CN107818209A - A kind of vibration analysis method of elastic plate - Google Patents

A kind of vibration analysis method of elastic plate Download PDF

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CN107818209A
CN107818209A CN201711016929.3A CN201711016929A CN107818209A CN 107818209 A CN107818209 A CN 107818209A CN 201711016929 A CN201711016929 A CN 201711016929A CN 107818209 A CN107818209 A CN 107818209A
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叶天贵
陈玉坤
靳国永
张春雨
张艳涛
王雪仁
缪旭弘
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Harbin Engineering University
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Abstract

A kind of vibration analysis method of elastic plate, comprises the following steps：Elastic plate thickness direction displacement is fitted using Kano high-order interception technology；Fourier space is improved using two dimension the expansion of full solution domain is carried out to elastic plate in-plane displacement；The global displacement of elastic plate is calculated by elastic plate section in-plane displacement and axial displacement；Calculate the strain vector and stress vector of elastic plate；The strain energy and kinetic energy equation of elastic plate are calculated, sets virtual spring border to obtain border energy；Structure Lagrange energy functional is established, the core mass matrix and stiffness matrix of elastic plate is calculated；The characteristic equation of overall mass matrix, stiffness matrix and structure is tried to achieve by iterative cycles kernel matrix；The intrinsic frequency of elastic plate is calculated, according to the vibration shape of characteristic vector export structure.The inventive method is applied to Multiple Shape, the elastic plate of multi-boundary Condition, and precision is high, convergence is fast, it is low to calculate cost.

Description

A kind of vibration analysis method of elastic plate
Technical field
The invention belongs to Structural Dynamics field, and in particular to a kind of vibration analysis method of elastic plate.
Background technology
Elastic plate is widely used in the engineering equipments such as Naval Architecture and Ocean Engineering, architectural engineering and Aero-Space, example Such as submarine, automobile and airframe.Vibration characteristics and its parameter affecting laws for furtheing investigate this class formation are early in equipment Design phase phase vibration noise level estimates and realized that Low Noise Design has important theory and practice directive significance.
Traditional elastic plate theory mainly has thin plate bending theoretical and shear deformable theory, and thin plate bending theory have ignored The strain and detrusion in structural thickness direction and the influence of rotary inertia, it is only applicable to solve the thin plate knot of micro-strain Structure；Shear deformable theory although it is contemplated that the influence of detrusion and rotary inertia, but still have ignored cross directional stretch deformation and Buckling deformation during torsion, it is only applicable to the transverse curvature problem of cut deal structure.When the thickness of harden structure is larger, both Often deviation is larger or even is not used to calculate analysis for the calculated results.Kano proposes Kano system based on three dimensional elasticity theory One theorem (CUF) (Carrera E.A class of two-dimensional theories for anisotropic multilayered plates analysis.[J].Mem.accad.sci.torino Cl.sci.fis.mat.natur, 1995:49-87.), theoretical based on this, the computational accuracy of elastic plate can be controlled by by the interpolating function of thickness direction, and The order for changing interpolating function does not have an impact to the theoretical kernel matrix, in this way, the elastic plate of corresponding different-thickness, its Kinetics equation can be come by same matrix core iteration, the number of the simply iteration of change.In addition, in the technology The number of known variables can determine according to the demand of problem in displacement, and most elastic plate vibration problem can pass through Select the quantity of suitable known variables and reach corresponding computational accuracy.
However, existing Kano high-order interception fitting technique is largely to solve elastic plate by embedded finite element method (typical document is Carrera E.Theories and finite elements for Structural Dynamics multilayered,anisotropic,composite plates and shells[J].Archives of Computational Methods in Engineering,2002,9(2):87-140.Cinefra M,Valvano S.A variable kinematic doubly-curved MITC9shell element for the analysis of laminated composites[J].Mechanics of Advanced Materials&Structures,2016,23 (11):P á gs.1312-1325.), change the shape of elastic plate or boundary condition generally requires adding unit quantity even again Modeling analysis, in addition finite element method generally have computationally intensive, computational accuracy is not high, and boundary condition applies relatively complicated grade and lacked Point.Therefore studying and establish a kind of can be applicable shaking for the fast elastic plate of Multiple Shape Parameters, multi-boundary Condition, calculating speed Dynamic analysis method is still focus on research direction.
The content of the invention
It is an object of the invention to provide a kind of applicable Multiple Shape Parameters and multi-boundary Condition, and precision is high, convergence is fast, Calculate that cost is low and the vibration analysis method of the simple elastic plate of computational methods.
The object of the present invention is achieved like this, comprises the following steps：
(1) elastic plate thickness direction displacement is fitted using Kano high-order interception technology, fitting form is as follows：
Wherein, x, y and z are the coordinate of space coordinates, and Φ (x, y, z) is the global displacement of elastic plate, Represent elastic plate in-plane displacement, i=1,2 ..., N+1, FiFor i-th in Taylor expansion, h is the thickness of elastic plate, N is the order of Taylor expansion.Φ takes U, V, W,U, v accordingly are taken, w corresponds to the displacement component on tri- directions of x, y and z respectively.
(2) Fourier space is improved using two dimension elastic plate in-plane displacement is carried out to solve domain expansion, specific shape entirely Formula is as follows：
WhereinWithFor the coefficient of corresponding expansion item, λm=m π/L1And λp=p π/L2(L1And L2Respectively For geometrical scale of the structure in x and y directions), M, P are to block item number.XmAnd YpRespectively x and y function,To be corresponding The coefficient of item, supplement function ξkAnd η (x)g(y) the displacement structure that is introduced for eliminating is launched into conventional Fourier cosine levels Itself and derivative are in the discontinuity of boundary during number, and so as to accelerate the convergence rate solved, supplement function concrete form is set It is set to：
(3) global displacement of elastic plate is calculated by elastic plate section in-plane displacement and axial displacement, has Body expression formula is as follows：
Wherein, U (x, y, z), V (x, y, z) and W (x, y, z) correspond to the position on tri- directions of space coordinates x, y and z respectively Move component, Ampi, BmpiAnd CmpiFor the coefficient of corresponding entry in displacement component.
(4) strain vector and stress vector of elastic plate are calculated；
The expression formula of the strain vector of involved elastic plate is：
ε=[εx,εy,εz,γxy,γyz,γxz]T
Wherein, ε represents the strain vector of elastic plate；Subscript T represents transposition；εx, εyAnd εzFor normal strain component； γxy, γyzAnd γxzFor shear strain component, and have
The expression formula of involved stress vector is：
σ=D ε
Wherein, σ represents the stress vector of elastic plate, and D is structural material coefficient matrix.
(5) strain energy and kinetic energy equation of elastic plate are calculated, meanwhile, set virtual spring border to obtain border Can, expression is as follows：
Wherein, Vs, TpAnd VpThe respectively strain energy of elastic plate, kinetic energy and border energy.T represents the time, and ρ is material Density.WithFor the virtual spring border set by x directions x=0 ends in plate face,WithFor x side in plate face To other end x=L1The virtual spring border that place is set；WithFor the virtual spring set by y directions y=0 ends in plate face Border,WithFor y directions other end y=L in plate face2The virtual spring border that place is set.
(6) structure Lagrange energy functional Ω=V is establisheds+Vp-Tp, then to coefficient A thereinmpi, BmpiAnd CmpiAsk Local derviation and to make its result be zero, you can obtain 3 × 3 rank core mass matrixes and stiffness matrix of elastic plate.Kernel matrix In element it is as follows：
Wherein KmnpqijFor core rigidity matrix, MmnpqijFor core mass matrix, superscript a, b and c are representing core Each element in matrix, such as the element that a rows b is arranged in ab representing matrixs；Subscript m, n=1 ..., M+3；P, q= 1,…,P+3；I, j=1 ..., N+1；X′m, Y 'pWith F 'iX is represented respectivelym, YpAnd FiFirst derivative, similarly X 'n, Y 'qWith F 'j X is represented respectivelyn, YqAnd FjFirst derivative.D11,…,D66For the element in structural material coefficient matrix D.
(7) overall mass matrix, stiffness matrix and oeverall quality matrix M are tried to achieve by iterative cycles kernel matrix, And then obtain the characteristic equation of structure；
The method for solving of the mass matrix and stiffness matrix is：Pointer i, j get N+1 circulation core rigidity matrixes by 1 KmnpqijObtain sub- submatrix Kmnpq, pointer p, q get P+3 by 1 and circulate sub- submatrix KmnpqObtain submatrix Kmn, pointer m, n by 1 gets M+3 circulation submatrixs KmnObtain global stiffness matrix K；
The characteristic equation expression formula of the structure is：
(K-ω2M) A=0
Wherein ω is circular frequency, and A is corresponding ω characteristic vector；
(8) intrinsic frequency of elastic plate is calculated, according to the vibration shape of characteristic vector export structure.
The present invention has the advantages that：
1. the present invention realizes the Parametric Analysis to elastic plate computational accuracy and convergence rate, the meter of method is improved Calculate precision or item number is blocked in increase, need to only simply increase the loop iteration number of kernel matrix；
2. the displacement of harden structure thickness direction is fitted with Taylor polynomial in the present invention, special specific aim is had no, it is theoretical Upper this method is applied to the elastic plate of any thickness；
3. harden structure cross-sectional displacement improves Fourier space with two dimension and carries out the fitting of full solution domain in the present invention, have with tradition Limit first method to compare, have the characteristics that fast convergence rate, computational accuracy are high；
4. the method for the present invention only needs to meet that the various boundary of structure will by controlling the rigidity of border spring Ask, without making any modification to program.
In summary, Multiple Shape Parameters, multi-boundary Condition, precision are high, convergence is fast, calculating with being applicable for method of the invention The features such as cost is low.
Brief description of the drawings
Fig. 1 is the flow chart of the present invention；
Fig. 2 is elastic plate schematic diagram；
Fig. 3 is stiffness matrix installation diagram.
Embodiment
For make present invention solves the technical problem that, the technical scheme that uses and the technique effect that reaches it is clearer, below The present invention is described further with reference to accompanying drawing.
The inventive method is performed shown in step reference picture 1.
A Rectangular Elastic harden structure is considered, as shown in Fig. 2 sectional dimension is L1=2m, L2=3m, thickness of elastic plates h= 0.2m, is isotropic material, Young's modulus E=75GPa, density p=7800kg/m3, Poisson's ratio μ=0.3.Harden structure four sides Freely-supported is without plus load.It is solved using the inventive method, comprised the following steps that：
(1) elastic plate thickness direction displacement is fitted using Kano high-order interception technology, by thickness and length Ratio choose Taylor expansion order N=2, fitting form it is as follows：
Wherein, x, y and z are the coordinate of space coordinates, and Φ (x, y, z) is the global displacement of elastic plate, Represent elastic plate in-plane displacement, i=1,2 ..., N+1, FiFor i-th in Taylor expansion, h is the thickness of elastic plate. Φ takes U, V and W,Accordingly u, v and w is taken to correspond to the displacement component on three directions of elastic plate respectively：
Wherein, U (x, y, z), V (x, y, z) and W (x, y, z) correspond to the position on tri- directions of space coordinates x, y and z respectively Move component.
(2) Fourier space is improved using two dimension elastic plate in-plane displacement is carried out to solve domain expansion, specific shape entirely Formula is as follows：
WhereinWithFor the coefficient of corresponding expansion item, λm=m π/L1And λp=p π/L2(L1And L2Respectively For geometrical scale of the structure in x and y directions), M, P are to block item number.XmAnd YpRespectively x and y function,To be corresponding The coefficient of item, supplement function ξkAnd η (x)g(y) the displacement structure that is introduced for eliminating is launched into conventional Fourier cosine levels Itself and derivative are in the discontinuity of boundary during number, and so as to accelerate the convergence rate solved, supplement function concrete form is set It is set to：
(3) elastic plate section in-plane displacement and axial displacement are combined, you can the global displacement of elastic plate is obtained, Expression is as follows：
Wherein, Ampi, BmpiAnd CmpiFor the coefficient of corresponding entry in displacement component.
(4) elastic plate is isotropic material, Young's modulus E=75GPa, Poisson's ratio μ=0.3.Calculate elastic plate The strain vector and stress vector of structure, vibration strains concrete form are as follows：
ε=[εx,εy,εz,γxy,γyz,γxz]T
Wherein, εx, εyAnd εzFor normal strain component, γxy, γyzAnd γxzFor shear strain component；ε represents elastic plate Strain vector；T represents transposition.Vibration stress concrete form is as follows：
σ=D ε
Wherein, σ represents the stress vector of elastic plate, and D is structural material coefficient matrix.
(5) strain energy and kinetic energy equation of elastic plate are calculated, meanwhile, set virtual spring border to obtain border Can, expression is as follows：
Wherein, Vs, TpAnd VpThe respectively strain energy of elastic plate, kinetic energy and border energy.T represent the time, material it is close Spend ρ=7800kg/m3。WithFor the virtual spring border set by x directions x=0 ends in plate face,WithFor X directions other end x=L in plate face1The virtual spring border that place is set；WithFor set by y directions y=0 ends in plate face Virtual spring border,WithFor y directions other end y=L in plate face2The virtual spring border that place is set.
(6) structure Lagrange energy functional Ω=V is establisheds+Vp-Tp, then to coefficient A thereinmpi, BmpiAnd CmpiAsk Local derviation and to make its result be zero, you can obtain 3 × 3 rank core mass matrixes and stiffness matrix of elastic plate.Kernel matrix In element it is as follows：
Wherein KmnpqijFor core rigidity matrix, MmnpqijFor core mass matrix, superscript a, b and c are representing core Each element in matrix, such as the element that a rows b is arranged in ab representing matrixs；Subscript m, n=1 ..., M+3；P, q= 1,…,P+3；I, j=1 ..., N+1；X′m, Y 'pWith F 'iX is represented respectivelym, YpAnd FiFirst derivative, similarly X 'n, Y 'qWith F 'j X is represented respectivelyn, YqAnd FjFirst derivative.D11,…,D66For the element in structural material coefficient matrix D.
(7) as shown in figure 3, trying to achieve overall mass matrix and stiffness matrix by iterative cycles kernel matrix：Pointer i, j N+1 circulation core rigidity matrix Ks are got by 1mnpqijObtain sub- submatrix Kmnpq, pointer p, q get P+3 by 1 and circulate sub- submatrix KmnpqObtain submatrix Kmn, pointer m, n get M+3 circulation submatrixs K by 1mnGlobal stiffness matrix K is obtained, by identical side Method circulation core mass matrix obtains oeverall quality matrix M, and then obtains the characteristic equation of structure：
(K-ω2M) A=0
Wherein ω is circular frequency, and A is corresponding ω characteristic vector.
(8) intrinsic frequency of MATLAB solvers output elastic plate is established using Arnoldi algorithm, and according to feature Each first order mode of vectorial export structure.
Finally it should be noted that：Implement example above to be merely illustrative of the technical solution of the present invention, rather than its limitations, this The technical staff in field should be understood：It modifies to the technical scheme described in foregoing embodiments, or to its middle part Divide or all technical characteristic carries out equivalent substitution, the essence of appropriate technical solution is departed from various embodiments of the present invention technology The scope of scheme.

Claims (1)

1. a kind of vibration analysis method of elastic plate, it is characterised in that comprise the following steps：
Step 1 is fitted using Kano high-order interception technology to elastic plate thickness direction displacement, and fitting form is as follows：
Wherein, x, y and z are the coordinate of space coordinates, and Φ (x, y, z) is the global displacement of elastic plate,Represent bullet Property harden structure in-plane displacement, i=1,2 ..., N+1, FiFor i-th in Taylor expansion, h is the thickness of elastic plate, and N is Thailand Strangle the order of expansion；Φ takes U, V, W,U, v accordingly are taken, w corresponds to the displacement component on tri- directions of x, y and z respectively；
Step 2 improves Fourier space using two dimension and elastic plate in-plane displacement is carried out to solve domain expansion, specific shape entirely Formula is as follows：
WhereinWithFor the coefficient of corresponding expansion item, λm=m π/L1And λp=p π/L2, L1And L2Respectively tie Geometrical scale of the structure in x and y directions；M, P are to block item number；XmAnd YpRespectively x and y function,For corresponding entry Coefficient, ξkAnd η (x)g(y) it is supplement function, supplement function expression is：
<mfenced open = "" close = ""> <mtable> <mtr> <mtd> <mrow> <msub> <mi>&amp;xi;</mi> <mi>k</mi> </msub> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> <mo>=</mo> <mfenced open = "{" close = ""> <mtable> <mtr> <mtd> <mrow> <mi>x</mi> <msup> <mrow> <mo>(</mo> <mrow> <mfrac> <mi>x</mi> <msub> <mi>L</mi> <mn>1</mn> </msub> </mfrac> <mo>-</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> <mn>2</mn> </msup> </mrow> </mtd> <mtd> <mrow> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mfrac> <msup> <mi>x</mi> <mn>2</mn> </msup> <msub> <mi>L</mi> <mn>1</mn> </msub> </mfrac> <mrow> <mo>(</mo> <mrow> <mfrac> <mi>x</mi> <msub> <mi>L</mi> <mn>1</mn> </msub> </mfrac> <mo>-</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> </mrow> </mtd> <mtd> <mrow> <mi>k</mi> <mo>=</mo> <mn>2</mn> </mrow> </mtd> </mtr> </mtable> </mfenced> </mrow> </mtd> <mtd> <mrow> <msub> <mi>&amp;eta;</mi> <mi>g</mi> </msub> <mrow> <mo>(</mo> <mi>y</mi> <mo>)</mo> </mrow> <mo>=</mo> <mfenced open = "{" close = ""> <mtable> <mtr> <mtd> <mrow> <mi>y</mi> <msup> <mrow> <mo>(</mo> <mfrac> <mi>y</mi> <msub> <mi>L</mi> <mn>2</mn> </msub> </mfrac> <mo>-</mo> <mn>1</mn> <mo>)</mo> </mrow> <mn>2</mn> </msup> </mrow> </mtd> <mtd> <mrow> <mi>g</mi> <mo>=</mo> <mn>1</mn> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mfrac> <msup> <mi>y</mi> <mn>2</mn> </msup> <msub> <mi>L</mi> <mn>2</mn> </msub> </mfrac> <mrow> <mo>(</mo> <mfrac> <mi>y</mi> <msub> <mi>L</mi> <mn>2</mn> </msub> </mfrac> <mo>-</mo> <mn>1</mn> <mo>)</mo> </mrow> </mrow> </mtd> <mtd> <mrow> <mi>g</mi> <mo>=</mo> <mn>2</mn> </mrow> </mtd> </mtr> </mtable> </mfenced> </mrow> </mtd> </mtr> </mtable> </mfenced>
Step 3 is calculated the global displacement of elastic plate, specific table by elastic plate section in-plane displacement and axial displacement It is as follows up to formula：
<mfenced open = "" close = ""> <mtable> <mtr> <mtd> <mrow> <mi>U</mi> <mrow> <mo>(</mo> <mrow> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>,</mo> <mi>z</mi> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <munderover> <mo>&amp;Sigma;</mo> <mrow> <mi>m</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow> <mi>M</mi> <mo>+</mo> <mn>3</mn> </mrow> </munderover> <munderover> <mo>&amp;Sigma;</mo> <mrow> <mi>p</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow> <mi>P</mi> <mo>+</mo> <mn>3</mn> </mrow> </munderover> <munderover> <mo>&amp;Sigma;</mo> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow> <mi>N</mi> <mo>+</mo> <mn>1</mn> </mrow> </munderover> <msub> <mi>A</mi> <mrow> <mi>m</mi> <mi>p</mi> <mi>i</mi> </mrow> </msub> <msub> <mi>X</mi> <mi>m</mi> </msub> <msub> <mi>Y</mi> <mi>p</mi> </msub> <msub> <mi>F</mi> <mi>i</mi> </msub> </mrow> </mtd> <mtd> <mrow> <mi>V</mi> <mrow> <mo>(</mo> <mrow> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>,</mo> <mi>z</mi> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <munderover> <mo>&amp;Sigma;</mo> <mrow> <mi>m</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow> <mi>M</mi> <mo>+</mo> <mn>3</mn> </mrow> </munderover> <munderover> <mo>&amp;Sigma;</mo> <mrow> <mi>p</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow> <mi>P</mi> <mo>+</mo> <mn>3</mn> </mrow> </munderover> <munderover> <mo>&amp;Sigma;</mo> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow> <mi>N</mi> <mo>+</mo> <mn>1</mn> </mrow> </munderover> <msub> <mi>B</mi> <mrow> <mi>m</mi> <mi>p</mi> <mi>i</mi> </mrow> </msub> <msub> <mi>X</mi> <mi>m</mi> </msub> <msub> <mi>Y</mi> <mi>p</mi> </msub> <msub> <mi>F</mi> <mi>i</mi> </msub> </mrow> </mtd> </mtr> </mtable> </mfenced>
<mrow> <mi>W</mi> <mrow> <mo>(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>,</mo> <mi>z</mi> <mo>)</mo> </mrow> <mo>=</mo> <munderover> <mo>&amp;Sigma;</mo> <mrow> <mi>m</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow> <mi>M</mi> <mo>+</mo> <mn>3</mn> </mrow> </munderover> <munderover> <mo>&amp;Sigma;</mo> <mrow> <mi>p</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow> <mi>P</mi> <mo>+</mo> <mn>3</mn> </mrow> </munderover> <munderover> <mo>&amp;Sigma;</mo> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow> <mi>N</mi> <mo>+</mo> <mn>1</mn> </mrow> </munderover> <msub> <mi>C</mi> <mrow> <mi>m</mi> <mi>p</mi> <mi>i</mi> </mrow> </msub> <msub> <mi>X</mi> <mi>m</mi> </msub> <msub> <mi>Y</mi> <mi>p</mi> </msub> <msub> <mi>F</mi> <mi>i</mi> </msub> </mrow>
Wherein, U (x, y, z), V (x, y, z) and W (x, y, z) correspond to the displacement point on tri- directions of space coordinates x, y and z respectively Amount, Ampi, BmpiAnd CmpiFor the coefficient of corresponding entry in displacement component；
Step 4 calculates the strain vector and stress vector of elastic plate；
The expression formula of the strain vector of involved elastic plate is：
ε=[εx,εy,εz,γxy,γyz,γxz]T
Wherein, ε represents the strain vector of elastic plate；Subscript T represents transposition；εx, εyAnd εzFor normal strain component；γxy, γyz And γxzFor shear strain component, and have
<mrow> <msub> <mi>&amp;epsiv;</mi> <mi>x</mi> </msub> <mo>=</mo> <mfrac> <mrow> <mo>&amp;part;</mo> <mi>U</mi> <mrow> <mo>(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>,</mo> <mi>z</mi> <mo>)</mo> </mrow> </mrow> <mrow> <mo>&amp;part;</mo> <mi>x</mi> </mrow> </mfrac> <mo>;</mo> <msub> <mi>&amp;gamma;</mi> <mrow> <mi>x</mi> <mi>y</mi> </mrow> </msub> <mo>=</mo> <mfrac> <mrow> <mo>&amp;part;</mo> <mi>U</mi> <mrow> <mo>(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>,</mo> <mi>z</mi> <mo>)</mo> </mrow> </mrow> <mrow> <mo>&amp;part;</mo> <mi>y</mi> </mrow> </mfrac> <mo>+</mo> <mfrac> <mrow> <mo>&amp;part;</mo> <mi>V</mi> <mrow> <mo>(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>,</mo> <mi>z</mi> <mo>)</mo> </mrow> </mrow> <mrow> <mo>&amp;part;</mo> <mi>x</mi> </mrow> </mfrac> <mo>;</mo> </mrow>
<mrow> <msub> <mi>&amp;epsiv;</mi> <mi>y</mi> </msub> <mo>=</mo> <mfrac> <mrow> <mo>&amp;part;</mo> <mi>V</mi> <mrow> <mo>(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>,</mo> <mi>z</mi> <mo>)</mo> </mrow> </mrow> <mrow> <mo>&amp;part;</mo> <mi>y</mi> </mrow> </mfrac> <mo>:</mo> <msub> <mi>&amp;gamma;</mi> <mrow> <mi>x</mi> <mi>z</mi> </mrow> </msub> <mo>=</mo> <mfrac> <mrow> <mo>&amp;part;</mo> <mi>U</mi> <mrow> <mo>(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>,</mo> <mi>z</mi> <mo>)</mo> </mrow> </mrow> <mrow> <mo>&amp;part;</mo> <mi>z</mi> </mrow> </mfrac> <mo>+</mo> <mfrac> <mrow> <mo>&amp;part;</mo> <mi>W</mi> <mrow> <mo>(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>,</mo> <mi>z</mi> <mo>)</mo> </mrow> </mrow> <mrow> <mo>&amp;part;</mo> <mi>x</mi> </mrow> </mfrac> <mo>;</mo> </mrow>
<mrow> <msub> <mi>&amp;epsiv;</mi> <mi>z</mi> </msub> <mo>=</mo> <mfrac> <mrow> <mo>&amp;part;</mo> <mi>W</mi> <mrow> <mo>(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>,</mo> <mi>z</mi> <mo>)</mo> </mrow> </mrow> <mrow> <mo>&amp;part;</mo> <mi>z</mi> </mrow> </mfrac> <mo>;</mo> <msub> <mi>&amp;gamma;</mi> <mrow> <mi>y</mi> <mi>z</mi> </mrow> </msub> <mo>=</mo> <mfrac> <mrow> <mo>&amp;part;</mo> <mi>V</mi> <mrow> <mo>(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>,</mo> <mi>z</mi> <mo>)</mo> </mrow> </mrow> <mrow> <mo>&amp;part;</mo> <mi>z</mi> </mrow> </mfrac> <mo>+</mo> <mfrac> <mrow> <mo>&amp;part;</mo> <mi>W</mi> <mrow> <mo>(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>,</mo> <mi>z</mi> <mo>)</mo> </mrow> </mrow> <mrow> <mo>&amp;part;</mo> <mi>y</mi> </mrow> </mfrac> <mo>;</mo> </mrow>
The expression formula of involved stress vector is：
σ=D ε
Wherein, σ represents the stress vector of elastic plate, and D is structural material coefficient matrix；
Step 5 calculates the strain energy and kinetic energy equation of elastic plate, and sets virtual spring border so as to obtain border energy, Expression is as follows：
<mrow> <msub> <mi>V</mi> <mi>s</mi> </msub> <mo>=</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <mo>&amp;Integral;</mo> <mo>&amp;Integral;</mo> <mo>&amp;Integral;</mo> <msup> <mi>&amp;epsiv;</mi> <mi>T</mi> </msup> <mi>&amp;sigma;</mi> <mi>d</mi> <mi>x</mi> <mi>d</mi> <mi>y</mi> <mi>d</mi> <mi>z</mi> </mrow>
<mrow> <msub> <mi>T</mi> <mi>p</mi> </msub> <mo>=</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <mi>&amp;rho;</mi> <mo>&amp;Integral;</mo> <mo>&amp;Integral;</mo> <mo>&amp;Integral;</mo> <mrow> <mo>(</mo> <msup> <mrow> <mo>(</mo> <mfrac> <mrow> <mo>&amp;part;</mo> <mi>U</mi> <mrow> <mo>(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>,</mo> <mi>z</mi> <mo>)</mo> </mrow> </mrow> <mrow> <mo>&amp;part;</mo> <mi>t</mi> </mrow> </mfrac> <mo>)</mo> </mrow> <mn>2</mn> </msup> <mo>+</mo> <msup> <mrow> <mo>(</mo> <mfrac> <mrow> <mo>&amp;part;</mo> <mi>V</mi> <mrow> <mo>(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>,</mo> <mi>z</mi> <mo>)</mo> </mrow> </mrow> <mrow> <mo>&amp;part;</mo> <mi>t</mi> </mrow> </mfrac> <mo>)</mo> </mrow> <mn>2</mn> </msup> <mo>+</mo> <msup> <mrow> <mo>(</mo> <mfrac> <mrow> <mo>&amp;part;</mo> <mi>W</mi> <mrow> <mo>(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>,</mo> <mi>z</mi> <mo>)</mo> </mrow> </mrow> <mrow> <mo>&amp;part;</mo> <mi>t</mi> </mrow> </mfrac> <mo>)</mo> </mrow> <mn>2</mn> </msup> <mo>)</mo> </mrow> <mi>d</mi> <mi>x</mi> <mi>d</mi> <mi>y</mi> <mi>d</mi> <mi>z</mi> </mrow>
<mfenced open = "" close = ""> <mtable> <mtr> <mtd> <mrow> <msub> <mi>V</mi> <mi>p</mi> </msub> <mo>=</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <msubsup> <mo>&amp;Integral;</mo> <mn>0</mn> <msub> <mi>L</mi> <mn>2</mn> </msub> </msubsup> <msubsup> <mo>&amp;Integral;</mo> <mrow> <mo>-</mo> <mfrac> <mi>h</mi> <mn>2</mn> </mfrac> </mrow> <mfrac> <mi>h</mi> <mn>2</mn> </mfrac> </msubsup> <mfenced open = "(" close = ")"> <mtable> <mtr> <mtd> <mrow> <mrow> <mo>(</mo> <mrow> <msubsup> <mi>k</mi> <mrow> <mi>x</mi> <mn>0</mn> </mrow> <mi>u</mi> </msubsup> <mi>U</mi> <msup> <mrow> <mo>(</mo> <mrow> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>,</mo> <mi>z</mi> </mrow> <mo>)</mo> </mrow> <mn>2</mn> </msup> <mo>+</mo> <msubsup> <mi>k</mi> <mrow> <mi>x</mi> <mn>0</mn> </mrow> <mi>v</mi> </msubsup> <mi>V</mi> <msup> <mrow> <mo>(</mo> <mrow> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>,</mo> <mi>z</mi> </mrow> <mo>)</mo> </mrow> <mn>2</mn> </msup> <mo>+</mo> <msubsup> <mi>k</mi> <mrow> <mi>x</mi> <mn>0</mn> </mrow> <mi>w</mi> </msubsup> <mi>W</mi> <msup> <mrow> <mo>(</mo> <mrow> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>,</mo> <mi>z</mi> </mrow> <mo>)</mo> </mrow> <mn>2</mn> </msup> </mrow> <mo>)</mo> </mrow> <msub> <mo>|</mo> <mrow> <mi>x</mi> <mo>=</mo> <mn>0</mn> </mrow> </msub> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>+</mo> <mrow> <mo>(</mo> <mrow> <msubsup> <mi>k</mi> <mrow> <msub> <mi>xL</mi> <mn>1</mn> </msub> </mrow> <mi>u</mi> </msubsup> <mi>U</mi> <msup> <mrow> <mo>(</mo> <mrow> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>,</mo> <mi>z</mi> </mrow> <mo>)</mo> </mrow> <mn>2</mn> </msup> <mo>+</mo> <msubsup> <mi>k</mi> <mrow> <msub> <mi>xL</mi> <mn>1</mn> </msub> </mrow> <mi>v</mi> </msubsup> <mi>V</mi> <msup> <mrow> <mo>(</mo> <mrow> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>,</mo> <mi>z</mi> </mrow> <mo>)</mo> </mrow> <mn>2</mn> </msup> <mo>+</mo> <msubsup> <mi>k</mi> <mrow> <msub> <mi>xL</mi> <mn>1</mn> </msub> </mrow> <mi>w</mi> </msubsup> <mi>W</mi> <msup> <mrow> <mo>(</mo> <mrow> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>,</mo> <mi>z</mi> </mrow> <mo>)</mo> </mrow> <mn>2</mn> </msup> </mrow> <mo>)</mo> </mrow> <msub> <mo>|</mo> <mrow> <mi>x</mi> <mo>=</mo> <msub> <mi>L</mi> <mn>1</mn> </msub> </mrow> </msub> </mrow> </mtd> </mtr> </mtable> </mfenced> <mi>d</mi> <mi>z</mi> <mi>d</mi> <mi>y</mi> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>+</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <msubsup> <mo>&amp;Integral;</mo> <mn>0</mn> <msub> <mi>L</mi> <mn>1</mn> </msub> </msubsup> <msubsup> <mo>&amp;Integral;</mo> <mrow> <mo>-</mo> <mfrac> <mi>h</mi> <mn>2</mn> </mfrac> </mrow> <mfrac> <mi>h</mi> <mn>2</mn> </mfrac> </msubsup> <mfenced open = "(" close = ")"> <mtable> <mtr> <mtd> <mrow> <mrow> <mo>(</mo> <mrow> <msubsup> <mi>k</mi> <mrow> <mi>y</mi> <mn>0</mn> </mrow> <mi>u</mi> </msubsup> <mi>U</mi> <msup> <mrow> <mo>(</mo> <mrow> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>,</mo> <mi>z</mi> </mrow> <mo>)</mo> </mrow> <mn>2</mn> </msup> <mo>+</mo> <msubsup> <mi>k</mi> <mrow> <mi>y</mi> <mn>0</mn> </mrow> <mi>v</mi> </msubsup> <mi>V</mi> <msup> <mrow> <mo>(</mo> <mrow> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>,</mo> <mi>z</mi> </mrow> <mo>)</mo> </mrow> <mn>2</mn> </msup> <mo>+</mo> <msubsup> <mi>k</mi> <mrow> <mi>y</mi> <mn>0</mn> </mrow> <mi>w</mi> </msubsup> <mi>W</mi> <msup> <mrow> <mo>(</mo> <mrow> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>,</mo> <mi>z</mi> </mrow> <mo>)</mo> </mrow> <mn>2</mn> </msup> </mrow> <mo>)</mo> </mrow> <msub> <mo>|</mo> <mrow> <mi>y</mi> <mo>=</mo> <mn>0</mn> </mrow> </msub> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>+</mo> <mrow> <mo>(</mo> <mrow> <msubsup> <mi>k</mi> <mrow> <msub> <mi>yL</mi> <mn>2</mn> </msub> </mrow> <mi>u</mi> </msubsup> <mi>U</mi> <msup> <mrow> <mo>(</mo> <mrow> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>,</mo> <mi>z</mi> </mrow> <mo>)</mo> </mrow> <mn>2</mn> </msup> <mo>+</mo> <msubsup> <mi>k</mi> <mrow> <msub> <mi>yL</mi> <mn>2</mn> </msub> </mrow> <mi>v</mi> </msubsup> <mi>V</mi> <msup> <mrow> <mo>(</mo> <mrow> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>,</mo> <mi>z</mi> </mrow> <mo>)</mo> </mrow> <mn>2</mn> </msup> <mo>+</mo> <msubsup> <mi>k</mi> <mrow> <msub> <mi>yL</mi> <mn>2</mn> </msub> </mrow> <mi>w</mi> </msubsup> <mi>W</mi> <msup> <mrow> <mo>(</mo> <mrow> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>,</mo> <mi>z</mi> </mrow> <mo>)</mo> </mrow> <mn>2</mn> </msup> </mrow> <mo>)</mo> </mrow> <msub> <mo>|</mo> <mrow> <mi>y</mi> <mo>=</mo> <msub> <mi>L</mi> <mn>2</mn> </msub> </mrow> </msub> </mrow> </mtd> </mtr> </mtable> </mfenced> <mi>d</mi> <mi>z</mi> <mi>d</mi> <mi>x</mi> </mrow> </mtd> </mtr> </mtable> </mfenced>
Wherein, Vs, TpAnd VpThe respectively strain energy of elastic plate, kinetic energy and border energy；T represents the time, and ρ is the close of material Degree；WithFor the virtual spring border set by x directions x=0 ends in plate face,WithIt is another for x directions in plate face One end x=L1The virtual spring border that place is set；WithFor the virtual spring side set by y directions y=0 ends in plate face Boundary,WithFor y directions other end y=L in plate face2The virtual spring border that place is set；
Step 6 establishes structure Lagrange energy functional Ω=Vs+Vp-Tp, then to coefficient A thereinmpi, BmpiAnd CmpiAsk Local derviation and to make its result be zero, is calculated the core mass matrix and stiffness matrix of elastic plate；Member in kernel matrix Element is as follows：
<mrow> <msubsup> <mi>K</mi> <mrow> <mi>m</mi> <mi>n</mi> <mi>p</mi> <mi>q</mi> <mi>i</mi> <mi>j</mi> </mrow> <mrow> <mi>a</mi> <mi>a</mi> </mrow> </msubsup> <mo>=</mo> <msub> <mi>D</mi> <mn>11</mn> </msub> <msubsup> <mi>X</mi> <mi>m</mi> <mo>&amp;prime;</mo> </msubsup> <msubsup> <mi>X</mi> <mi>n</mi> <mo>&amp;prime;</mo> </msubsup> <msub> <mi>Y</mi> <mi>p</mi> </msub> <msub> <mi>Y</mi> <mi>q</mi> </msub> <msub> <mi>F</mi> <mi>i</mi> </msub> <msub> <mi>F</mi> <mi>j</mi> </msub> <mo>+</mo> <msub> <mi>D</mi> <mn>44</mn> </msub> <msub> <mi>X</mi> <mi>m</mi> </msub> <msub> <mi>X</mi> <mi>n</mi> </msub> <msubsup> <mi>Y</mi> <mi>p</mi> <mo>&amp;prime;</mo> </msubsup> <msubsup> <mi>Y</mi> <mi>q</mi> <mo>&amp;prime;</mo> </msubsup> <msub> <mi>F</mi> <mi>i</mi> </msub> <msub> <mi>F</mi> <mi>j</mi> </msub> <mo>+</mo> <msub> <mi>D</mi> <mn>55</mn> </msub> <msub> <mi>X</mi> <mi>m</mi> </msub> <msub> <mi>X</mi> <mi>n</mi> </msub> <msub> <mi>Y</mi> <mi>p</mi> </msub> <msub> <mi>Y</mi> <mi>q</mi> </msub> <msubsup> <mi>F</mi> <mi>i</mi> <mo>&amp;prime;</mo> </msubsup> <msubsup> <mi>F</mi> <mi>j</mi> <mo>&amp;prime;</mo> </msubsup> </mrow>
<mrow> <msubsup> <mi>K</mi> <mrow> <mi>m</mi> <mi>n</mi> <mi>p</mi> <mi>q</mi> <mi>i</mi> <mi>j</mi> </mrow> <mrow> <mi>a</mi> <mi>b</mi> </mrow> </msubsup> <mo>=</mo> <msub> <mi>D</mi> <mn>12</mn> </msub> <msubsup> <mi>X</mi> <mi>m</mi> <mo>&amp;prime;</mo> </msubsup> <msub> <mi>X</mi> <mi>n</mi> </msub> <msub> <mi>Y</mi> <mi>p</mi> </msub> <msubsup> <mi>Y</mi> <mi>q</mi> <mo>&amp;prime;</mo> </msubsup> <msub> <mi>F</mi> <mi>i</mi> </msub> <msub> <mi>F</mi> <mi>j</mi> </msub> <mo>+</mo> <msub> <mi>D</mi> <mn>44</mn> </msub> <msub> <mi>X</mi> <mi>m</mi> </msub> <msubsup> <mi>X</mi> <mi>n</mi> <mo>&amp;prime;</mo> </msubsup> <msubsup> <mi>Y</mi> <mi>p</mi> <mo>&amp;prime;</mo> </msubsup> <msub> <mi>Y</mi> <mi>q</mi> </msub> <msub> <mi>F</mi> <mi>i</mi> </msub> <msub> <mi>F</mi> <mi>j</mi> </msub> </mrow>
<mrow> <msubsup> <mi>K</mi> <mrow> <mi>m</mi> <mi>n</mi> <mi>p</mi> <mi>q</mi> <mi>i</mi> <mi>j</mi> </mrow> <mrow> <mi>a</mi> <mi>c</mi> </mrow> </msubsup> <mo>=</mo> <msub> <mi>D</mi> <mn>13</mn> </msub> <msubsup> <mi>X</mi> <mi>m</mi> <mo>&amp;prime;</mo> </msubsup> <msub> <mi>X</mi> <mi>n</mi> </msub> <msub> <mi>Y</mi> <mi>p</mi> </msub> <msub> <mi>Y</mi> <mi>q</mi> </msub> <msub> <mi>F</mi> <mi>i</mi> </msub> <msubsup> <mi>F</mi> <mi>j</mi> <mo>&amp;prime;</mo> </msubsup> <mo>+</mo> <msub> <mi>D</mi> <mn>55</mn> </msub> <msub> <mi>X</mi> <mi>m</mi> </msub> <msubsup> <mi>X</mi> <mi>n</mi> <mo>&amp;prime;</mo> </msubsup> <msub> <mi>Y</mi> <mi>p</mi> </msub> <msub> <mi>Y</mi> <mi>q</mi> </msub> <msubsup> <mi>F</mi> <mi>i</mi> <mo>&amp;prime;</mo> </msubsup> <msub> <mi>F</mi> <mi>j</mi> </msub> </mrow>
<mrow> <msubsup> <mi>K</mi> <mrow> <mi>m</mi> <mi>n</mi> <mi>p</mi> <mi>q</mi> <mi>i</mi> <mi>j</mi> </mrow> <mrow> <mi>b</mi> <mi>a</mi> </mrow> </msubsup> <mo>=</mo> <msub> <mi>D</mi> <mn>21</mn> </msub> <msub> <mi>X</mi> <mi>m</mi> </msub> <msubsup> <mi>X</mi> <mi>n</mi> <mo>&amp;prime;</mo> </msubsup> <msubsup> <mi>Y</mi> <mi>p</mi> <mo>&amp;prime;</mo> </msubsup> <msub> <mi>Y</mi> <mi>q</mi> </msub> <msub> <mi>F</mi> <mi>i</mi> </msub> <msub> <mi>F</mi> <mi>j</mi> </msub> <mo>+</mo> <msub> <mi>D</mi> <mn>44</mn> </msub> <msubsup> <mi>X</mi> <mi>m</mi> <mo>&amp;prime;</mo> </msubsup> <msub> <mi>X</mi> <mi>n</mi> </msub> <msub> <mi>Y</mi> <mi>p</mi> </msub> <msubsup> <mi>Y</mi> <mi>q</mi> <mo>&amp;prime;</mo> </msubsup> <msub> <mi>F</mi> <mi>i</mi> </msub> <msub> <mi>F</mi> <mi>j</mi> </msub> </mrow>
<mrow> <msubsup> <mi>K</mi> <mrow> <mi>m</mi> <mi>n</mi> <mi>p</mi> <mi>q</mi> <mi>i</mi> <mi>j</mi> </mrow> <mrow> <mi>b</mi> <mi>b</mi> </mrow> </msubsup> <mo>=</mo> <msub> <mi>D</mi> <mn>22</mn> </msub> <msub> <mi>X</mi> <mi>m</mi> </msub> <msub> <mi>X</mi> <mi>n</mi> </msub> <msubsup> <mi>Y</mi> <mi>p</mi> <mo>&amp;prime;</mo> </msubsup> <msubsup> <mi>Y</mi> <mi>q</mi> <mo>&amp;prime;</mo> </msubsup> <msub> <mi>F</mi> <mi>i</mi> </msub> <msub> <mi>F</mi> <mi>j</mi> </msub> <mo>+</mo> <msub> <mi>D</mi> <mn>44</mn> </msub> <msubsup> <mi>X</mi> <mi>m</mi> <mo>&amp;prime;</mo> </msubsup> <msubsup> <mi>X</mi> <mi>n</mi> <mo>&amp;prime;</mo> </msubsup> <msub> <mi>Y</mi> <mi>p</mi> </msub> <msub> <mi>Y</mi> <mi>q</mi> </msub> <msub> <mi>F</mi> <mi>i</mi> </msub> <msub> <mi>F</mi> <mi>j</mi> </msub> <mo>+</mo> <msub> <mi>D</mi> <mn>66</mn> </msub> <msub> <mi>X</mi> <mi>m</mi> </msub> <msub> <mi>X</mi> <mi>n</mi> </msub> <msub> <mi>Y</mi> <mi>p</mi> </msub> <msub> <mi>Y</mi> <mi>q</mi> </msub> <msubsup> <mi>F</mi> <mi>i</mi> <mo>&amp;prime;</mo> </msubsup> <msubsup> <mi>F</mi> <mi>j</mi> <mo>&amp;prime;</mo> </msubsup> </mrow>
<mrow> <msubsup> <mi>K</mi> <mrow> <mi>m</mi> <mi>n</mi> <mi>p</mi> <mi>q</mi> <mi>i</mi> <mi>j</mi> </mrow> <mrow> <mi>b</mi> <mi>c</mi> </mrow> </msubsup> <mo>=</mo> <msub> <mi>D</mi> <mn>23</mn> </msub> <msub> <mi>X</mi> <mi>m</mi> </msub> <msub> <mi>X</mi> <mi>n</mi> </msub> <msubsup> <mi>Y</mi> <mi>p</mi> <mo>&amp;prime;</mo> </msubsup> <msub> <mi>Y</mi> <mi>q</mi> </msub> <msub> <mi>F</mi> <mi>i</mi> </msub> <msubsup> <mi>F</mi> <mi>j</mi> <mo>&amp;prime;</mo> </msubsup> <mo>+</mo> <msub> <mi>D</mi> <mn>66</mn> </msub> <msub> <mi>X</mi> <mi>m</mi> </msub> <msub> <mi>X</mi> <mi>n</mi> </msub> <msub> <mi>Y</mi> <mi>p</mi> </msub> <msubsup> <mi>Y</mi> <mi>q</mi> <mo>&amp;prime;</mo> </msubsup> <msubsup> <mi>F</mi> <mi>i</mi> <mo>&amp;prime;</mo> </msubsup> <msub> <mi>F</mi> <mi>j</mi> </msub> </mrow>
<mrow> <msubsup> <mi>K</mi> <mrow> <mi>m</mi> <mi>n</mi> <mi>p</mi> <mi>q</mi> <mi>i</mi> <mi>j</mi> </mrow> <mrow> <mi>c</mi> <mi>a</mi> </mrow> </msubsup> <mo>=</mo> <msub> <mi>D</mi> <mn>31</mn> </msub> <msub> <mi>X</mi> <mi>m</mi> </msub> <msubsup> <mi>X</mi> <mi>n</mi> <mo>&amp;prime;</mo> </msubsup> <msub> <mi>Y</mi> <mi>p</mi> </msub> <msub> <mi>Y</mi> <mi>q</mi> </msub> <msubsup> <mi>F</mi> <mi>i</mi> <mo>&amp;prime;</mo> </msubsup> <msub> <mi>F</mi> <mi>j</mi> </msub> <mo>+</mo> <msub> <mi>D</mi> <mn>55</mn> </msub> <msubsup> <mi>X</mi> <mi>m</mi> <mo>&amp;prime;</mo> </msubsup> <msub> <mi>X</mi> <mi>n</mi> </msub> <msub> <mi>Y</mi> <mi>p</mi> </msub> <msub> <mi>Y</mi> <mi>q</mi> </msub> <msub> <mi>F</mi> <mi>i</mi> </msub> <msubsup> <mi>F</mi> <mi>j</mi> <mo>&amp;prime;</mo> </msubsup> </mrow>
<mrow> <msubsup> <mi>K</mi> <mrow> <mi>m</mi> <mi>n</mi> <mi>p</mi> <mi>q</mi> <mi>i</mi> <mi>j</mi> </mrow> <mrow> <mi>c</mi> <mi>b</mi> </mrow> </msubsup> <mo>=</mo> <msub> <mi>D</mi> <mn>32</mn> </msub> <msub> <mi>X</mi> <mi>m</mi> </msub> <msub> <mi>X</mi> <mi>n</mi> </msub> <msub> <mi>Y</mi> <mi>p</mi> </msub> <msubsup> <mi>Y</mi> <mi>q</mi> <mo>&amp;prime;</mo> </msubsup> <msubsup> <mi>F</mi> <mi>i</mi> <mo>&amp;prime;</mo> </msubsup> <msub> <mi>F</mi> <mi>j</mi> </msub> <mo>+</mo> <msub> <mi>D</mi> <mn>66</mn> </msub> <msub> <mi>X</mi> <mi>m</mi> </msub> <msub> <mi>X</mi> <mi>n</mi> </msub> <msubsup> <mi>Y</mi> <mi>p</mi> <mo>&amp;prime;</mo> </msubsup> <msub> <mi>Y</mi> <mi>q</mi> </msub> <msub> <mi>F</mi> <mi>i</mi> </msub> <msubsup> <mi>F</mi> <mi>j</mi> <mo>&amp;prime;</mo> </msubsup> </mrow>
<mrow> <msubsup> <mi>K</mi> <mrow> <mi>m</mi> <mi>n</mi> <mi>p</mi> <mi>q</mi> <mi>i</mi> <mi>j</mi> </mrow> <mrow> <mi>c</mi> <mi>c</mi> </mrow> </msubsup> <mo>=</mo> <msub> <mi>D</mi> <mn>33</mn> </msub> <msub> <mi>X</mi> <mi>m</mi> </msub> <msub> <mi>X</mi> <mi>n</mi> </msub> <msub> <mi>Y</mi> <mi>p</mi> </msub> <msub> <mi>Y</mi> <mi>q</mi> </msub> <msubsup> <mi>F</mi> <mi>i</mi> <mo>&amp;prime;</mo> </msubsup> <msubsup> <mi>F</mi> <mi>j</mi> <mo>&amp;prime;</mo> </msubsup> <mo>+</mo> <msub> <mi>D</mi> <mn>55</mn> </msub> <msubsup> <mi>X</mi> <mi>m</mi> <mo>&amp;prime;</mo> </msubsup> <msubsup> <mi>X</mi> <mi>n</mi> <mo>&amp;prime;</mo> </msubsup> <msub> <mi>Y</mi> <mi>p</mi> </msub> <msub> <mi>Y</mi> <mi>q</mi> </msub> <msub> <mi>F</mi> <mi>i</mi> </msub> <msub> <mi>F</mi> <mi>j</mi> </msub> <mo>+</mo> <msub> <mi>D</mi> <mn>66</mn> </msub> <msub> <mi>X</mi> <mi>m</mi> </msub> <msub> <mi>X</mi> <mi>n</mi> </msub> <msubsup> <mi>Y</mi> <mi>p</mi> <mo>&amp;prime;</mo> </msubsup> <msubsup> <mi>Y</mi> <mi>q</mi> <mo>&amp;prime;</mo> </msubsup> <msub> <mi>F</mi> <mi>i</mi> </msub> <msub> <mi>F</mi> <mi>j</mi> </msub> </mrow>
<mrow> <msubsup> <mi>M</mi> <mrow> <mi>m</mi> <mi>n</mi> <mi>p</mi> <mi>q</mi> <mi>i</mi> <mi>j</mi> </mrow> <mrow> <mi>a</mi> <mi>a</mi> </mrow> </msubsup> <mo>=</mo> <msubsup> <mi>M</mi> <mrow> <mi>m</mi> <mi>n</mi> <mi>p</mi> <mi>q</mi> <mi>i</mi> <mi>j</mi> </mrow> <mrow> <mi>b</mi> <mi>b</mi> </mrow> </msubsup> <mo>=</mo> <msubsup> <mi>M</mi> <mrow> <mi>m</mi> <mi>n</mi> <mi>p</mi> <mi>q</mi> <mi>i</mi> <mi>j</mi> </mrow> <mrow> <mi>c</mi> <mi>c</mi> </mrow> </msubsup> <mo>=</mo> <msub> <mi>&amp;rho;X</mi> <mi>m</mi> </msub> <msub> <mi>X</mi> <mi>n</mi> </msub> <msub> <mi>Y</mi> <mi>p</mi> </msub> <msub> <mi>Y</mi> <mi>q</mi> </msub> <msub> <mi>F</mi> <mi>i</mi> </msub> <msub> <mi>F</mi> <mi>j</mi> </msub> </mrow>
<mrow> <msubsup> <mi>M</mi> <mrow> <mi>m</mi> <mi>n</mi> <mi>p</mi> <mi>q</mi> <mi>i</mi> <mi>j</mi> </mrow> <mrow> <mi>a</mi> <mi>b</mi> </mrow> </msubsup> <mo>=</mo> <msubsup> <mi>M</mi> <mrow> <mi>m</mi> <mi>n</mi> <mi>p</mi> <mi>q</mi> <mi>i</mi> <mi>j</mi> </mrow> <mrow> <mi>a</mi> <mi>c</mi> </mrow> </msubsup> <mo>=</mo> <msubsup> <mi>M</mi> <mrow> <mi>m</mi> <mi>n</mi> <mi>p</mi> <mi>q</mi> <mi>i</mi> <mi>j</mi> </mrow> <mrow> <mi>b</mi> <mi>a</mi> </mrow> </msubsup> <mo>=</mo> <msubsup> <mi>M</mi> <mrow> <mi>m</mi> <mi>n</mi> <mi>p</mi> <mi>q</mi> <mi>i</mi> <mi>j</mi> </mrow> <mrow> <mi>b</mi> <mi>c</mi> </mrow> </msubsup> <mo>=</mo> <msubsup> <mi>M</mi> <mrow> <mi>m</mi> <mi>n</mi> <mi>p</mi> <mi>q</mi> <mi>i</mi> <mi>j</mi> </mrow> <mrow> <mi>c</mi> <mi>a</mi> </mrow> </msubsup> <mo>=</mo> <msubsup> <mi>M</mi> <mrow> <mi>m</mi> <mi>n</mi> <mi>p</mi> <mi>q</mi> <mi>i</mi> <mi>j</mi> </mrow> <mrow> <mi>c</mi> <mi>b</mi> </mrow> </msubsup> <mo>=</mo> <mn>0</mn> </mrow>
Wherein KmnpqijFor core rigidity matrix, MmnpqijFor core mass matrix, superscript a, b and c are element in kernel matrix Label；Subscript m, n=1 ..., M+3；P, q=1 ..., P+3；I, j=1 ..., N+1；X′m, Y 'pWith F 'iX is represented respectivelym, YpAnd FiFirst derivative, similarly X 'n, Y 'qWith F 'jX is represented respectivelyn, YqAnd FjFirst derivative；D11,…,D66For structural material Element in coefficient matrix D；
Step 7 tries to achieve overall mass matrix, stiffness matrix and oeverall quality matrix M by iterative cycles kernel matrix, And then obtain the characteristic equation of structure；
The method for solving of the mass matrix and stiffness matrix is：Pointer i, j get N+1 circulation core rigidity matrix Ks by 1mnpqij Obtain sub- submatrix Kmnpq, pointer p, q get P+3 by 1 and circulate sub- submatrix KmnpqObtain submatrix Kmn, pointer m, n are got by 1 M+3 circulation submatrixs KmnObtain global stiffness matrix K；
The characteristic equation expression formula of the structure is：
(K-ω2M) A=0
Wherein ω is circular frequency, and A is corresponding ω characteristic vector；
Step 8 calculates the intrinsic frequency of elastic plate, according to the vibration shape of characteristic vector export structure.
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