CN102607695B - Method for calculating natural frequency of honeycomb sandwich plate - Google Patents

Abstract

The invention relates to a method for calculating a natural frequency of a honeycomb sandwich plate, which comprises the following steps of: by measuring bending rigidities D11, D12, D22 and D66 of the sandwich plate, shear moduli Gxz and Gyz of a core layer in x-z and y-z planes, a length a and a width b of the sandwich plate, a panel thickness t, a core layer thickness h and a panel density rho<f> and a core layer density rho<c> of the sandwich plate, calculating to obtain shear stiffnesses Cxz and Cyz of the core layer and an equivalent face density rho; and substituting the measured data into a formula to calculate so as to solve the natural frequency omega<m> of an mth order of the sandwich plate. According to the invention, a bending vibration equation system form and the process of solving the natural frequency of the sandwich plate are simplified, the influence of the transverse shear deformation of the core layer and the integral orthogonal anisotropy of the sandwich plate on the mechanical property of the sandwich plate is also considered, the calculating process of the natural frequency is simplified, the calculating efficiency is improved, the calculating difficulty is reduced and the calculating accuracy of the natural frequency is improved.

Description

A kind of measuring method of honeycomb sandwich panel natural frequency

Technical field

The present invention relates to a kind of measuring method of sandwich plate natural frequency, particularly a kind of measuring method for honeycomb sandwich panel natural frequency.

Background technology

Bending Vibration Analysis is the important foundation of carrying out structural design and engineering application thereof.Along with the widespread use of honeycomb sandwich construction in various fields such as space flight, aviation, high-speed transit means of transport and modern structure engineerings, its vibration problem becomes increasingly conspicuous, generally attention and the concern of engineering circles have been caused, vibration analysis and natural frequency design have become a key link in product design, are also one of major mechanical problems in Important Project.In the past few decades, Chinese scholars uses analytic method, numerical method and experimental technique from various angles, the vibration characteristics of honeycomb sandwich panel to be studied.The vibration problem of isotropy panel sandwich construction that the people such as Allen, Vinson has used classical laminated plate theoretical research, but this theory is based upon on the straight normal hypothesis of Kirchhoff basis, because the sandwich layer of sandwich construction is softer and thickness is larger, during flexural vibrations, under cross shear effect, larger detrusion will be there is, so straight normal hypothesis is no longer applicable when analyzing sandwich plate crooked, its computational accuracy has much room for improvement.In order to improve the computational accuracy of sandwich plate vibration problem, the people such as Reissner and Mindlin have proposed the shear deformable theory of slab, and have derived the bending vibration control system of equations of sandwich plate.Owing to containing 3 generalized displacements in Orthotropic Sandwich Plate bending vibration control system of equations the most conventional in engineering, for the general employing of solving of its natural frequency Rayleigh-Ritz method, equation form and solution procedure are all comparatively complicated, and this has limited this application of system of equations in the engineering practices such as composite Materials Design to a certain extent.Many for sandwich plate bending vibration control system of equations displacement function, equation form is complicated, by problems such as analytical method solving difficulties, Inst. of Mechanics, CAS is in the bending of monograph < < Laminated Plate/Shell, stable and vibration > > (Science Press, 1977) in provided a kind of engineering simplification solution, but this kind of method is only limited to the situation of isotropy panel, ignored the anisotropic impact of sandwich plate whole-body quadrature; The people such as the Wu Hui of Lanzhou University are at the natural frequency > > (authorship of the sandwich clip rectangle laminate of paper < < simply supported on four sides orthotropy ripple type, the 22nd the 9th phase of volume of calendar year 2001,919 pages-926 pages) in a kind of short-cut method of sandwich plate bending vibration control system of equations has been proposed, but this method has been ignored the impact of sandwich layer transverse shear deformation, and computational accuracy still has much room for improvement.

Summary of the invention

Object of the present invention is just to provide a kind of measuring method of honeycomb sandwich panel natural frequency, and it can simplify the computation process of natural frequency, improves counting yield, reduces difficulty in computation, and improves the accuracy in computation of natural frequency.

The object of the invention is to realize by such technical scheme, concrete steps are as follows:

1) use compound substance universal testing machine to measure sandwich plate to be measured, obtain sandwich plate bending stiffness D ₁₁, D ₁₂, D ₂₂, D ₆₆with the shear modulus G of sandwich layer in x-z and y-z plane _xz, G _yz;

2) utilize ruler measurement measured panel sandwich plate length a, width b, thickness t and core layer thickness h;

3) by formula C _xz=G _xzh (1+t/h) ²and C _yz=G _yzh (1+t/h) ², calculate the shearing rigidity C of sandwich layer _xzand C _yz;

4) by material handbook, check in the panel density p of clamping plate to be measured _fwith sandwich layer density p _c, by formula ρ=ρ _ch+2 ρ _ft calculates the equivalent face density p of clamping plate;

5) data substitution formula above-mentioned steps being recorded calculates, and in conjunction with the boundary condition of sandwich plate, solves the natural frequency ω on sandwich plate m rank _m, its formula is as follows:

$\begin{array}{c}\frac{D_{11}+D_{66}}{C_{\mathrm{yz}}}\frac{{\&PartialD;}^6\mathrm{\&chi;}}{\&PartialD;x^6}+Q_1\frac{{\&PartialD;}^6\mathrm{\&chi;}}{\&PartialD;x^4\&PartialD;y^2}+Q_2\frac{{\&PartialD;}^6\mathrm{\&chi;}}{\&PartialD;x^2{\&PartialD;y}^4}-2(D_{12}+{2D}_{66})\frac{{\&PartialD;}^4\mathrm{\&chi;}}{\&PartialD;x^2{\&PartialD;y}^2}-D_{11}\frac{{\&PartialD;}^4\mathrm{\&chi;}}{\&PartialD;x^4}-D_{22}\frac{{\&PartialD;}^4\mathrm{\&chi;}}{\&PartialD;y^4}+\frac{D_{22}{D}_{66}}{C_{\mathrm{xz}}}\frac{{\&PartialD;}^6\mathrm{\&chi;}}{\&PartialD;y^6}\\ +\mathrm{\&rho;}\frac{{\&PartialD;}^2}{\&PartialD;t^2}\{\mathrm{\&chi;}-(\frac{D_{11}}{C_{\mathrm{xz}}}+\frac{D_{66}}{C_{\mathrm{yz}}})\frac{{\&PartialD;}^2\mathrm{\&chi;}}{\&PartialD;x^2}-\frac{D_{11}{D}_{66}}{C_{\mathrm{xz}}{C}_{\mathrm{yz}}}\frac{{\&PartialD;}^4\mathrm{\&chi;}}{\&PartialD;x^4}-(\frac{D_{22}}{C_{\mathrm{yz}}}+\frac{D_{66}}{C_{\mathrm{xz}}})\frac{{\&PartialD;}^2\mathrm{\&chi;}}{\&PartialD;y^2}+Q_3\frac{{\&PartialD;}^4\mathrm{\&chi;}}{\&PartialD;x^2\&PartialD;y^2}+Q_4\frac{{\&PartialD;}^4\mathrm{\&chi;}}{\&PartialD;y^4}\}=0\end{array}$

In formula

$Q_1=\frac{D_{11}{D}_{66}}{C_{\mathrm{xz}}}+\frac{D_{11}{D}_{22}-2D_{12}{D}_{66}-D_{12}^2}{C_{\mathrm{yz}}},Q_2=\frac{D_{11}{D}_{22}-D_{12}^2}{C_{\mathrm{xz}}}+\frac{D_{22}{D}_{66}}{C_{\mathrm{yz}}}$

$Q_3=\frac{2D_{11}{D}_{66}+2D_{11}{D}_{12}}{C_{\mathrm{xz}}^2}-\frac{{(D_{12}+D_{66})}^2+D_{66}^2+D_{11}{D}_{22}}{C_{\mathrm{xz}}{C}_{\mathrm{yz}}},Q_4=\frac{2D_{66}^2}{C_{\mathrm{xz}}^2}+\frac{D_{22}{D}_{66}}{C_{\mathrm{xz}}{C}_{\mathrm{yz}}}$

m wherein ₁, m ₂for mode ordinal number, a is sandwich plate length, and b is width, x, and y is denotation coordination respectively.

Further, the derivation of formula described in step 5) is as follows:

The middle face of sandwich is x-y plane, and under z is axial, the one side of z>0 is top panel, and the one side of z<0 is lower panel, and plate length is a, and width is b, and plate thickness is t, and sandwich thickness is h, and sandwich plate gross thickness is H=h+2t;

The free vibration governing equation group of introducing orthotropy clip rectangle laminate is:

$D_{11}\frac{{\&PartialD;}^2{\mathrm{\&psi;}}_x}{\&PartialD;x^2}+(D_{12}+D_{66})\frac{{\&PartialD;}^2{\mathrm{\&psi;}}_y}{\&PartialD;x\&PartialD;y}+D_{66}\frac{{\&PartialD;}^2{\mathrm{\&psi;}}_x}{\&PartialD;y^2}+C_{\mathrm{xz}}(\frac{\&PartialD;w}{\&PartialD;x}-{\mathrm{\&psi;}}_x)=0---\left(1\right)$

$(D_{12}+D_{66})\frac{{\&PartialD;}^2{\mathrm{\&psi;}}_x}{\&PartialD;x\&PartialD;y}+D_{66}\frac{{\&PartialD;}^2{\mathrm{\&psi;}}_y}{\&PartialD;x^2}+D_{22}\frac{{\&PartialD;}^2{\mathrm{\&psi;}}_y}{\&PartialD;y^2}+C_{\mathrm{yz}}(\frac{\&PartialD;w}{\&PartialD;y}-{\mathrm{\&psi;}}_y)=0---\left(2\right)$

$C_{\mathrm{xz}}(\frac{{\&PartialD;}^{2}w}{\&PartialD;x^2}-\frac{\&PartialD;{\mathrm{\&psi;}}_x}{\&PartialD;x})+C_{\mathrm{yz}}(\frac{{\&PartialD;}^{2}w}{\&PartialD;y^2}-\frac{\&PartialD;{\mathrm{\&psi;}}_y}{\&PartialD;y})+\mathrm{\&rho;}\frac{{\&PartialD;}^{2}w}{\&PartialD;t^2}=0---\left(3\right)$

Wherein w is the amount of deflection of sandwich plate under action of lateral load, ψ _x, ψ _yfor before sandwich plate distortion perpendicular to the straight-line segment of middle the corner in x-z and y-z plane;

System of equations Chinese style (1) is asked to local derviation to y, and formula (2) is asked local derviation to x, and then two formulas are subtracted each other, and can obtain

$\frac{D_{11}}{C_{\mathrm{xz}}}\frac{{\&PartialD;}^3{\mathrm{\&psi;}}_x}{\&PartialD;x^2\&PartialD;y}+\frac{D_{66}}{C_{\mathrm{xz}}}\frac{{\&PartialD;}^3{\mathrm{\&psi;}}_x}{\&PartialD;y^3}-\frac{D_{12}+D_{66}}{C_{\mathrm{yz}}}\frac{{\&PartialD;}^3{\mathrm{\&psi;}}_x}{{\&PartialD;}^{2}x\&PartialD;y}-\frac{\&PartialD;{\mathrm{\&psi;}}_x}{\&PartialD;y}+\frac{D_{12}+D_{66}}{C_{\mathrm{xz}}}\frac{{\&PartialD;}^3{\mathrm{\&psi;}}_y}{\&PartialD;x\&PartialD;y^2}-\frac{D_{66}}{C_{\mathrm{yz}}}\frac{{\&PartialD;}^3{\mathrm{\&psi;}}_y}{\&PartialD;x^3}-\frac{D_{22}}{C_{\mathrm{yz}}}\frac{{\&PartialD;}^3{\mathrm{\&psi;}}_y}{\&PartialD;x\&PartialD;y^2}+\frac{\&PartialD;{\mathrm{\&psi;}}_y}{\&PartialD;x}=0---\left(4\right)$

Introduce partial differential operator, formula (4) can be rewritten as

$L_1\frac{\&PartialD;{\mathrm{\&psi;}}_x}{\&PartialD;y}=L_2{\mathrm{\&psi;}}_y---\left(5\right)$

Wherein L1, L2 are partial differential operator:

$L_1=-\frac{D_{11}}{C_{\mathrm{xz}}}\frac{{\&PartialD;}^2}{\&PartialD;x^2}-\frac{D_{66}}{C_{\mathrm{xz}}}\frac{{\&PartialD;}^2}{\&PartialD;y^2}+\frac{D_{12}+D_{66}}{C_{\mathrm{yz}}}\frac{{\&PartialD;}^2}{\&PartialD;x^2}+1---\left(6\right)$

$L_2=\frac{D_{12}+D_{66}}{C_{\mathrm{xz}}}\frac{{\&PartialD;}^3}{\&PartialD;x\&PartialD;y^2}-\frac{D_{66}}{C_{\mathrm{yz}}}\frac{{\&PartialD;}^3}{\&PartialD;x^3}-\frac{D_{22}}{C_{\mathrm{yz}}}\frac{{\&PartialD;}^3}{\&PartialD;x\&PartialD;y^2}+\frac{\&PartialD;}{\&PartialD;x}---\left(7\right)$

Introduce displacement function χ, and order

ψ _x＝L ₂χ (8)

By formula (8) substitution formula (5), obtain

$L_2{\mathrm{\&psi;}}_y=L_1\frac{\&PartialD;}{\&PartialD;y}\left(L_2\mathrm{\&chi;}\right)=L_1{L}_2\frac{\&PartialD;\mathrm{\&chi;}}{\&PartialD;y}---\left(9\right)$

By (9) formula, obtained

${\mathrm{\&psi;}}_y=L_1\frac{\&PartialD;\mathrm{\&chi;}}{\&PartialD;y}---\left(10\right)$

By in formula (6), (7) substitution formula (8), (10), generalized displacement ψ _xand ψ _yavailable displacement function χ is expressed as

${\mathrm{\&psi;}}_x=[1-D_{66}(\frac{1}{C_{\mathrm{yz}}}\frac{{\&PartialD;}^2}{\&PartialD;x^2}-\frac{1}{C_{\mathrm{xz}}}\frac{{\&PartialD;}^2}{\&PartialD;y^2})+(\frac{D_{12}}{C_{\mathrm{xz}}}-\frac{D_{22}}{C_{\mathrm{yz}}})\frac{{\&PartialD;}^2}{\&PartialD;y^2}]\frac{\&PartialD;\mathrm{\&chi;}}{\&PartialD;x}---\left(11\right)$

${\mathrm{\&psi;}}_y=[1+D_{66}(\frac{1}{C_{\mathrm{yz}}}\frac{{\&PartialD;}^2}{\&PartialD;x^2}-\frac{1}{C_{\mathrm{xz}}}\frac{{\&PartialD;}^2}{\&PartialD;y^2})+(\frac{D_{12}}{C_{\mathrm{yz}}}-\frac{D_{11}}{C_{\mathrm{xz}}})\frac{{\&PartialD;}^2}{\&PartialD;x^2}]\frac{\&PartialD;\mathrm{\&chi;}}{\&PartialD;y}---\left(12\right)$

By formula (11), (12) substitution formula (1), through abbreviation, obtain

$\begin{array}{c}\frac{\&PartialD;w}{\&PartialD;x}=[1-D_{66}(\frac{1}{C_{\mathrm{yz}}}\frac{{\&PartialD;}^2}{\&PartialD;x^2}-\frac{1}{C_{\mathrm{xz}}}\frac{{\&PartialD;}^2}{\&PartialD;y^2})+(\frac{D_{12}}{C_{\mathrm{xz}}}-\frac{D_{22}}{C_{\mathrm{yz}}})\frac{{\&PartialD;}^2}{\&PartialD;y^2}](1-\frac{D_{11}}{C_{\mathrm{xz}}}\frac{{\&PartialD;}^2}{\&PartialD;x^2}-\frac{D_{66}}{C_{\mathrm{xz}}}\frac{{\&PartialD;}^2}{\&PartialD;y^2})\frac{\&PartialD;\mathrm{\&chi;}}{\&PartialD;x}\\ -\frac{D_{12}+D_{66}}{C_{\mathrm{xz}}}[1+D_{66}(\frac{1}{C_{\mathrm{yz}}}\frac{{\&PartialD;}^2}{\&PartialD;x^2}-\frac{1}{C_{\mathrm{xz}}}\frac{{\&PartialD;}^2}{\&PartialD;y^2})+(\frac{D_{12}}{C_{\mathrm{yz}}}-\frac{D_{11}}{C_{\mathrm{xz}}})\frac{{\&PartialD;}^2}{\&PartialD;x^2}]\frac{{\&PartialD;}^3\mathrm{\&chi;}}{\&PartialD;x\&PartialD;y^2}\end{array}---\left(13\right)$

Indefinite integral is carried out to x in formula (13) both sides, and generalized displacement w is expressed as by displacement function χ

$\begin{array}{c}w=[1-D_{66}(\frac{1}{C_{\mathrm{yz}}}\frac{{\&PartialD;}^2}{\&PartialD;x^2}-\frac{1}{C_{\mathrm{xz}}}\frac{{\&PartialD;}^2}{\&PartialD;y^2})+(\frac{D_{12}}{C_{\mathrm{xz}}}-\frac{D_{22}}{C_{\mathrm{yz}}})\frac{{\&PartialD;}^2}{\&PartialD;y^2}](\mathrm{\&chi;}-\frac{D_{11}}{C_{\mathrm{xz}}}\frac{{\&PartialD;}^2\mathrm{\&chi;}}{\&PartialD;x^2}-\frac{D_{66}}{C_{\mathrm{xz}}}\frac{{\&PartialD;}^2\mathrm{\&chi;}}{\&PartialD;y^2})\\ -\frac{D_{12}+D_{66}}{C_{\mathrm{xz}}}[1+D_{66}(\frac{1}{C_{\mathrm{yz}}}\frac{{\&PartialD;}^2}{\&PartialD;x^2}-\frac{1}{C_{\mathrm{xz}}}\frac{{\&PartialD;}^2}{\&PartialD;y^2})+(\frac{D_{12}}{C_{\mathrm{yz}}}-\frac{D_{11}}{C_{\mathrm{xz}}})\frac{{\&PartialD;}^2}{\&PartialD;x^2}]\frac{{\&PartialD;}^2\mathrm{\&chi;}}{\&PartialD;y^2}\end{array}---\left(14\right)$

By formula (11), (12) and formula (14) substitution system of equations formula (3), through merging, abbreviation, obtain the vibration control equation about displacement function χ

$U_1\frac{{\&PartialD;}^2\mathrm{\&chi;}}{\&PartialD;x^2}{+U}_2\frac{{\&PartialD;}^4\mathrm{\&chi;}}{\&PartialD;x^2\&PartialD;y^2}+U_3\frac{{\&PartialD;}^2\mathrm{\&chi;}}{\&PartialD;y^2}+U_4\frac{{\&PartialD;}^4\mathrm{\&chi;}}{\&PartialD;y^4}+\mathrm{\&rho;}\frac{{\&PartialD;}^2}{\&PartialD;t^2}{U}_5=0---\left(15\right)$

Wherein

$U_1=-C_{\mathrm{xz}}[1-D_{66}(\frac{1}{C_{\mathrm{yz}}}\frac{{\&PartialD;}^2}{\&PartialD;x^2}-\frac{1}{C_{\mathrm{xz}}}\frac{{\&PartialD;}^2}{\&PartialD;y^2})+(\frac{D_{12}}{C_{\mathrm{xz}}}-\frac{D_{22}}{C_{\mathrm{yz}}})\frac{{\&PartialD;}^2}{\&PartialD;y^2}](\frac{D_{11}}{C_{\mathrm{xz}}}\frac{{\&PartialD;}^2}{\&PartialD;x^2}+\frac{D_{66}}{C_{\mathrm{xz}}}\frac{{\&PartialD;}^2}{\&PartialD;y^2})---\left(16\right)$

$U_2=-(D_{12}+D_{66})[1+D_{66}(\frac{1}{C_{\mathrm{yz}}}\frac{{\&PartialD;}^2}{\&PartialD;x^2}-\frac{1}{C_{\mathrm{xz}}}\frac{{\&PartialD;}^2}{\&PartialD;y^2})+(\frac{D_{12}}{C_{\mathrm{yz}}}-\frac{D_{11}}{C_{\mathrm{xz}}})\frac{{\&PartialD;}^2}{\&PartialD;x^2}]---\left(17\right)$

$\begin{array}{c}{U}_3=C_{\mathrm{yz}}[1-D_{66}(\frac{1}{C_{\mathrm{yz}}}\frac{{\&PartialD;}^2}{\&PartialD;x^2}-\frac{1}{C_{\mathrm{xz}}}\frac{{\&PartialD;}^2}{\&PartialD;y^2})+(\frac{D_{12}}{C_{\mathrm{xz}}}-\frac{D_{22}}{C_{\mathrm{yz}}})\frac{{\&PartialD;}^2}{\&PartialD;y^2}](1-\frac{D_{11}}{C_{\mathrm{xz}}}\frac{{\&PartialD;}^2}{\&PartialD;x^2}-\frac{D_{66}}{C_{\mathrm{xz}}}\frac{{\&PartialD;}^2}{\&PartialD;y^2})\\ -C_{\mathrm{yz}}[1+D_{66}(\frac{1}{C_{\mathrm{yz}}}\frac{{\&PartialD;}^2}{\&PartialD;x^2}-\frac{1}{C_{\mathrm{xz}}}\frac{{\&PartialD;}^2}{\&PartialD;y^2})+(\frac{D_{12}}{C_{\mathrm{yz}}}-\frac{D_{11}}{C_{\mathrm{xz}}})\frac{{\&PartialD;}^2}{\&PartialD;x^2}]\end{array}---\left(18\right)$

$U_4=\frac{C_{\mathrm{yz}}}{C_{\mathrm{xz}}}(D_{12}+D_{66})[1+D_{66}(\frac{1}{C_{\mathrm{yz}}}\frac{{\&PartialD;}^2}{\&PartialD;x^2}-\frac{1}{C_{\mathrm{xz}}}\frac{{\&PartialD;}^2}{\&PartialD;y^2})+(\frac{D_{12}}{C_{\mathrm{yz}}}-\frac{D_{11}}{C_{\mathrm{xz}}})\frac{{\&PartialD;}^2}{\&PartialD;x^2}]---\left(19\right)$

$\begin{array}{c}{U}_5=[1-D_{66}(\frac{1}{C_{\mathrm{yz}}}\frac{{\&PartialD;}^2}{\&PartialD;x^2}-\frac{1}{C_{\mathrm{xz}}}\frac{{\&PartialD;}^2}{\&PartialD;y^2})+(\frac{D_{12}}{C_{\mathrm{xz}}}-\frac{D_{22}}{C_{\mathrm{yz}}})\frac{{\&PartialD;}^2}{\&PartialD;y^2}](\mathrm{\&chi;}-\frac{D_{11}}{C_{\mathrm{xz}}}\frac{{\&PartialD;}^2\mathrm{\&chi;}}{\&PartialD;x^2}-\frac{D_{66}}{C_{\mathrm{xz}}}\frac{{\&PartialD;}^2\mathrm{\&chi;}}{\&PartialD;y^2})\\ -\frac{D_{12}+D_{66}}{C_{\mathrm{xz}}}[1+D_{66}(\frac{1}{C_{\mathrm{yz}}}\frac{{\&PartialD;}^2}{\&PartialD;x^2}-\frac{1}{C_{\mathrm{xz}}}\frac{{\&PartialD;}^2}{\&PartialD;y^2})+(\frac{D_{12}}{C_{\mathrm{yz}}}-\frac{D_{11}}{C_{\mathrm{xz}}})\frac{{\&PartialD;}^2}{\&PartialD;x^2}]\frac{{\&PartialD;}^2\mathrm{\&chi;}}{\&PartialD;y^2}\end{array}---\left(20\right)$

To formula (16)-(20) merge, abbreviation, and substitution formula (15), just can obtain only containing the Orthotropic Sandwich Plate free vibration governing equation of displacement function χ

$\begin{array}{c}\frac{D_{11}+D_{66}}{C_{\mathrm{yz}}}\frac{{\&PartialD;}^6\mathrm{\&chi;}}{\&PartialD;x^6}+Q_1\frac{{\&PartialD;}^6\mathrm{\&chi;}}{\&PartialD;x^4\&PartialD;y^2}+Q_2\frac{{\&PartialD;}^6\mathrm{\&chi;}}{\&PartialD;x^2{\&PartialD;y}^4}-2(D_{12}+{2D}_{66})\frac{{\&PartialD;}^4\mathrm{\&chi;}}{\&PartialD;x^2{\&PartialD;y}^2}-D_{11}\frac{{\&PartialD;}^4\mathrm{\&chi;}}{\&PartialD;x^4}-D_{22}\frac{{\&PartialD;}^4\mathrm{\&chi;}}{\&PartialD;y^4}+\frac{D_{22}{D}_{66}}{C_{\mathrm{xz}}}\frac{{\&PartialD;}^6\mathrm{\&chi;}}{\&PartialD;y^6}\\ +\mathrm{\&rho;}\frac{{\&PartialD;}^2}{\&PartialD;t^2}\{\mathrm{\&chi;}-(\frac{D_{11}}{C_{\mathrm{xz}}}+\frac{D_{66}}{C_{\mathrm{yz}}})\frac{{\&PartialD;}^2\mathrm{\&chi;}}{\&PartialD;x^2}-\frac{D_{11}{D}_{66}}{C_{\mathrm{xz}}{C}_{\mathrm{yz}}}\frac{{\&PartialD;}^4\mathrm{\&chi;}}{\&PartialD;x^4}-(\frac{D_{22}}{C_{\mathrm{yz}}}+\frac{D_{66}}{C_{\mathrm{xz}}})\frac{{\&PartialD;}^2\mathrm{\&chi;}}{\&PartialD;y^2}+Q_3\frac{{\&PartialD;}^4\mathrm{\&chi;}}{\&PartialD;x^2\&PartialD;y^2}+Q_4\frac{{\&PartialD;}^4\mathrm{\&chi;}}{\&PartialD;y^4}\}=0\end{array}---\left(21\right)$

Wherein

$Q_1=\frac{D_{11}{D}_{66}}{C_{\mathrm{xz}}}+\frac{D_{11}{D}_{22}-2D_{12}{D}_{66}-D_{12}^2}{C_{\mathrm{yz}}},Q_2=\frac{D_{11}{D}_{22}-D_{12}^2}{C_{\mathrm{xz}}}+\frac{D_{22}{D}_{66}}{C_{\mathrm{yz}}}$

$Q_3=\frac{2D_{11}{D}_{66}+2D_{11}{D}_{12}}{C_{\mathrm{xz}}^2}-\frac{{(D_{12}+D_{66})}^2+D_{66}^2+D_{11}{D}_{22}}{C_{\mathrm{xz}}{C}_{\mathrm{yz}}},Q_4=\frac{2D_{66}^2}{C_{\mathrm{xz}}^2}+\frac{D_{22}{D}_{66}}{C_{\mathrm{xz}}{C}_{\mathrm{yz}}}.$

Owing to having adopted technique scheme, the present invention has advantages of as follows:

The present invention carries out simply sandwich plate flexural vibrations system of equations form and natural frequency solution procedure, and consider sandwich layer transverse shear deformation and the impact of sandwich plate whole-body quadrature anisotropy on sandwich plate mechanical characteristic simultaneously, simplify the computation process of natural frequency, improve counting yield, reduce difficulty in computation, and improve the accuracy in computation of natural frequency.

Other advantages of the present invention, target and feature will be set forth to a certain extent in the following description, and to a certain extent, based on will be apparent to those skilled in the art to investigating below, or can be instructed from the practice of the present invention.Target of the present invention and other advantages can be realized and be obtained by instructions and claims below.

Accompanying drawing explanation

Accompanying drawing of the present invention is described as follows.

Fig. 1 is the structural representation of orthotropy honeycomb sandwich panel;

The comparison diagram of Fig. 2 sandwich plate calculation on Natural Frequency result and this method computation structure when not considering sandwich layer transverse shear deformation;

The comparison diagram of Fig. 3 sandwich plate calculation on Natural Frequency result and this method computation structure when not considering orthotropy.

Embodiment

Below in conjunction with drawings and Examples, the invention will be further described.

A measuring method for honeycomb sandwich panel natural frequency, concrete steps are as follows:

1) use compound substance universal testing machine to measure sandwich plate to be measured, obtain sandwich plate bending stiffness D ₁₁, D ₁₂, D ₂₂, D ₆₆with the shear modulus G of sandwich layer in x-z and y-z plane _xz, G _yz;

2) utilize ruler measurement measured panel sandwich plate length a, width b, thickness t and core layer thickness h;

3) by formula C _xz=G _xzh (1+t/h) ²and C _yz=G _yzh (1+t/h) ², calculate the shearing rigidity C of sandwich layer _xzand C _yz;

4) by material handbook, check in the panel density p of clamping plate to be measured _fwith sandwich layer density p _c, by formula ρ=ρ _ch+2 ρ _ft calculates the equivalent face density p of clamping plate;

5) data substitution formula above-mentioned steps being recorded calculates, and in conjunction with the boundary condition of sandwich plate, solves the natural frequency ω on sandwich plate m rank _m, its formula is as follows:

$\begin{array}{c}\frac{D_{11}+D_{66}}{C_{\mathrm{yz}}}\frac{{\&PartialD;}^6\mathrm{\&chi;}}{\&PartialD;x^6}+Q_1\frac{{\&PartialD;}^6\mathrm{\&chi;}}{\&PartialD;x^4\&PartialD;y^2}+Q_2\frac{{\&PartialD;}^6\mathrm{\&chi;}}{\&PartialD;x^2{\&PartialD;y}^4}-2(D_{12}+{2D}_{66})\frac{{\&PartialD;}^4\mathrm{\&chi;}}{\&PartialD;x^2{\&PartialD;y}^2}-D_{11}\frac{{\&PartialD;}^4\mathrm{\&chi;}}{\&PartialD;x^4}-D_{22}\frac{{\&PartialD;}^4\mathrm{\&chi;}}{\&PartialD;y^4}+\frac{D_{22}{D}_{66}}{C_{\mathrm{xz}}}\frac{{\&PartialD;}^6\mathrm{\&chi;}}{\&PartialD;{\mathrm{<figure-callout\; id="6"\; label="y"\; filenames="120215154407.png"\; state="\{\{state\}\}">y</figure-callout>}}^6}\\ +\mathrm{\&rho;}\frac{{\&PartialD;}^2}{\&PartialD;t^2}\{\mathrm{\&chi;}-(\frac{D_{11}}{C_{\mathrm{xz}}}+\frac{D_{66}}{C_{\mathrm{yz}}})\frac{{\&PartialD;}^2\mathrm{\&chi;}}{\&PartialD;x^2}-\frac{D_{11}{D}_{66}}{C_{\mathrm{xz}}{C}_{\mathrm{yz}}}\frac{{\&PartialD;}^4\mathrm{\&chi;}}{\&PartialD;x^4}-(\frac{D_{22}}{C_{\mathrm{yz}}}+\frac{D_{66}}{C_{\mathrm{xz}}})\frac{{\&PartialD;}^2\mathrm{\&chi;}}{\&PartialD;y^2}+Q_3\frac{{\&PartialD;}^4\mathrm{\&chi;}}{\&PartialD;x^2\&PartialD;y^2}+Q_4\frac{{\&PartialD;}^4\mathrm{\&chi;}}{\&PartialD;y^4}\}=0\end{array}$

In formula

$Q_1=\frac{D_{11}{D}_{66}}{C_{\mathrm{xz}}}+\frac{D_{11}{D}_{22}-2D_{12}{D}_{66}-D_{12}^2}{C_{\mathrm{yz}}},Q_2=\frac{D_{11}{D}_{22}-D_{12}^2}{C_{\mathrm{xz}}}+\frac{D_{22}{D}_{66}}{C_{\mathrm{yz}}}$

$Q_3=\frac{2D_{11}{D}_{66}+2D_{11}{D}_{12}}{C_{\mathrm{xz}}^2}-\frac{{(D_{12}+D_{66})}^2+D_{66}^2+D_{11}{D}_{22}}{C_{\mathrm{xz}}{C}_{\mathrm{yz}}},Q_4=\frac{2D_{66}^2}{C_{\mathrm{xz}}^2}+\frac{D_{22}{D}_{66}}{C_{\mathrm{xz}}{C}_{\mathrm{yz}}}$

m wherein ₁, m ₂for mode ordinal number, a is sandwich plate length, and b is width, x, and y is denotation coordination respectively.

The derivation of formula described in step 5) is as follows:

The middle face of sandwich is x-y plane, and under z is axial, the one side of z>0 is top panel, and the one side of z<0 is lower panel, and plate length is a, and width is b, and plate thickness is t, and sandwich thickness is h, and sandwich plate gross thickness is H=h+2t;

The free vibration governing equation group of introducing orthotropy clip rectangle laminate is:

$D_{11}\frac{{\&PartialD;}^2{\mathrm{\&psi;}}_x}{\&PartialD;x^2}+(D_{12}+D_{66})\frac{{\&PartialD;}^2{\mathrm{\&psi;}}_y}{\&PartialD;x\&PartialD;y}+D_{66}\frac{{\&PartialD;}^2{\mathrm{\&psi;}}_x}{\&PartialD;y^2}+C_{\mathrm{xz}}(\frac{\&PartialD;w}{\&PartialD;x}-{\mathrm{\&psi;}}_x)=0---\left(1\right)$

$(D_{12}+D_{66})\frac{{\&PartialD;}^2{\mathrm{\&psi;}}_x}{\&PartialD;x\&PartialD;y}+D_{66}\frac{{\&PartialD;}^2{\mathrm{\&psi;}}_y}{\&PartialD;x^2}+D_{22}\frac{{\&PartialD;}^2{\mathrm{\&psi;}}_y}{\&PartialD;y^2}+C_{\mathrm{yz}}(\frac{\&PartialD;w}{\&PartialD;y}-{\mathrm{\&psi;}}_y)=0---\left(2\right)$

$C_{\mathrm{xz}}(\frac{{\&PartialD;}^{2}w}{\&PartialD;x^2}-\frac{\&PartialD;{\mathrm{\&psi;}}_x}{\&PartialD;x})+C_{\mathrm{yz}}(\frac{{\&PartialD;}^{2}w}{\&PartialD;y^2}-\frac{\&PartialD;{\mathrm{\&psi;}}_y}{\&PartialD;y})+\mathrm{\&rho;}\frac{{\&PartialD;}^{2}w}{\&PartialD;t^2}=0---\left(3\right)$

Wherein w is the amount of deflection of sandwich plate under action of lateral load, ψ _x, ψ _yfor before sandwich plate distortion perpendicular to the straight-line segment of middle the corner in x-z and y-z plane;

System of equations Chinese style (1) is asked to local derviation to y, and formula (2) is asked local derviation to x, and then two formulas are subtracted each other, and can obtain

$\frac{D_{11}}{C_{\mathrm{xz}}}\frac{{\&PartialD;}^3{\mathrm{\&psi;}}_x}{\&PartialD;x^2\&PartialD;y}+\frac{D_{66}}{C_{\mathrm{xz}}}\frac{{\&PartialD;}^3{\mathrm{\&psi;}}_x}{\&PartialD;y^3}-\frac{D_{12}+D_{66}}{C_{\mathrm{yz}}}\frac{{\&PartialD;}^3{\mathrm{\&psi;}}_x}{{\&PartialD;}^{2}x\&PartialD;y}-\frac{\&PartialD;{\mathrm{\&psi;}}_x}{\&PartialD;y}+\frac{D_{12}+D_{66}}{C_{\mathrm{xz}}}\frac{{\&PartialD;}^3{\mathrm{\&psi;}}_y}{\&PartialD;x\&PartialD;y^2}-\frac{D_{66}}{C_{\mathrm{yz}}}\frac{{\&PartialD;}^3{\mathrm{\&psi;}}_y}{\&PartialD;x^3}-\frac{D_{22}}{C_{\mathrm{yz}}}\frac{{\&PartialD;}^3{\mathrm{\&psi;}}_y}{\&PartialD;x\&PartialD;y^2}+\frac{\&PartialD;{\mathrm{\&psi;}}_y}{\&PartialD;x}=0---\left(4\right)$

Introduce partial differential operator, formula (4) can be rewritten as

$L_1\frac{\&PartialD;{\mathrm{\&psi;}}_x}{\&PartialD;y}=L_2{\mathrm{\&psi;}}_y---\left(5\right)$

Wherein L1, L2 are partial differential operator:

$L_1=-\frac{D_{11}}{C_{\mathrm{xz}}}\frac{{\&PartialD;}^2}{\&PartialD;x^2}-\frac{D_{66}}{C_{\mathrm{xz}}}\frac{{\&PartialD;}^2}{\&PartialD;y^2}+\frac{D_{12}+D_{66}}{C_{\mathrm{yz}}}\frac{{\&PartialD;}^2}{\&PartialD;x^2}+1---\left(6\right)$

$L_2=\frac{D_{12}+D_{66}}{C_{\mathrm{xz}}}\frac{{\&PartialD;}^3}{\&PartialD;x\&PartialD;y^2}-\frac{D_{66}}{C_{\mathrm{yz}}}\frac{{\&PartialD;}^3}{\&PartialD;x^3}-\frac{D_{22}}{C_{\mathrm{yz}}}\frac{{\&PartialD;}^3}{\&PartialD;x\&PartialD;y^2}+\frac{\&PartialD;}{\&PartialD;x}---\left(7\right)$

Introduce displacement function χ, and order

ψ _x＝L ₂χ (8)

By formula (8) substitution formula (5), obtain

$L_2{\mathrm{\&psi;}}_y=L_1\frac{\&PartialD;}{\&PartialD;y}\left(L_2\mathrm{\&chi;}\right)=L_1{L}_2\frac{\&PartialD;\mathrm{\&chi;}}{\&PartialD;y}---\left(9\right)$

By (9) formula, obtained

${\mathrm{\&psi;}}_y=L_1\frac{\&PartialD;\mathrm{\&chi;}}{\&PartialD;y}---\left(10\right)$

By in formula (6), (7) substitution formula (8), (10), generalized displacement ψ _xand ψ _yavailable displacement function χ is expressed as

${\mathrm{\&psi;}}_x=[1-D_{66}(\frac{1}{C_{\mathrm{yz}}}\frac{{\&PartialD;}^2}{\&PartialD;x^2}-\frac{1}{C_{\mathrm{xz}}}\frac{{\&PartialD;}^2}{\&PartialD;y^2})+(\frac{D_{12}}{C_{\mathrm{xz}}}-\frac{D_{22}}{C_{\mathrm{yz}}})\frac{{\&PartialD;}^2}{\&PartialD;y^2}]\frac{\&PartialD;\mathrm{\&chi;}}{\&PartialD;x}---\left(11\right)$

${\mathrm{\&psi;}}_y=[1+D_{66}(\frac{1}{C_{\mathrm{yz}}}\frac{{\&PartialD;}^2}{\&PartialD;x^2}-\frac{1}{C_{\mathrm{xz}}}\frac{{\&PartialD;}^2}{\&PartialD;y^2})+(\frac{D_{12}}{C_{\mathrm{yz}}}-\frac{D_{11}}{C_{\mathrm{xz}}})\frac{{\&PartialD;}^2}{\&PartialD;x^2}]\frac{\&PartialD;\mathrm{\&chi;}}{\&PartialD;y}---\left(12\right)$

By formula (11), (12) substitution formula (1), through abbreviation, obtain

$\begin{array}{c}\frac{\&PartialD;w}{\&PartialD;x}=[1-D_{66}(\frac{1}{C_{\mathrm{yz}}}\frac{{\&PartialD;}^2}{\&PartialD;x^2}-\frac{1}{C_{\mathrm{xz}}}\frac{{\&PartialD;}^2}{\&PartialD;y^2})+(\frac{D_{12}}{C_{\mathrm{xz}}}-\frac{D_{22}}{C_{\mathrm{yz}}})\frac{{\&PartialD;}^2}{\&PartialD;y^2}](1-\frac{D_{11}}{C_{\mathrm{xz}}}\frac{{\&PartialD;}^2}{\&PartialD;x^2}-\frac{D_{66}}{C_{\mathrm{xz}}}\frac{{\&PartialD;}^2}{\&PartialD;y^2})\frac{\&PartialD;\mathrm{\&chi;}}{\&PartialD;x}\\ -\frac{D_{12}+D_{66}}{C_{\mathrm{xz}}}[1+D_{66}(\frac{1}{C_{\mathrm{yz}}}\frac{{\&PartialD;}^2}{\&PartialD;x^2}-\frac{1}{C_{\mathrm{xz}}}\frac{{\&PartialD;}^2}{\&PartialD;y^2})+(\frac{D_{12}}{C_{\mathrm{yz}}}-\frac{D_{11}}{C_{\mathrm{xz}}})\frac{{\&PartialD;}^2}{\&PartialD;x^2}]\frac{{\&PartialD;}^3\mathrm{\&chi;}}{\&PartialD;x\&PartialD;y^2}\end{array}---\left(13\right)$

Indefinite integral is carried out to x in formula (13) both sides, and generalized displacement w is expressed as by displacement function χ

$\begin{array}{c}w=[1-D_{66}(\frac{1}{C_{\mathrm{yz}}}\frac{{\&PartialD;}^2}{\&PartialD;x^2}-\frac{1}{C_{\mathrm{xz}}}\frac{{\&PartialD;}^2}{\&PartialD;y^2})+(\frac{D_{12}}{C_{\mathrm{xz}}}-\frac{D_{22}}{C_{\mathrm{yz}}})\frac{{\&PartialD;}^2}{\&PartialD;y^2}](\mathrm{\&chi;}-\frac{D_{11}}{C_{\mathrm{xz}}}\frac{{\&PartialD;}^2\mathrm{\&chi;}}{\&PartialD;x^2}-\frac{D_{66}}{C_{\mathrm{xz}}}\frac{{\&PartialD;}^2\mathrm{\&chi;}}{\&PartialD;y^2})\\ -\frac{D_{12}+D_{66}}{C_{\mathrm{xz}}}[1+D_{66}(\frac{1}{C_{\mathrm{yz}}}\frac{{\&PartialD;}^2}{\&PartialD;x^2}-\frac{1}{C_{\mathrm{xz}}}\frac{{\&PartialD;}^2}{\&PartialD;y^2})+(\frac{D_{12}}{C_{\mathrm{yz}}}-\frac{D_{11}}{C_{\mathrm{xz}}})\frac{{\&PartialD;}^2}{\&PartialD;x^2}]\frac{{\&PartialD;}^2\mathrm{\&chi;}}{\&PartialD;y^2}\end{array}---\left(14\right)$

By formula (11), (12) and formula (14) substitution system of equations formula (3), through merging, abbreviation, obtain the vibration control equation about displacement function χ

$U_1\frac{{\&PartialD;}^2\mathrm{\&chi;}}{\&PartialD;x^2}{+U}_2\frac{{\&PartialD;}^4\mathrm{\&chi;}}{\&PartialD;x^2\&PartialD;y^2}+U_3\frac{{\&PartialD;}^2\mathrm{\&chi;}}{\&PartialD;y^2}+U_4\frac{{\&PartialD;}^4\mathrm{\&chi;}}{\&PartialD;y^4}+\mathrm{\&rho;}\frac{{\&PartialD;}^2}{\&PartialD;t^2}{U}_5=0---\left(15\right)$

Wherein

$U_1=-C_{\mathrm{xz}}[1-D_{66}(\frac{1}{C_{\mathrm{yz}}}\frac{{\&PartialD;}^2}{\&PartialD;x^2}-\frac{1}{C_{\mathrm{xz}}}\frac{{\&PartialD;}^2}{\&PartialD;y^2})+(\frac{D_{12}}{C_{\mathrm{xz}}}-\frac{D_{22}}{C_{\mathrm{yz}}})\frac{{\&PartialD;}^2}{\&PartialD;y^2}](\frac{D_{11}}{C_{\mathrm{xz}}}\frac{{\&PartialD;}^2}{\&PartialD;x^2}+\frac{D_{66}}{C_{\mathrm{xz}}}\frac{{\&PartialD;}^2}{\&PartialD;y^2})---\left(16\right)$

$U_2=-(D_{12}+D_{66})[1+D_{66}(\frac{1}{C_{\mathrm{yz}}}\frac{{\&PartialD;}^2}{\&PartialD;x^2}-\frac{1}{C_{\mathrm{xz}}}\frac{{\&PartialD;}^2}{\&PartialD;y^2})+(\frac{D_{12}}{C_{\mathrm{yz}}}-\frac{D_{11}}{C_{\mathrm{xz}}})\frac{{\&PartialD;}^2}{\&PartialD;x^2}]---\left(17\right)$

$\begin{array}{c}{U}_3=C_{\mathrm{yz}}[1-D_{66}(\frac{1}{C_{\mathrm{yz}}}\frac{{\&PartialD;}^2}{\&PartialD;x^2}-\frac{1}{C_{\mathrm{xz}}}\frac{{\&PartialD;}^2}{\&PartialD;y^2})+(\frac{D_{12}}{C_{\mathrm{xz}}}-\frac{D_{22}}{C_{\mathrm{yz}}})\frac{{\&PartialD;}^2}{\&PartialD;y^2}](1-\frac{D_{11}}{C_{\mathrm{xz}}}\frac{{\&PartialD;}^2}{\&PartialD;x^2}-\frac{D_{66}}{C_{\mathrm{xz}}}\frac{{\&PartialD;}^2}{\&PartialD;y^2})\\ -C_{\mathrm{yz}}[1+D_{66}(\frac{1}{C_{\mathrm{yz}}}\frac{{\&PartialD;}^2}{\&PartialD;x^2}-\frac{1}{C_{\mathrm{xz}}}\frac{{\&PartialD;}^2}{\&PartialD;y^2})+(\frac{D_{12}}{C_{\mathrm{yz}}}-\frac{D_{11}}{C_{\mathrm{xz}}})\frac{{\&PartialD;}^2}{\&PartialD;x^2}]\end{array}---\left(18\right)$

$U_4=\frac{C_{\mathrm{yz}}}{C_{\mathrm{xz}}}(D_{12}+D_{66})[1+D_{66}(\frac{1}{C_{\mathrm{yz}}}\frac{{\&PartialD;}^2}{\&PartialD;x^2}-\frac{1}{C_{\mathrm{xz}}}\frac{{\&PartialD;}^2}{\&PartialD;y^2})+(\frac{D_{12}}{C_{\mathrm{yz}}}-\frac{D_{11}}{C_{\mathrm{xz}}})\frac{{\&PartialD;}^2}{\&PartialD;x^2}]---\left(19\right)$

$\begin{array}{c}{U}_5=[1-D_{66}(\frac{1}{C_{\mathrm{yz}}}\frac{{\&PartialD;}^2}{\&PartialD;x^2}-\frac{1}{C_{\mathrm{xz}}}\frac{{\&PartialD;}^2}{\&PartialD;y^2})+(\frac{D_{12}}{C_{\mathrm{xz}}}-\frac{D_{22}}{C_{\mathrm{yz}}})\frac{{\&PartialD;}^2}{\&PartialD;y^2}](\mathrm{\&chi;}-\frac{D_{11}}{C_{\mathrm{xz}}}\frac{{\&PartialD;}^2\mathrm{\&chi;}}{\&PartialD;x^2}-\frac{D_{66}}{C_{\mathrm{xz}}}\frac{{\&PartialD;}^2\mathrm{\&chi;}}{\&PartialD;y^2})\\ -\frac{D_{12}+D_{66}}{C_{\mathrm{xz}}}[1+D_{66}(\frac{1}{C_{\mathrm{yz}}}\frac{{\&PartialD;}^2}{\&PartialD;x^2}-\frac{1}{C_{\mathrm{xz}}}\frac{{\&PartialD;}^2}{\&PartialD;y^2})+(\frac{D_{12}}{C_{\mathrm{yz}}}-\frac{D_{11}}{C_{\mathrm{xz}}})\frac{{\&PartialD;}^2}{\&PartialD;x^2}]\frac{{\&PartialD;}^2\mathrm{\&chi;}}{\&PartialD;y^2}\end{array}---\left(20\right)$

To formula (16)-(20) merge, abbreviation, and substitution formula (15), just can obtain only containing the Orthotropic Sandwich Plate free vibration governing equation of displacement function χ

$\begin{array}{c}\frac{D_{11}+D_{66}}{C_{\mathrm{yz}}}\frac{{\&PartialD;}^6\mathrm{\&chi;}}{\&PartialD;x^6}+Q_1\frac{{\&PartialD;}^6\mathrm{\&chi;}}{\&PartialD;x^4\&PartialD;y^2}+Q_2\frac{{\&PartialD;}^6\mathrm{\&chi;}}{\&PartialD;x^2{\&PartialD;y}^4}-2(D_{12}+{2D}_{66})\frac{{\&PartialD;}^4\mathrm{\&chi;}}{\&PartialD;x^2{\&PartialD;y}^2}-D_{11}\frac{{\&PartialD;}^4\mathrm{\&chi;}}{\&PartialD;x^4}-D_{22}\frac{{\&PartialD;}^4\mathrm{\&chi;}}{\&PartialD;y^4}+\frac{D_{22}{D}_{66}}{C_{\mathrm{xz}}}\frac{{\&PartialD;}^6\mathrm{\&chi;}}{\&PartialD;{\mathrm{<figure-callout\; id="6"\; label="y"\; filenames="120215154407.png"\; state="\{\{state\}\}">y</figure-callout>}}^6}\\ +\mathrm{\&rho;}\frac{{\&PartialD;}^2}{\&PartialD;t^2}\{\mathrm{\&chi;}-(\frac{D_{11}}{C_{\mathrm{xz}}}+\frac{D_{66}}{C_{\mathrm{yz}}})\frac{{\&PartialD;}^2\mathrm{\&chi;}}{\&PartialD;x^2}-\frac{D_{11}{D}_{66}}{C_{\mathrm{xz}}{C}_{\mathrm{yz}}}\frac{{\&PartialD;}^4\mathrm{\&chi;}}{\&PartialD;x^4}-(\frac{D_{22}}{C_{\mathrm{yz}}}+\frac{D_{66}}{C_{\mathrm{xz}}})\frac{{\&PartialD;}^2\mathrm{\&chi;}}{\&PartialD;y^2}+Q_3\frac{{\&PartialD;}^4\mathrm{\&chi;}}{\&PartialD;x^2\&PartialD;y^2}+Q_4\frac{{\&PartialD;}^4\mathrm{\&chi;}}{\&PartialD;y^4}\}=0\end{array}---\left(21\right)$

Wherein

$Q_1=\frac{D_{11}{D}_{66}}{C_{\mathrm{xz}}}+\frac{D_{11}{D}_{22}-2D_{12}{D}_{66}-D_{12}^2}{C_{\mathrm{yz}}},Q_2=\frac{D_{11}{D}_{22}-D_{12}^2}{C_{\mathrm{xz}}}+\frac{D_{22}{D}_{66}}{C_{\mathrm{yz}}}$

$Q_3=\frac{2D_{11}{D}_{66}+2D_{11}{D}_{12}}{C_{\mathrm{xz}}^2}-\frac{{(D_{12}+D_{66})}^2+D_{66}^2+D_{11}{D}_{22}}{C_{\mathrm{xz}}{C}_{\mathrm{yz}}},Q_4=\frac{2D_{66}^2}{C_{\mathrm{xz}}^2}+\frac{D_{22}{D}_{66}}{C_{\mathrm{xz}}{C}_{\mathrm{yz}}}.$

The example that is calculated as with simply supported on four sides clip rectangle laminate natural frequency:

The clip rectangle laminate that is simply supported on four sides for boundary condition, its boundary condition is that amount of deflection is 0, moment of flexure is 0,

At x=0, a place

At y=0, b place

Utilize formula (11), (12) and (14), boundary condition can be rewritten as:

At x=0, a place $\mathrm{\&chi;}=0,\frac{{\&PartialD;}^2\mathrm{\&chi;}}{\&PartialD;x^2}=0,\frac{{\&PartialD;}^4\mathrm{\&chi;}}{\&PartialD;x^4}=0---\left(24\right)$

At y=0, b place $\mathrm{\&chi;}=0,\frac{{\&PartialD;}^2\mathrm{\&chi;}}{\&PartialD;y^2}=0,\frac{{\&PartialD;}^4\mathrm{\&chi;}}{\&PartialD;y^4}=0---\left(25\right)$

For the sandwich plate of simply supported on four sides, the solution of formula (21) has following form:

$\mathrm{\&chi;}=A\sin k_{m_1}x\sin k_{m_2}y{e}^{j{\mathrm{\&omega;}}_{m}t}---\left(26\right)$

ω wherein _mthe m rank natural frequency that represents sandwich plate,

m ₁and m ₂for mode ordinal number.

By formula (26) substitution formula (21), obtain natural frequency equation

$\begin{array}{c}\frac{D_{11}+D_{66}}{C_{\mathrm{yz}}}{k}_{m1}^6+Q_1{k}_{m1}^4{k}_{m2}^2+Q_2{k}_{m1}^2{k}_{m2}^4+2(D_{12}+{2D}_{66})k_{m1}^2{k}_{m2}^2+D_{11}{k}_{m1}^4+D_{22}{k}_{m2}^4+\frac{D_{22}{D}_{66}}{C_{\mathrm{xz}}}{k}_{m1}^6\\ -\mathrm{\&rho;}{\mathrm{\&omega;}}_m^2[1+(\frac{D_{11}}{C_{\mathrm{xz}}}+\frac{D_{66}}{C_{\mathrm{yz}}})k_{m1}^2-\frac{D_{11}{D}_{66}}{C_{\mathrm{xz}}{C}_{\mathrm{yz}}}{k}_{m1}^4+(\frac{D_{22}}{C_{\mathrm{yz}}}+\frac{D_{66}}{C_{\mathrm{xz}}})k_{m2}^2+Q_3{k}_{m1}^2{k}_{m2}^2+Q_4{k}_{m2}^4]=0\end{array}---\left(27\right)$

By formula (27), can be solved the natural frequency of simply supported on four sides Orthotropic Sandwich Plate

$\begin{array}{c}{\mathrm{\&omega;}}_m={[\frac{D_{11}+D_{66}}{C_{\mathrm{yz}}}{k}_{m1}^6+Q_1{k}_{m1}^4{k}_{m2}^2+Q_2{k}_{m1}^2{k}_{m2}^4+2(D_{12}+2D_{66})k_{m1}^2{k}_{m2}^2+D_{11}{k}_{m1}^4+D_{22}{k}_{m2}^4+\frac{D_{22}{D}_{66}}{C_{\mathrm{xz}}}{k}_{m1}^6]}^{\frac{1}{2}}\\ \&times;\frac{1}{\sqrt{\mathrm{\&rho;}}}{[1+(\frac{D_{11}}{C_{\mathrm{xz}}}+\frac{D_{66}}{C_{\mathrm{yz}}})k_{m1}^2-\frac{D_{11}{D}_{66}}{C_{\mathrm{xz}}{C}_{\mathrm{yz}}}{k}_{m1}^4+(\frac{D_{22}}{C_{\mathrm{yz}}}+\frac{D_{66}}{C_{\mathrm{xz}}}){k_{m2}^2+}^{}{Q}_3{k}_{m1}^2{k}_{m2}^2+Q_4{k}_{m2}^4]}^{-\frac{1}{2}}\end{array}---\left(28\right)$

Table 1 is sandwich plate parameter to be measured:

Table 2 is the parameter that measuring method of the present invention and traditional measuring method calculate:

In table, FEM is that finite element method, SFPM are that the finite point method, HSDPT are that Higher-order Shear is theoretical.

From table 2 data, relatively can find out, it is fine that the sandwich plate natural frequency calculating by the inventive method and experimental result are coincide, maximum error is no more than 1.8%, least error is only 0.32%, average error is only 1.18%, the computational accuracy of the Orthotropic Sandwich Plate bending vibration control equation that explanation obtains by the inventive method and the calculation on Natural Frequency formula being obtained by governing equation is obviously better than finite element method, quite theoretical with the Higher-order Shear that computational accuracy is very high, therefore can think that the sandwich plate bending vibration control system of equations short-cut method that the present invention proposes has enough computational accuracies.

When not considering the impact of sandwich layer transverse shear deformation, suppose the shearing rigidity C of sandwich in x-z, y-z plane _xz, C _yzduring for infinity, the calculation on Natural Frequency formula (28) of simply supported on four sides sandwich plate deteriorates to

${\mathrm{\&omega;}}_m=\sqrt{\frac{2(D_{12}+2D_{66})k_{m1}^2{k}_{m2}^2+D_{11}{k}_{m1}^4+D_{22}{k}_{m2}^4}{\mathrm{\&rho;}}}---\left(29\right)$

Get identical sandwich plate parameter, with formula (28) and (29), calculate respectively the front 25 rank natural frequencys more as shown in Figure 2 of sandwich plate.As can be seen from Figure 2, sandwich layer transverse shear deformation has comparatively significantly impact to sandwich plate natural frequency, do not consider that sandwich layer transverse shear deformation will cause sandwich plate natural frequency overvalued, and this impact is along with the rising of order and natural frequency has the trend of continuous enhancing, as the 21st rank, the 23rd rank natural frequency, do not consider that the natural frequency of sandwich layer transverse shear deformation exceeds approximately 17% than the natural frequency of considering sandwich layer transverse shear deformation.Therefore the impact of, considering sandwich layer transverse shear deformation in sandwich plate bending vibration control equation and calculation on Natural Frequency process is very necessary.

When the panel of sandwich plate is isotropic material, the whole-body quadrature anisotropy of sandwich plate is very weak consequently can be ignored.But when panel is compound substance, sandwich plate presents comparatively significantly whole-body quadrature anisotropy.The orthotropic degree available orthogonal of sandwich plate anisotropy factor α represents

$\mathrm{\&alpha;}=(D_{12}+2D_{66})/\sqrt{D_{11}{D}_{22}}---\left(30\right)$

When panel is isotropy, α=1 and D ₁₁=D ₂₂=D, now the calculation on Natural Frequency formula (28) of sandwich plate deteriorates to

$\begin{array}{c}{\mathrm{\&omega;}}_m={[\frac{D+D_{66}}{C_{\mathrm{yz}}}{k}_{m1}^6+Q_1^{\&prime;}{k}_{m1}^4{k}_{m2}^2+Q_2^{\&prime;}{k}_{m1}^2{k}_{m2}^4+2{\mathrm{Dk}}_{m1}^2{k}_{m2}^2+D{k}_{m1}^4+D{k}_{m2}^4+\frac{D{D}_{66}}{C_{\mathrm{xz}}}{k}_{m1}^6]}^{\frac{1}{2}}\\ \&times;\frac{1}{\sqrt{\mathrm{\&rho;}}}{[1+(\frac{D}{C_{\mathrm{xz}}}+\frac{D_{66}}{C_{\mathrm{yz}}})k_{m1}^2-\frac{D{D}_{66}}{C_{\mathrm{xz}}{C}_{\mathrm{yz}}}{k}_{m1}^4+(\frac{D}{C_{\mathrm{yz}}}+\frac{D_{66}}{C_{\mathrm{xz}}}){k_{m2}^2+}^{}{Q}_3^{\&prime;}{k}_{m1}^2{k}_{m2}^2+Q_4^{\&prime;}{k}_{m2}^4]}^{-\frac{1}{2}}\end{array}---\left(31\right)$

Wherein $Q_1^{\&prime;}=\frac{D{D}_{66}}{C_{\mathrm{xz}}}+\frac{2D{D}_{66}}{C_{\mathrm{yz}}},Q_2^{\&prime;}=\frac{4D_{66}(D-D_{66})}{C_{\mathrm{xz}}}+\frac{D{D}_{66}}{C_{\mathrm{yz}}}$

$Q_3^{\&prime;}=\frac{2D(D-D_{66})}{C_{\mathrm{xz}}^2}-\frac{2D(D-D_{66})+2D_{66}^2}{C_{\mathrm{xz}}{C}_{\mathrm{yz}}},Q_4^{\&prime;}=\frac{2D_{66}^2}{C_{\mathrm{xz}}^2}+\frac{D{D}_{66}}{C_{\mathrm{xz}}{C}_{\mathrm{yz}}}$

Get identical sandwich plate parameter, with formula (28) and (31), calculate respectively the front 25 rank natural frequencys more as shown in Figure 3 of sandwich plate.As can be seen from Figure 3, sandwich plate whole-body quadrature anisotropy has comparatively significantly impact to its natural frequency, do not consider that orthotropy will cause the valuation of sandwich plate natural frequency obviously too high, as the 1st rank, the 4th rank, the 10th rank, the 23rd rank natural frequency, do not consider that the natural frequency value of orthotropy impact is than considering that orthotropic natural frequency value exceeds approximately 30%.Therefore, in sandwich plate bending vibration control equation and calculation on Natural Frequency process, consider that orthotropic impact is very necessary.

Finally explanation is, above embodiment is only unrestricted in order to technical scheme of the present invention to be described, although the present invention is had been described in detail with reference to preferred embodiment, those of ordinary skill in the art is to be understood that, can modify or be equal to replacement technical scheme of the present invention, and not departing from aim and the scope of the technical program, it all should be encompassed in the middle of claim scope of the present invention.

Claims (2)

1. a measuring method for honeycomb sandwich panel natural frequency, is characterized in that, concrete steps are as follows:

1) use compound substance universal testing machine to measure sandwich plate to be measured, obtain sandwich plate bending stiffness d ₁₁, d ₁₂, d ₂₂, d ₆₆exist with sandwich layer x- zwith y- zmodulus of shearing in plane g _xz, g _yz;

2) utilize ruler measurement measured panel sandwich plate length a, width b, thickness tand core layer thickness h;

3) pass through formula

with

, calculate the shearing rigidity of sandwich layer c _xzwith c _yz;

4) by material handbook, check in the panel density of clamping plate to be measured

with sandwich layer density , pass through formula

calculate the equivalent face density of clamping plate

;

5) data substitution formula above-mentioned steps being recorded calculates, and in conjunction with the boundary condition of sandwich plate, solves the natural frequency on sandwich plate m rank

, its formula is as follows:

In formula

，

，

X, y is denotation coordination respectively.

2. the measuring method of a kind of honeycomb sandwich panel natural frequency as claimed in claim 1, is characterized in that: the derivation of formula described in step 5) is as follows:

The middle face of sandwich is x-y plane, and under z is axial, the one side of z>0 is top panel, and the one side of z<0 is lower panel, and plate length is a, and width is b, and plate thickness is t, and sandwich thickness is h, and sandwich plate gross thickness is H=h+2t;

The free vibration governing equation group of introducing orthotropy clip rectangle laminate is:

(1)

(2)

(3)

Wherein w is the amount of deflection of sandwich plate under action of lateral load,

,

for before sandwich plate distortion perpendicular to the straight-line segment of middle the corner in x-z and y-z plane;

System of equations Chinese style (1) is right

ask local derviation, formula (2) is right

ask local derviation, then two formulas are subtracted each other, and can obtain

(4)

Introduce partial differential operator, formula (4) can be rewritten as

(5)

L wherein ₁, L ₂for partial differential operator:

(6)

(7)

Introduce displacement function

, and order

(8)

By formula (8) substitution formula (5), obtain

(9)

By (9) formula, obtained

(10)

By in formula (6), (7) substitution formula (8), (10), generalized displacement

with

available displacement function be expressed as

(11)

(12)

By formula (11), (12) substitution formula (1), through abbreviation, obtain

(13)

By formula (13) both sides pair xcarry out indefinite integral, generalized displacement wby displacement function

be expressed as

(14)

By formula (11), (12) and formula (14) substitution system of equations formula (3), through merging, abbreviation, obtain about displacement function

vibration control equation

(15)

Wherein

(16)

(17)

(18)

(19)

(20)

To formula (16)-(20) merge, abbreviation, and substitution formula (15), just can obtain only containing displacement function

orthotropic Sandwich Plate free vibration governing equation

(21)

Wherein

，

，

。

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