A vibration calculation method for sandwich panels with three-dimensional elastic core materials
Through the vibration calculation method of the sandwich plate of the three-dimensional elastomeric core material, the motion differential equation and the neutral surface displacement equation are established, which solves the calculation error problem caused by ignoring the normal deformation of the core material in the prior art, and improves the accuracy of vibration calculation , suitable for a variety of sandwich plate applications.
Patent Information
- Application Number
- CN202310911116.X
- Authority / Receiving Office
- CN · China
- Patent Type
- Patents(China)
- Current Assignee / Owner
- Filing Date
- 2023-07-21
- Publication Date
- 2024-12-13
- Estimated Expiration
- 2043-07-21
Smart Images
Figure CN116882247B_ABST
Abstract
Description
Technical Field
[0001] The present application relates to the technical field of structural vibration, and in particular to a vibration calculation method for a sandwich panel with a three-dimensional elastic core material. Background Art
[0002] The core structure of sandwich panels is a type of composite layer structure, usually consisting of two layers of high-strength thin surface panels and a soft and light core material. Sandwich panels have the characteristics of light weight, high strength and high rigidity. If appropriate panels and core materials are selected, excellent properties such as vibration reduction, sound insulation and heat insulation can also be obtained. Due to the above advantages of sandwich panels, sandwich panels are increasingly widely used in various military and civilian fields.
[0003] Most of the current sandwich panel theories, such as Reissner theory and Mead theory, have introduced some artificial simplified assumptions, such as ignoring the influence of normal extrusion deformation of the core material, assuming that the deflection of the panels on both sides of the core material is consistent, etc. However, since the core material modulus of sandwich panels is usually small, when medium and high frequency vibrations are performed, a variety of vibration modes will be reflected. At this time, the normal deformation of the core material cannot be ignored, and the vibration mode of the core material directly affects the vibration response of the sandwich panel, which leads to errors in the calculation of medium and high frequency vibrations in various existing theories. Summary of the invention
[0004] In view of the technical problem and technical demand of ignoring the normal deformation of the core material, which leads to errors in the calculation of high-frequency vibration of the sandwich panel, the applicant proposes a vibration calculation method for a sandwich panel with a three-dimensional elastic core material. The technical solution of this application is as follows:
[0005] The vibration calculation method of sandwich panels with three-dimensional elastic core materials includes:
[0006] A motion differential equation of the core material of the three-dimensional elastic material in the sandwich panel is established, and based on the motion differential equation of the core material, a core material displacement equation is obtained. The core material displacement equation indicates the functional relationship between the displacement of the core material during vibration and the modal order and contains unknown coefficients.
[0007] The two side panels on both sides of the core material in the sandwich panel are equivalent to thin plates respectively. For each panel, the neutral plane displacement equation of the panel is established based on the thin plate assumption. The neutral plane displacement equation of each panel indicates the functional relationship between the displacement of the neutral plane of the panel and the modal displacement of the panel during vibration. The modal displacement of the panel is related to the modal order. The neutral plane of each panel is a virtual plane in the panel that is neither subjected to tension nor pressure. The modal displacement of each panel includes the modal displacement of the panel in the x, y, and z directions of the virtual three-dimensional coordinate system at the corresponding modal order. The z direction is the normal direction perpendicular to the plane where the sandwich panel is located. The x direction and the y direction are perpendicular to each other on the plane where the sandwich panel is located, and the xoy plane is located on the middle surface of the core material.
[0008] Taking the displacement continuity on the interface formed by the core material and the panel on each side as the boundary condition, the functional relationship between the undetermined coefficients in the core material displacement equation and the modal displacements of the panels on both sides is determined according to the core material displacement equation and the neutral plane displacement equation of the corresponding panel on one side.
[0009] According to the core material displacement equation combined with the functional relationship between the unknown coefficients in the core material displacement equation and the modal displacement of the two side panels, the functional relationship between the shear stress and normal stress at the interface between the core material and each side panel and the modal displacement of the two side panels is obtained respectively.
[0010] Substitute the functional relationships between the shear stress and normal stress of the two interfaces and the modal displacement of the two side panels into the normal vibration motion equations of the two side panels and the in-plane vibration motion equations of the two side panels to obtain the vibration motion equations of the sandwich panel. The vibration motion equations of the sandwich panel indicate the functional relationship between the external load vector of the sandwich panel and the modal displacement of the two side panels.
[0011] The vibration equations of sandwich panels are used to calculate the vibration of sandwich panels.
[0012] The beneficial technical effects of this application are:
[0013] The present application discloses a vibration calculation method for a sandwich panel with a three-dimensional elastic body core material. The sandwich panel vibration calculation method treats the core material of the sandwich panel as equivalent to a three-dimensional elastic body and treats the face plate of the sandwich panel as equivalent to a thin plate during calculation, thereby taking into account the normal extrusion of the core material on the face plate, and significantly improving the accuracy of the vibration calculation results of the sandwich panel under vibrations in various frequency bands.
[0014] The present application also takes into account the inconsistency of thickness and material of the panels on both sides of the core material in the mechanical equation of the panels on both sides of the core material, so that vibration calculation can be performed on sandwich panels with inconsistent disturbance of the panels on both sides, making the application scope of the present application wider. BRIEF DESCRIPTION OF THE DRAWINGS
[0015] Figure 1 It is a flowchart of an embodiment of the present application.
[0016] Figure 2 It is a schematic diagram of a model of a simply supported rectangular sandwich panel according to an embodiment of the present application.
[0017] Figure 3 It is a stress diagram of a simply supported rectangular sandwich panel according to an embodiment of the present application.
[0018] Figure 4 It is a schematic diagram of the calculation results of the forced vibration response of a sandwich panel of an example of the present application.
[0019] Explanation of reference numerals: 1. core material; 2. first panel; 3. second panel. DETAILED DESCRIPTION
[0020] The specific implementation of the present application is further described below in conjunction with the accompanying drawings.
[0021] like Figure 1 As shown, in one embodiment, the vibration calculation method of the sandwich panel with three-dimensional elastic core material of the present application includes:
[0022] Step S110, establishing a differential equation of motion for a core material of a three-dimensional elastic material in a sandwich panel, and obtaining a core material displacement equation based on the differential equation of motion for the core material, wherein the core material displacement equation indicates a functional relationship between the displacement of the core material during vibration and the modal order and contains undetermined coefficients.
[0023] Step S120, the two side panels located on both sides of the core material in the sandwich panel are respectively equivalent to thin plates. For each panel, the neutral plane displacement equation of the panel is established based on the thin plate assumption. The neutral plane displacement equation of each panel indicates the functional relationship between the displacement of the neutral plane of the panel during vibration and the modal displacement of the panel. The modal displacement of the panel is related to the modal order. The neutral plane of each panel is a virtual plane in the panel that is neither subjected to tension nor pressure. The modal displacement of each panel includes the modal displacement of the panel in the x, y, and z directions of the virtual three-dimensional coordinate system at the corresponding modal order. The z direction is the normal direction perpendicular to the plane where the sandwich panel is located, and the x direction and the y direction are perpendicular to each other on the plane where the sandwich panel is located.
[0024] Step S130, taking the displacement continuity on the interface formed by the core material contacting the panel on each side as a boundary condition, according to the core material displacement equation and the neutral plane displacement equation of the panel on the corresponding side, determine the functional relationship between the undetermined coefficients in the core material displacement equation and the modal displacement of the two side panels.
[0025] Step S140, according to the core material displacement equation combined with the functional relationship between the undetermined coefficients in the core material displacement equation and the modal displacement of the two side panels, the functional relationship between the shear stress and normal stress of the interface between the core material and each side panel and the modal displacement of the two side panels is obtained respectively.
[0026] Step S150, substitute the functional relationship between the shear stress and normal stress of the two interfaces and the modal displacement of the two side panels into the normal vibration motion equation of the two side panels and the in-plane vibration motion equation of the two side panels to obtain the vibration motion equation group of the sandwich panel, which indicates the functional relationship between the external load vector of the sandwich panel and the modal displacement of the two side panels.
[0027] Step S160, performing vibration calculation on the sandwich panel using the vibration motion equations of the sandwich panel.
[0028] During calculation, the present application treats the core material of the sandwich panel as equivalent to a three-dimensional elastic body, constructs the core material displacement equation through three-dimensional elastic mechanics, and treats the face plate of the sandwich panel as equivalent to a thin plate, thereby taking into account the normal extrusion of the core material on the face plate, which significantly improves the accuracy of the vibration calculation results of the sandwich panel under vibrations in various frequency bands.
[0029] In order to more clearly illustrate the vibration calculation method of the sandwich panel with three-dimensional elastic core material of the present application, another embodiment of the present application is described in detail below with reference to the accompanying drawings.
[0030] Step S210, establishing a differential equation of motion of a core material of a three-dimensional elastic body in a sandwich panel.
[0031] In this embodiment, the sandwich panel is a simply supported rectangular sandwich panel. The structure of the simply supported rectangular sandwich panel is as follows: Figure 2 As shown, the sandwich panel includes a core material 1 and a first panel 2 and a second panel 3 located on both sides of the core material. The contact surface between the core material 1 and the first panel 2 is the first interface, and the contact surface between the core material 1 and the second panel 3 is the second interface. The sandwich panel is located in a virtual three-dimensional coordinate system composed of the x, y, and z axes. The z direction is the normal direction perpendicular to the plane where the sandwich panel is located. The x direction and the y direction are perpendicular to each other on the plane where the sandwich panel is located, and the xoy plane is located on the middle surface of the core material. The dimensions of the sandwich panel in the x and y directions are a and b respectively, the thickness of the core material 1 in the z direction is h, the thickness of the first panel 2 in the z direction is t1, and the thickness of the second panel 3 in the z direction is t2.
[0032] The core material is equivalent to a three-dimensional elastic body. In this embodiment, the core material is equivalent to a three-dimensional isotropic elastic body, and the influence of body force (such as gravity) is not considered, so the original motion differential equation of the core material is established as follows:
[0033]
[0034] Among them, σ x is the normal stress of the core material parallel to the x direction, σ y is the normal stress of the core material parallel to the y direction, σ z is the normal stress of the core material parallel to the z direction, τ xy is the shear stress of the core material perpendicular to the x-direction plane and parallel to the y-direction, τ yx is the shear stress of the core material perpendicular to the y-direction plane and parallel to the x-direction, τ xz is the shear stress of the core material perpendicular to the x-direction plane and parallel to the z-direction, τ zx is the shear stress of the core material perpendicular to the z-direction plane and parallel to the x-direction, τ yz is the shear stress of the core material perpendicular to the y-direction plane and parallel to the z-direction, τ zyis the shear stress of the core material perpendicular to the z-direction plane and parallel to the y-direction, ρ is the density of the core material, u(x,y,z,t) is the displacement of the core material at the coordinate (x,y,z) in the x-direction at time t, v(x,y,z,t) is the displacement of the core material at the coordinate (x,y,z) in the y-direction at time t, w(x,y,z,t) is the displacement of the core material at the coordinate (x,y,z) in the z-direction at time t, and t is the time parameter.
[0035] From the physical equation of the core material, we know that:
[0036]
[0037]
[0038] in, μ is the Poisson's ratio of the core material, G is the shear modulus of the core material,
[0039] The time domain expression of core displacement is:
[0040]
[0041] in, is the displacement of the microelement of the core material at the coordinate (x, y, z) in the x direction, is the displacement of the microelement of the core material at the coordinate (x, y, z) in the y direction, is the displacement of the microelement of the core material at the coordinate (x, y, z) in the z direction, j is the imaginary unit, ω=2πf, ω is the circular frequency of vibration, f is the vibration frequency,
[0042] In this embodiment, for convenience of representation, the time term e is not considered. jωt , so the core displacement is directly expressed as:
[0043]
[0044] Combining formulas (1)(2)(3)(5), the differential equation of motion of the core material is obtained as follows:
[0045]
[0046] in, is the three-dimensional Laplace operator.
[0047] Step S220, obtaining a core material displacement equation based on the core material's motion differential equation.
[0048] Using the displacement function method, two displacement functions F(x, y, z) and Φ(x, y, z) are introduced. Similarly, the two displacement functions omit the time term e jωt The expression for constructing the displacement function in series form is:
[0049]
[0050] Among them, f mn (z) is the general solution of the displacement function F(x,y,z), is the general solution of the displacement function Φ(x,y,z), X m (x) is the shape function of the first panel and the second panel in the x-coordinate direction (m, n), Y n (y) is the shape function of the first panel and the second panel in the y-coordinate direction (m, n), X m (x) = sin(η m x), Y n (y) = sin(ν n y), η m is the wave number of the (m,n)-order x-direction shape function, ν n is the wave number of the (m,n)-order shape function in the y direction, η m =mπ / a, ν n =nπ / b, m and n are the modal orders.
[0051] Based on the characteristics of the core material using three-dimensional elastic material, it is determined that the displacement function satisfies the following relationship:
[0052]
[0053] in, is the two-dimensional Laplace operator.
[0054] Solving for this yields:
[0055]
[0056] Among them, k mn is the total wave number of the sandwich panel vibrating in the xy plane,
[0057] Determine the general solution in the expression for the displacement function and f mn The expression of (z) is:
[0058]
[0059] Among them, A mn , B mn , C mn , D mn、E mn 、F mn are all undetermined coefficients under the (m,n)th order mode, k s is the wave number of shear waves in an infinite three-dimensional elastic body, k l is the wave number of the expansion and contraction wave in the infinite three-dimensional elastic body.
[0060] Substituting formula (10) into formula (7), we can obtain the expression of the displacement function expressed in terms of the modal order and its corresponding unknown coefficients:
[0061]
[0062] The relationship between the displacement of the core material during vibration and the displacement function is determined as follows:
[0063]
[0064] Where, α = 2(1-μ),
[0065] Substituting formula (11) (12) into formula (6), the core displacement equation is obtained as follows:
[0066]
[0067] in, i is 1 or 2, γ = 1-2μ, X′ m (x) is X m The derivative of (x), Y n ′(y) is Y n The derivative of (y).
[0068] Step S230: Equivalent two side panels located on both sides of the core material in the sandwich panel to thin plates respectively, and for each panel, establish a neutral plane displacement equation of the panel based on the thin plate assumption.
[0069] In this embodiment, the first panel and the second panel are equivalent to thin plates, that is, the thickness of the first panel and the second panel remains unchanged during vibration bending, and the rotational inertia force in the x and y directions during vibration of the first panel and the second panel is not considered.
[0070] The equation for determining the neutral plane displacement of the first panel is:
[0071]
[0072] The equation for determining the neutral plane displacement of the second panel is:
[0073]
[0074] in, is the displacement of the microelement at the coordinate (x, y, z) on the neutral plane of the first panel in the x direction, is the displacement of the infinitesimal element at the coordinate (x, y, z) on the neutral plane of the first panel in the y direction, is the displacement of the microelement at the coordinate (x, y, z) on the neutral plane of the first panel in the z direction; is the modal displacement of the first panel in the (m,n)th mode in the x direction, is the modal displacement of the first panel in the (m,n)th mode in the y direction, is the modal displacement of the first panel in the (m,n)th mode in the z direction; is the displacement of the microelement at the coordinate (x, y, z) on the neutral plane of the second panel in the x direction, is the displacement of the microelement at the coordinate (x, y, z) on the neutral plane of the second panel in the y direction, is the displacement of the microelement at the coordinate (x, y, z) on the neutral plane of the second panel in the z direction; is the modal displacement of the second panel in the (m,n)th mode in the x direction, is the modal displacement of the second panel in the (m,n)th mode in the y direction, is the modal displacement of the second panel in the (m, n)th mode in the z direction.
[0075] Step S240, taking the displacement continuity on the interface formed by the core material contacting the panel on each side as a boundary condition, according to the core material displacement equation and the neutral plane displacement equation of the panel on the corresponding side, determine the functional relationship between the undetermined coefficients in the core material displacement equation and the modal displacement of the panels on both sides.
[0076] In this embodiment, displacement continuity boundary conditions are established between the core material and the first panel and the second panel at the first interface and the second interface respectively. Based on the displacement continuity boundary conditions on the first interface, the first boundary condition equation group is constructed:
[0077]
[0078] Based on the displacement continuity boundary condition on the second interface, the second boundary condition equation group is constructed:
[0079]
[0080] in, The core material is located at coordinate The displacement of the infinitesimal element at in the x direction is, The core material is located at coordinate The displacement of the infinitesimal element at in the y direction is, The core material is located at coordinate The displacement of the infinitesimal element in the z direction, coordinate The infinitesimal element at is located on the first interface; The core material is located at coordinate The displacement of the infinitesimal element at in the x direction is, The core material is located at coordinate The displacement of the infinitesimal element at in the y direction is, The core material is located at coordinate The displacement of the infinitesimal element in the z direction, coordinate The infinitesimal element at is located on the second interface.
[0081] Substituting formula (14) (15) into formula (16) (17), the functional relationship between the unknown coefficients in the core displacement equation and the modal displacements of the two side panels is as follows:
[0082]
[0083]
[0084]
[0085]
[0086]
[0087]
[0088] in,
[0089] Step S250, according to the core material displacement equation combined with the functional relationship between the undetermined coefficients in the core material displacement equation and the modal displacement of the two side panels, the functional relationship between the shear stress and normal stress of the interface between the core material and each side panel and the modal displacement of the two side panels is obtained respectively.
[0090] like Figure 3 The stress diagram of a simply supported rectangular sandwich panel is shown to determine the shear stress in the core material and its interface with any side panel. and The expression is:
[0091]
[0092] in, is the shear stress in the first interface perpendicular to the x-direction plane and parallel to the z-direction, is the shear stress in the first interface perpendicular to the y-direction plane and parallel to the z-direction, is the shear stress in the second interface perpendicular to the x-direction plane and parallel to the z-direction, is the shear stress in the second interface that is perpendicular to the y-direction plane and parallel to the z-direction.
[0093] Determine the normal stress at the interface between the core and any side face sheets The expression is in, is the normal stress parallel to the z direction in the first interface, is the normal stress parallel to the z direction in the second interface.
[0094] Substituting formula (13) and formula (18)-(23) into formula (24), we obtain:
[0095]
[0096]
[0097] Substituting formula (13) and formula (18)-(23) into the expression of normal stress between the core material and the interface on both sides, we get:
[0098]
[0099] Step S260, substitute the functional relationship between the shear stress and the normal stress of the two interfaces and the modal displacement of the two side panels into the normal vibration motion equations of the two side panels and the in-plane vibration motion equations of the two side panels to obtain the vibration motion equation group of the sandwich panel.
[0100] In this embodiment, taking the first panel as an example, since the first panel is equivalent to a thin plate, the rotational inertia force in the x and y directions during vibration is not considered, and thus the mechanical equation is obtained:
[0101]
[0102] in, is the bending moment of the first panel rotating around the y-axis in the plane perpendicular to the x-direction, is the bending moment of the first panel rotating around the x-axis in the plane perpendicular to the y-direction, is the torque of the first panel rotating around the x-axis in the plane perpendicular to the x-direction, is the shear force parallel to the z direction on the xz plane of the first panel, is the shear force parallel to the z direction on the yz plane of the first panel, q1 is the total force in the z direction on the first panel, including the external load, the normal inertia force and the normal stress of the core material on the first panel, P1(x,y) is the external load on the microelement of the first panel at the coordinate (x,y), and ρ1 is the density of the first panel.
[0103] Known is the modal load of the first panel in the (m, n)th mode. Substituting these two known formulas into formula (28), the expression of the normal vibration motion equation of the first panel is obtained: D1 is the bending stiffness of the first panel, E1 is the Young's modulus of the first panel, and μ1 is the Poisson's ratio of the first panel.
[0104] The shear stress between the core material and the first interface and And the normal stress between the core material and the first interface Substituting the expression of the normal vibration motion equation of the first panel into the equation, the normal vibration motion equation of the first panel (m, n)th mode is obtained as follows:
[0105]
[0106] Similarly, the expression of the normal vibration motion equation of the second panel is obtained: D2 is the bending stiffness of the second panel, E2 is the Young's modulus of the second panel, μ2 is the Poisson's ratio of the second panel, ρ2 is the density of the second panel, and the calculation method of the normal vibration motion equation of the second panel is the same as that of the first panel. Those skilled in the art can calculate it by analogy, so it will not be repeated here. and And the normal stress between the core material and the second interface Substituting the expression of the normal vibration motion equation of the second panel into the equation, the normal vibration motion equation of the second panel (m, n)th order mode is obtained as follows:
[0107]
[0108] in, is the modal load of the second panel in the (m,n)th mode.
[0109] Determined based on the thin plate assumption in, is the normal stress parallel to the x-direction at the neutral surface of the first panel, is the shear stress perpendicular to the x-direction plane and parallel to the y-direction at the neutral surface of the first panel; combined with the shear stress of the core material and the first interface Substitute the equation of motion for the first panel's in-plane vibration in the x direction into The in-plane vibration motion equation of the first panel in the (m, n)th mode in the x direction is obtained as follows:
[0110]
[0111] Determined based on the thin plate assumption in, is the normal stress parallel to the x-direction at the neutral surface of the second panel, is the shear stress perpendicular to the x-direction plane and parallel to the y-direction at the neutral surface of the second panel; combined with the shear stress of the core material and the second interface Substitute the equation of motion for the second panel's in-plane vibration in the x direction into the equation The in-plane vibration motion equation of the (m, n)th mode in the x direction of the second panel is obtained as follows:
[0112]
[0113] Determined based on the thin plate assumption in, is the normal stress parallel to the y direction at the neutral surface of the first panel; combined with the shear stress between the core material and the first interface Substitute the equation of motion for the first panel's in-plane vibration in the y direction into the equation: The in-plane vibration motion equation of the (m, n)th order mode in the y direction of the first panel is obtained as follows:
[0114]
[0115] Determined based on the thin plate assumption in, is the normal stress parallel to the y direction at the neutral surface of the second panel; combined with the shear stress between the core material and the second interface Substitute the equation of motion for the second panel's in-plane vibration in the y direction into the equation The in-plane vibration motion equation of the (m, n)th order mode of the second panel in the y direction is obtained as follows:
[0116]
[0117] in,
[0118]
[0119]
[0120]
[0121]
[0122]
[0123]
[0124]
[0125]
[0126] Step S260, performing vibration calculation on the sandwich panel using the vibration motion equations of the sandwich panel.
[0127] Since the above vibration motion equations exist These six unknowns are solved by combining formulas (29)-(34). In this embodiment, for the convenience of solving, the vibration motion equations are formed into the following matrix control form:
[0128] L mn ·d mn =P mn (35)
[0129] Among them, the displacement vector of the (m,n)th order mode is External load vector of the (m,n)th mode Parameter matrix L mn It is a matrix obtained based on the vibration motion equations and is independent of the template displacement of the two side panels and the external loads they are subjected to.
[0130] The vibration motion equations of sandwich panels are used to calculate the vibration of sandwich panels, including:
[0131] 1. Determine the parameter matrix L of the (m,n)th order mode mn , according to the matrix control form combined with the external load vector P of the (m,n)th order mode mn Determine the natural frequency of the (m,n)th mode of the sandwich panel in free vibration:
[0132] Determine that all elements in the external load vector of the (m, n)th order mode of the sandwich panel during free vibration are 0, and solve the parameter matrix L of the (m, n)th order mode mn The determinant of |L mn |=0 to obtain the natural frequency of the (m,n)th mode of the sandwich panel during free vibration.
[0133] In this embodiment, the parameter matrix L mn Each element in is the natural circular frequency ω of the (m,n)th mode. mn The relevant quantity is, let the determinant |L mn |=0, and the result contains only the unknown quantity ω mn After determining the modal order, the natural circular frequency ω corresponding to the modal order is obtained by solving the one-variable equation group. mn ; Through the natural frequency expression Solve to obtain the natural frequency f of the (m,n)th order mode mn .
[0134] 2. According to the matrix control form combined with the external load vector P of the (m,n)th order mode mnDetermine the displacement of the core and the two side panels of the sandwich panel when it is subjected to forced vibration:
[0135] Determine the external load vector P obtained when the sandwich panel is subjected to forced vibration mn , the external load vector P mn and the parameter matrix L mn Substitute into the vibration motion equation and solve to obtain the displacement vector of the (m,n)th order mode The modal displacements of the two side panels in the (m, n)th order mode included in are substituted into formulas (18)-(23) and (13) to obtain the displacement of the core material; the modal displacements of the two side panels in the (m, n)th order mode are substituted into formulas (14) and (15) to obtain the displacements of the two side panels.
[0136] In this embodiment, in order to verify that the sandwich panel vibration calculation method of the present application is more accurate than the calculation results of the prior art, three examples are used below to calculate and compare the natural frequency and forced vibration response results of the sandwich panel under various methods:
[0137] 1. Natural frequency calculation and comparison
[0138] In the first example, a model of a simply supported rectangular sandwich panel is established. The length of the sandwich panel is 0.6m, the width is 0.5m, the thickness of the first panel and the second panel are both 0.005m, and the thickness of the core material is 0.1m. Table 1 shows the materials and mechanical properties of the sandwich panel.
[0139] Table 1 Materials and mechanical properties of sandwich panels
[0140] Material Modulus / GPa Poisson's ratio Density / kg·m-3 Two side panels Fiberglass 20 0.15 2000 core material PVC Foam 0.055 0.32 100
[0141] The finite element model of the sandwich panel was established using finite element software. The core material used three-dimensional solid elements, and the two side panels used shell elements to calculate its natural frequency. In order to better reflect the vibration phenomenon of core material expansion and shear coexistence, multi-layer grids were divided in the thickness direction of the sandwich panel, and second-order elements were used for calculation.
[0142] The sandwich panel vibration calculation method of the present application is compared with the calculation results of the Reissner theory and the Mead theory in the prior art and the finite element calculation results, as shown in Table 2.
[0143] Table 2 Calculation of natural frequency of sandwich panels
[0144]
[0145] In Table 2, the Reissner theory only considers the tensile and compressive capacity of the sandwich panel face sheet and the shear capacity of the core material, and assumes that the sandwich panel cannot deform in the direction of vibration, so its calculation error is large, and the error increases with the increase of the modal order (increase of frequency). The Mead theory considers the bending resistance of the face sheet itself and the interaction between the core material and the face sheet of the sandwich panel, but also assumes that the sandwich panel cannot deform in the direction of vibration. The error of its calculation result is smaller than that of the Reissner theory, but it still increases with the increase of the modal order. It can be seen that the assumption of ignoring the normal deformation of the sandwich panel is not applicable in the medium and high frequency bands. The calculation process of the sandwich panel vibration calculation method of the present application is not only simpler, but also can maintain stable calculation accuracy at high modal orders. This proves the effectiveness of the method of the present application in predicting the natural frequency of simply supported rectangular sandwich panels, and at the same time proves that the assumption of "the face sheet is a thin plate and the core material is a three-dimensional elastic body" is applicable to the vibration prediction of sandwich panels in the medium and high frequency bands.
[0146] In addition, the present application also takes into account the situation that the thickness or material of the panels on both sides of the core material is inconsistent, and can perform vibration calculation on sandwich panels with inconsistent deflections of the panels on both sides. The following is another example to describe:
[0147] In the second example, a model of a simply supported rectangular sandwich panel is established. The sandwich panel is 1.2 m long, 1 m wide, the first panel thickness is 0.002 m, the second panel thickness is 0.008 m, and the core material thickness is 0.05 m. Table 1 shows the materials and mechanical properties of the sandwich panel.
[0148] The finite element model of the sandwich panel was established using finite element software. The core material used three-dimensional solid elements, and the two side panels used shell elements to calculate its natural frequency. In order to better reflect the vibration phenomenon of core material expansion and shear coexistence, multi-layer grids were divided in the thickness direction of the sandwich panel, and second-order elements were used for calculation.
[0149] Since the Reissner theory and the Mead theory in the prior art cannot calculate the natural frequency of a sandwich panel with inconsistent thickness or material of the two side panels, the calculation results of this example are only compared with the finite element calculation results, as shown in Table 3.
[0150] Table 3 Calculation of natural frequency of sandwich panels
[0151]
[0152] Table 3 compares the natural frequencies calculated in this example with those calculated by finite element method. The errors are both within 5%, which further illustrates the validity of the theoretical calculation in this paper.
[0153] 2. Calculation and comparison of forced vibration response of sandwich panels
[0154] The calculation is performed using the sandwich panel model in the first example. A unit excitation force is applied at the center of the first panel. The equation L mn ·d mn =P mn , solve the vibration acceleration response of the excitation point, the reference level of acceleration is 10 -6 m / s 2 .
[0155] Comparison of calculation results of sandwich panel forced vibration response Figure 4 As shown, in the frequency range of 1Hz to 1000Hz, the frequencies of the first few vibration peaks of Mead theory, the theory of this application and the finite element calculation results are very close, and they are all bending vibration modes, while the frequency of the vibration peak of Reissner theory is low, and the deviation increases with the increase of frequency. In the frequency range of 1000 to 2000Hz, the theory of this application is in good agreement with the finite element calculation results, and has several more vibration peaks than Reissner theory and Mead theory. After analysis, most of the peaks are affected by the expansion vibration mode, and because Reissner theory and Mead theory ignore the normal deformation of the sandwich panel, they cannot reflect the response part caused by the expansion vibration of the sandwich panel. This proves the effectiveness of the method of this application in predicting the forced vibration response of simply supported rectangular sandwich panels, especially in the medium and high frequency bands, with more accurate forced vibration response prediction results.
[0156] The above is only a preferred embodiment of the present application, and the present application is not limited to the above embodiments. It is understood that other improvements and changes directly derived or associated by those skilled in the art without departing from the spirit and concept of the present application should be considered to be included in the protection scope of the present application.
Claims
1. A vibration calculation method for sandwich panels with three-dimensional elastic core materials, It is characterized in that The sandwich panel vibration calculation method comprises: Establishing a differential equation of motion of a core material of a three-dimensional elastic material in a sandwich panel, and obtaining a core material displacement equation based on the differential equation of motion of the core material, wherein the core material displacement equation indicates a functional relationship between the displacement of the core material and the modal order during vibration and contains undetermined coefficients; The core material displacement equation is: in, is the displacement of the microelement of the core material at the coordinate (x, y, z) in the x direction, is the displacement of the microelement of the core material at the coordinate (x, y, z) in the y direction, is the displacement of the microelement of the core material at the coordinate (x, y, z) in the z direction, i is 1 or 2, γ=1-2μ, ω=2πf, ω is the circular frequency of vibration, f is the vibration frequency, μ is the Poisson's ratio of the core material, ρ is the density of the core material, G is the shear modulus of the core material, α=2(1-μ), X m (x) = sin(η m x), Y n (y) = sin(ν n y), X′ m (x) is X m The derivative of (x), Y′ n (y) is Y n The derivative of (y), η m =mπ / a, ν n =nπ / b, a, b are the dimensions of the sandwich panel in the x-direction and y-direction respectively, m, n are the modal orders, A mn , B mn , C mn , D mn 、E mn 、F mn are all undetermined coefficients under the (m,n)th order mode, The core material displacement equation is obtained based on the motion differential equation of the core material, including: Based on the characteristics of the core material being a three-dimensional elastic material, the motion differential equation of the core material is constructed, and the expression is: The expression for constructing the displacement function expressed by the modal order and its corresponding undetermined coefficients is: The relationship between the displacement of the core material during vibration and the displacement function is determined as follows: Substituting the relationship between the displacement of the core material during vibration and the displacement function and the expression of the displacement function into the expression of the differential equation of motion of the core material to obtain the displacement equation of the core material; in, is the two-dimensional Laplace operator, t is the time parameter; The expression for constructing the displacement function represented by the modal order and its corresponding undetermined coefficients includes: The expression for constructing the displacement function in series form is: Based on the characteristics of the core material being a three-dimensional elastic material, it is determined that the displacement function satisfies the following relationship: Solving for this yields: Determine the general solution in the expression for the displacement function and f mn The expression of (z) is: The general solution and f mn Substituting the expression of (z) into the expression of the displacement function, an expression of the displacement function represented by the modal order and its corresponding undetermined coefficients is obtained; The two side panels located on both sides of the core material in the sandwich panel are respectively equivalent to thin plates. For each of the panels, the neutral plane displacement equation of the panel is established based on the thin plate assumption. The neutral plane displacement equation of each panel indicates the functional relationship between the displacement of the neutral plane of the panel and the modal displacement of the panel during vibration. The modal displacement of the panel is related to the modal order. The neutral plane of each panel is a virtual plane in the panel that is neither subjected to tension nor pressure. The modal displacement of each panel includes the modal displacement of the panel in the x, y, and z directions of the virtual three-dimensional coordinate system oxyz at the corresponding modal order. The z direction is the normal direction perpendicular to the plane where the sandwich panel is located. The x direction and the y direction are perpendicular to each other on the plane where the sandwich panel is located, and the xoy plane is located on the middle surface of the core material. Taking the displacement continuity on the interface formed by the core material contacting the panel on each side as a boundary condition, according to the core material displacement equation and the neutral plane displacement equation of the panel on the corresponding side, determining the functional relationship between the undetermined coefficients in the core material displacement equation and the modal displacements of the panels on both sides; According to the core material displacement equation combined with the functional relationship between the undetermined coefficients in the core material displacement equation and the modal displacements of the two side panels, the functional relationship between the shear stress and the normal stress at the interface between the core material and each side panel and the modal displacements of the two side panels is obtained respectively; Substituting the functional relationships between the shear stress and the normal stress of the two interfaces and the modal displacement of the two side panels into the normal vibration motion equations of the two side panels and the in-plane vibration motion equations of the two side panels, respectively, to obtain a group of vibration motion equations of the sandwich panel, wherein the group of vibration motion equations of the sandwich panel indicates the functional relationship between the external load vector of the sandwich panel and the modal displacement of the two side panels; The vibration motion equations of the sandwich panel are used to perform vibration calculation on the sandwich panel.
2. The sandwich panel vibration calculation method according to claim 1, It is characterized in that The neutral plane displacement equation of the panel is established based on the thin plate assumption, including: The neutral plane displacement equation of the first panel on one side of the core material is determined as: The equation for determining the neutral plane displacement of the second panel on the other side of the core material is: in, is the displacement of the microelement at the coordinate (x, y, z) on the neutral plane of the first panel in the x direction, is the displacement of the infinitesimal element at the coordinate (x, y, z) on the neutral plane of the first panel in the y direction, is the displacement of the microelement at the coordinate (x, y, z) on the neutral plane of the first panel in the z direction; is the modal displacement of the first panel in the (m,n)th mode in the x direction, is the modal displacement of the first panel in the (m,n)th mode in the y direction, is the modal displacement of the first panel in the (m,n)th mode in the z direction; is the displacement of the microelement at the coordinate (x, y, z) on the neutral plane of the second panel in the x direction, is the displacement of the microelement at the coordinate (x, y, z) on the neutral plane of the second panel in the y direction, is the displacement of the microelement at the coordinate (x, y, z) on the neutral plane of the second panel in the z direction; is the modal displacement of the second panel in the (m,n)th mode in the x direction, is the modal displacement of the second panel in the (m,n)th mode in the y direction, is the modal displacement of the second panel in the (m, n)th mode in the z direction.
3. The sandwich panel vibration calculation method according to claim 2, It is characterized in that Determining the functional relationship between the undetermined coefficients in the core material displacement equation and the modal displacements of the two side panels includes: Based on the displacement continuity boundary condition on the first interface formed by the contact between the core material and the first panel, a first boundary condition equation group is constructed: Based on the displacement continuity boundary condition on the second interface formed by the contact between the core material and the second panel, a second boundary condition equation group is constructed: in, The core material is located at coordinates The displacement of the infinitesimal element at in the x direction is, The core material is located at coordinates The displacement of the infinitesimal element at in the y direction is, The core material is located at coordinates The displacement of the infinitesimal element in the z direction, coordinate The microelement at is located on the first interface; The core material is located at coordinates The displacement of the infinitesimal element at in the x direction is, The core material is located at coordinates The displacement of the infinitesimal element at in the y direction is, The core material is located at coordinates The displacement of the infinitesimal element in the z direction, coordinate The microelement at is located on the second interface; h is the thickness of the core material in the z direction, t 1 is the thickness of the first panel in the z direction, t 2 is the thickness of the second panel in the z direction; Substituting the core material displacement equation and the neutral plane displacement equation of the two side panels into two boundary condition equation groups, determining the functional relationship between the undetermined coefficients in the core material displacement equation and the modal displacement of the two side panels includes: in, 4. The sandwich panel vibration calculation method according to claim 3, It is characterized in that The functional relationship between the shear stress and normal stress of the interface between the core material and each side panel and the modal displacement of the two side panels is obtained according to the core material displacement equation combined with the functional relationship between the undetermined coefficients in the core material displacement equation and the modal displacement of the two side panels, including: Determine the shear stress at the core and at its interface with any side panels and The expression is: Determine the normal stress at the interface between the core and any side panels The expression is Substituting the core material displacement equation and the functional relationship between the undetermined coefficients in the core material displacement equation and the modal displacements of the two side panels into the expression of the shear stress between the core material and the two side interfaces, we obtain: in, is the shear stress in the first interface perpendicular to the x-direction plane and parallel to the z-direction, is the shear stress in the first interface perpendicular to the y-direction plane and parallel to the z-direction, is the shear stress in the second interface perpendicular to the x-direction plane and parallel to the z-direction, is the shear stress in the second interface perpendicular to the y-direction plane and parallel to the z-direction; Substituting the core material displacement equation and the functional relationship between the undetermined coefficients in the core material displacement equation and the modal displacement of the two side panels into the expression of the normal stress between the core material and the two side interfaces, we obtain: in, is the normal stress parallel to the z direction in the first interface, is the normal stress parallel to the z direction in the second interface.
5. The sandwich panel vibration calculation method according to claim 4, It is characterized in that The vibration motion equation group includes: the normal vibration motion equation of the first panel (m, n)th order mode, the normal vibration motion equation of the second panel (m, n)th order mode, the in-plane vibration motion equation of the first panel in the x-direction (m, n)th order mode, the in-plane vibration motion equation of the second panel in the x-direction (m, n)th order mode, the in-plane vibration motion equation of the first panel in the y-direction (m, n)th order mode, and the in-plane vibration motion equation of the second panel in the y-direction (m, n)th order mode; the vibration motion equation group of the sandwich panel is obtained, including: The shear stress between the core material and the first interface and and the normal stress between the core material and the first interface Substitute the expression of the normal vibration motion equation of the first panel into The normal vibration motion equation of the (m, n)th order mode of the first panel is obtained as follows: The shear stress between the core material and the second interface and and the normal stress between the core material and the second interface Substitute the expression of the normal vibration motion equation of the second panel into The normal vibration motion equation of the (m, n)th order mode of the second panel is obtained as follows: Determined based on the thin plate assumption Combining the shear stress of the core material and the first interface Substitute the equation of motion of the first panel in the x direction into the expression of the in-plane vibration motion equation The in-plane vibration motion equation of the first panel in the (m, n)th order mode in the x direction is obtained as follows: Determined based on the thin plate assumption Combined shear stress of the core material and the second interface Substitute the equation of motion of the second panel in the x direction into the expression The in-plane vibration motion equation of the (m, n)th order mode of the second panel in the x direction is obtained as follows: Determined based on the thin plate assumption Combining the shear stress of the core material and the first interface Substitute the equation of motion of the first panel in the y direction into the expression of the in-plane vibration equation The in-plane vibration motion equation of the (m, n)th order mode of the first panel in the y direction is obtained as follows: Determined based on the thin plate assumption Combined shear stress of the core material and the second interface Substitute the equation of motion of the second panel in the y direction into the expression The in-plane vibration motion equation of the (m, n)th order mode of the second panel in the y direction is obtained as follows: in, is the modal load of the first panel in the (m,n)th mode, is the modal load of the second panel in the (m, n)th mode; P 1 (x, y) is the external load on the microelement of the first panel at the coordinate (x, y), P 2 (x, y) is the external load on the microelement of the second panel at the coordinate (x, y); ρ 1 is the density of the first panel, ρ 2 is the density of the second panel; E 1 is the Young's modulus of the first panel, E 2 is the Young's modulus of the second panel; t is the time parameter, μ 1 is the Poisson's ratio of the first panel, μ 2 is the Poisson's ratio of the second panel; is the normal stress parallel to the x-direction at the neutral plane of the first panel, is the normal stress parallel to the x-direction at the neutral plane of the second panel, is the normal stress parallel to the y direction at the neutral plane of the first panel, is the normal stress parallel to the y direction at the neutral plane of the second panel, is the shear stress perpendicular to the x-direction plane and parallel to the y-direction at the neutral plane of the first panel, is the shear stress perpendicular to the x-direction plane and parallel to the y-direction at the neutral plane of the second panel, D 1 is the bending stiffness of the first panel, D 2 is the bending stiffness of the second panel; 6. The sandwich panel vibration calculation method according to claim 5, It is characterized in that The vibration motion equations are formed into the following matrix control form: L mn ·d mn =P mn ; The method of performing vibration calculation on the sandwich panel using the vibration motion equations of the sandwich panel comprises: Determine the parameter matrix L of the (m,n)th mode mn , according to the matrix control form combined with the external load vector P of the (m,n)th order mode mn Determine the natural frequency of the (m, n)th mode of the sandwich panel during free vibration, and / or, according to the matrix control form combined with the external load vector P of the (m, n)th mode mn Determining the displacement of the core material and the two side panels when the sandwich panel is subjected to forced vibration; Among them, the displacement vector of the (m,n)th order mode is External load vector of the (m,n)th mode Parameter matrix L mn It is a matrix obtained based on the vibration motion equations and is independent of the template displacement of the two side panels and the external loads they are subjected to.
7. The sandwich panel vibration calculation method according to claim 6, It is characterized in that Parameter matrix L mn Each element in is related to the natural circular frequency ω mn Related quantities, determining the natural frequency of the (m, n)th order mode of the sandwich panel during free vibration and the displacement of the core material and the two side panels during forced vibration of the sandwich panel, include: Determine the external load vector P of the (m, n)th mode of the sandwich panel during free vibration mn All elements in are 0, let the parameter matrix L mn The determinant of |L mn |=0, solve to get the natural circular frequency ω of the (m,n)th order mode mn ;pass Solve to obtain the natural frequency f of the (m,n)th mode of the sandwich panel during free vibration mn ; Determine the external load vector P obtained by the sandwich panel when it is subjected to forced vibration mn , the external load vector P mn and the parameter matrix L mn Substitute into the vibration motion equation and solve to obtain the displacement vector of the (m,n)th order mode: The modal displacement of the two side panels in the (m, n)th order mode included in is substituted into the functional relationship between the undetermined coefficients in the core material displacement equation and the modal displacement of the two side panels and the core material displacement equation to obtain the displacement of the core material; the modal displacement of the two side panels in the (m, n)th order mode is substituted into the neutral plane displacement equation of the two side panels to obtain the displacement of the two side panels.