenvalue of the vibration mode function, and draw the amplitude image of the lateral free vibration of the rectangular thin plate according to the vibration mode function. Finally, the natural frequency of the rectangular thin plate is solved by the frequency parameter.

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

[0186] Physical simulation results:

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

[0188] Secondly, after solving the eigenvalue equation, the natural frequencies corresponding to the first fourteenth-order vibration modes of the rectangular thin plate in the transverse free vibration under the four-side clamped boundary condition can be obtained by calculation, as shown in Table 2.

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

[0190]

[0191] Under the four-side clamped boundary condition, the first-order natural frequency (i.e., fundamental frequency) of the rectangular thin plate is equal to 617.1660 Hz. This fundamental frequency is the most critical natural frequency of the rectangular thin plate, and the fundamental frequency data is required in subsequent test 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 order vibration mode is dimensionless, with the maximum value being 1, and the rest being distributed proportionally; and there are positive and negative vibration modes, which are reflected in the vibration mode diagram as the vibration mode in the positive direction is green-yellow, and the vibration mode in the negative direction is green-blue. 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 images 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 then decreases in circles to the surroundings until the boundary of the rectangular thin plate. Due to the four-side fixed boundary condition, the amplitude on the boundary is zero.

[0197] Taking the fifth-order mode as an example, there is not only symmetry in the positive and negative directions of the Z direction, but also symmetry within the XY plane by observing the XY plane projection diagram. For example, the fifth-order mode presents a centrally symmetrical vibration mode distribution with one diagonal being positive and the other diagonal being negative.

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

[0199] Fourth, since all modes are superimposed together to obtain the final vibration shape when the rectangular thin plate responds to lateral free vibration, by comparing the vibration shapes of the first few modal modes with the subsequent modal modes, it can be found that the main role is played by the low-order modes. Since the high-order modes have more peaks and valleys, and the corresponding modes with similar shapes can be approximately considered to cancel out the peaks and valleys corresponding to the previous low-order modes, the main role is still played by the first few modes, especially the fundamental frequency of the rectangular thin plate, which is the first-order mode.

[0200] Fifth, it is calculated that the fundamental frequency of the lateral free vibration of the 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 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 examples of the principles and applications of the present invention. It should therefore be understood that many modifications may be made to the exemplary embodiments and that other arrangements may be devised without departing from the spirit and scope of the present invention as defined by the appended claims. It should be understood that the various dependent claims and features described herein may be combined in a manner different from that described in the original claims. It should also be understood that the features described in conjunction with a single embodiment may be used in 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, 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 eigenvalue differential equation of the rectangular thin plate; Considering the separation of variables method, the vibration mode function in the form of separation of variables is established; based on the vibration 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; Constraint equations based on three boundary conditions of rectangular thin plates, namely, simply supported, fixedly supported and free boundaries, are established. Combined with the eigenfunction expression, the mode functions corresponding to the three boundary conditions and the corresponding natural vibration frequencies of the rectangular thin plates are calculated.

2. The method for analyzing the lateral free vibration of a rectangular thin plate structure of a flat-panel satellite according to claim 1 is characterized in that: The method to obtain the differential equation at the neutral surface of a rectangular thin plate under static equilibrium conditions is: Assume that the deflection of the rectangular thin plate under static equilibrium conditions 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: D▽ 4 w st =q st (1), Among them, D 4 w st 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; ▽ 4 is a biharmonic operator: ▽ 4 ＝▽ 2 ▽ 2 ， Where ▽ 2 is the Laplace operator: Wherein 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 disturbance 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 a small disturbance, the elastic force D▽ per unit area when the rectangular thin plate vibrates freely 4 w free With lateral static load q st and the inertia force per unit volume of the rectangular plate Phase equilibrium: Subtracting formula (2) from formula (1), we get: 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 Where γ is the frequency parameter; The final form of the eigendifferential equation for the rectangular plate is obtained: ▽ 4 W-γ 4 W=0(9)。 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), Where μ is the spatial eigenvalue in the X-axis direction, and λ is the spatial eigenvalue in the Y-axis direction.

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 vibration mode function in the form of separated variables into formula (9), we can 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 , 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 formula (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), Where X1 is the eigenfunction in the X-axis direction, and Y1 is the eigenfunction in the Y-axis direction; 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 eigenfunction coefficient one in the Y-axis direction, B2 is the eigenfunction coefficient two in the Y-axis direction, B3 is the eigenfunction coefficient three in the Y-axis direction, and B4 is the eigenfunction coefficient four in the Y-axis direction.

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 constraint equations based on three boundary conditions of rectangular thin plates: simply supported, fixed, and free boundaries is: The boundary conditions of the simply supported edge are: w = 0, M n =0; The boundary condition of the fixed edge is: w = 0, The boundary condition of the free edge is: 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 thin plate in the X-axis direction is x=a, the constraint equation is obtained: Where SS represents the corresponding simply supported edge boundary condition, C represents the corresponding fixed edge boundary condition, and F represents the corresponding free edge boundary condition; 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 cases where the four sides of the rectangular thin plate are simply supported, fixedly supported, and free, based on the boundary conditions, the four boundary conditions corresponding to the two sides x=0 and x=a give the homogeneous linear equations about A1~A4; similarly, the four boundary conditions corresponding to the two sides y=0 and y=b give the homogeneous linear equations about B1~B4; where b is the length of the side of the rectangular thin plate in the Y-axis direction; By setting the determinants of the two homogeneous linear equations equal to zero, we obtain the transcendental equation about (λ,μ). Then, by solving the transcendental equation, we obtain the vibration mode function W(x,y) and the natural vibration frequency.

10. The method for analyzing the lateral free vibration of a rectangular thin plate structure of a flat-panel satellite according to claim 9, characterized in that: For the four sides of the rectangular thin plate as fixed 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), According to formula (23), the constraint equation of the rectangular thin plate with four sides as fixed boundary conditions is obtained as follows: Substituting formula (19) and (20) into formula (28), we can obtain: Solving equations (29) and (30), we obtain expressions for coefficients A1~A4 and B1~B4; Let the determinant of the coefficient matrix corresponding to coefficients A1~A4 and B1~B4 be equal to zero, and we get: According to formulas (17) and (18), we can obtain: Formulas (31) to (33) are combined and the coefficients A1~A4 and B1~B4 are solved using the Newton-Raphson iterative numerical method to determine the eigenfunction and obtain the spatial eigenvalue (λ,μ), and then the mode function and the corresponding natural vibration frequency of the rectangular thin plate are obtained.

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