CN105675484A - Method for determining elastic energy of circular membrane with rigid slab in center and under uniformly distributed load - Google Patents

Abstract

The invention discloses a method for determining elastic energy of a circular membrane with a rigid slab in the center and under a uniformly distributed load. The method is characterized in that a clamping device with the internal radius of a is adopted to fixedly clamp a circular membrane with the thickness of h, the Young's elastic modulus of E and the Poisson's ratio of v, so as to form an axisymmetric circular membrane with the external radius of a and the internal radius of b, wherein a rigid slab with the radius of b is arranged in the center of the circular membrane, the periphery of the axisymmetric circular membrane is fixedly clamped, and the rigid slab is arranged in the center of the axisymmetric circular membrane; a uniformly distributed load q is transversely exerted on the axisymmetric circular membrane; based on static equilibrium analysis and energy equilibrium analysis of the axisymmetrical deformation problem, and by use of the measured value of the uniformly distributed load q, the elastic energy U of the circular membrane can be determined.

Description

The defining method of the annular membrane elasticity energy of center band rigid plate under uniform load

Technical field

The present invention relates to the defining method of the annular membrane elasticity of center band rigid plate under uniform load energy.

Background technology

Under Uniform Loads, the analytic solutions of the annular membrane On Axisymmetric Deformation of A of center band rigid plate, significant to the development of sensor and instrument, instrument. Owing to thin film is flexible material, typically exhibiting out bigger amount of deflection under Uniform Loads, thus its problem on deformation has stronger non-linear, these nonlinear problems are generally difficult to Analytical Solution. This invention address that the analysis research of the annular membrane On Axisymmetric Deformation of A of center band rigid plate under uniform load, it is thus achieved that the analytic solutions of this problem, and give the defining method of the annular membrane elasticity energy of center band rigid plate under uniform load on this basis.

Summary of the invention

The defining method of the annular membrane elasticity energy of center band rigid plate under uniform load: adopting inside radius is the clamping device of a, it is h by thickness, Young's modulus of elasticity is E, Poisson's ratio is ν, the annular membrane that center band radius is the rigid plate of b fixes to clamp, forming an outer radius is a, inside radius is the axial symmetry annular membrane of the center band rigid plate that the periphery of b fixes to clamp, it is laterally applied to a uniform load q, standing balance analysis based on this On Axisymmetric Deformation of A, can be obtained by the analytic solutions of this problem, owing to heat consumption produced in deformation of thin membrane process is very little, therefore can be approximately considered uniform load q work done and be completely converted into the elasticity energy of thin film, so according to principle of energy balance, utilize the analytic solutions that standing balance analysis obtains, the parsing relation of the elasticity energy U and uniform load q that then can obtain this annular membrane is

$U=\mathrm{\π}{\left(\frac{a^4{q}^4}{hE}\right)}^{1/3}\underset{n=0}{\overset{10}{\Σ}}\[\frac{4a^2{c}_n}{n+2}{\left(\frac{a-b}{2a}\right)}^{n+2}+\frac{2a(a+b)c_n}{n+1}{\left(\frac{a-b}{2a}\right)}^{n+1}+b^2{c}_n{\left(\frac{b-a}{2a}\right)}^n\],$

Wherein,

$c_3=\frac{1}{6}\frac{1}{{\mathrm{\β}}^3{c}_1}(8{\mathrm{\β}}^3{c}_2^2-10{\mathrm{\β}}^2{c}_1{c}_2+3{\mathrm{\βc}}_1^2-c_1^5),$

$c_4=\frac{1}{24}\frac{1}{{\mathrm{\β}}^4{c}_1^2}(48{\mathrm{\β}}^4{c}_2^3-120{\mathrm{\β}}^3{c}_1{c}_2^2+90{\mathrm{\β}}^2{c}_1^2{c}_2-16{\mathrm{\βc}}_1^5{c}_2-21{\mathrm{\βc}}_1^3+8c_1^6),$

$\begin{array}{c}{c}_5=\frac{1}{120}\frac{1}{{\mathrm{\β}}^6{c}_1^3}(384{\mathrm{\β}}^6{c}_2^4-1440{\mathrm{\β}}^5{c}_1{c}_2^3+1896{\mathrm{\β}}^4{c}_1^2{c}_2^2-232{\mathrm{\β}}^3{c}_1^5{c}_2^2-1032{\mathrm{\β}}^3{c}_1^3{c}_2\\ +312{\mathrm{\β}}^2{c}_1^6{c}_2+198{\mathrm{\β}}^2{c}_1^4-101{\mathrm{\βc}}_1^7+8c_1^{10})\end{array},$

$\begin{array}{c}{c}_6=\frac{1}{720}\frac{1}{{\mathrm{\β}}^7{c}_1^4}(3840{\mathrm{\β}}^7{c}_2^5-19200{\mathrm{\β}}^6{c}_1{c}_2^4+3670{\mathrm{\β}}^5{c}_1^2{c}_2^3-3552{\mathrm{\β}}^5{c}_1^5{c}_2^3\\ -33360{\mathrm{\β}}^4{c}_1^3{c}_2^2+8296{\mathrm{\β}}^3{c}_1^6{c}_2^2+14340{\mathrm{\β}}^3{c}_1^4{c}_2-6208{\mathrm{\β}}^2{c}_1^7{c}_2\\ -2340{\mathrm{\β}}^2{c}_1^5+344{\mathrm{\βc}}_1^{10}{c}_2+1489{\mathrm{\βc}}_1^8-204c_1^{11})\end{array},$

$\begin{array}{c}{c}_7=\frac{1}{2520}\frac{1}{{\mathrm{\β}}^9{c}_1^5}(23040{\mathrm{\β}}^9{c}_2^6-144000{\mathrm{\β}}^8{c}_1{c}_2^5+362880{\mathrm{\β}}^7{c}_1^2{c}_2^4-29664{\mathrm{\β}}^6{c}_1^5{c}_2^4\\ -469800{\mathrm{\β}}^6{c}_1^3{c}_2^3+100944{\mathrm{\β}}^5{c}_1^6{c}_2^3+328140{\mathrm{\β}}^5{c}_1^4{c}_2^2-124644{\mathrm{\β}}^4{c}_1^7{c}_2^2\\ -116910{\mathrm{\β}}^4{c}_1^5{c}_2+5416{\mathrm{\β}}^3{c}_1^{10}{c}_2^2+66120{\mathrm{\β}}^3{c}_1^8{c}_2+16605{\mathrm{\β}}^3{c}_1^6\\ -7468{\mathrm{\β}}^2{c}_1^{11}{c}_2-12708{\mathrm{\β}}^2{c}_1^9+2524{\mathrm{\βc}}_1^{12}-86c_1^{15})\end{array},$

$\begin{array}{c}{c}_8=\frac{1}{20160}\frac{1}{{\mathrm{\β}}^{10}{c}_1^6}(322560{\mathrm{\β}}^{10}{c}_2^7-2419200{\mathrm{\β}}^9{c}_1{c}_2^6+7580160{\mathrm{\β}}^8{c}_1^2{c}_2^5-543744{\mathrm{\β}}^7{c}_1^5{c}_2^5\\ -12821760{\mathrm{\β}}^7{c}_1^3{c}_2^4+2455488{\mathrm{\β}}^6{c}_1^6{c}_2^4+12602520{\mathrm{\β}}^6{c}_1^4{c}_2^3-4317408{\mathrm{\β}}^5{c}_1^7{c}_2^3\\ -7175700{\mathrm{\β}}^5{c}_1^5{c}_2^2+156816{\mathrm{\β}}^4{c}_1^{10}{c}_2^3+3689808{\mathrm{\β}}^4{c}_1^8{c}_2^2+2186730{\mathrm{\β}}^4{c}_1^6{c}_2\\ -357560{\mathrm{\β}}^3{c}_1^{11}{c}_2^2-1531056{\mathrm{\β}}^3{c}_1^9{c}_2-274995{\mathrm{\β}}^3{c}_1^7+265844{\mathrm{\β}}^2{c}_1^{12}{c}_2\\ +246591{\mathrm{\β}}^2{c}_1^{10}-7136{\mathrm{\βc}}_1^{15}{c}_2-64454{\mathrm{\βc}}_1^{13}+4508c_1^{16})\end{array},$

$\begin{array}{c}{c}_9=\frac{1}{181440}\frac{1}{{\mathrm{\β}}^{12}{c}_1^7}(5160960{\mathrm{\β}}^{12}{c}_2^8-45158400{\mathrm{\β}}^{11}{c}_1{c}_2^7+169344000{\mathrm{\β}}^{10}{c}_1^2{c}_2^6\\ -10916352{\mathrm{\β}}^9{c}_1^5{c}_2^6-354654720{\mathrm{\β}}^9{c}_1^3{c}_2^5+61770240{\mathrm{\β}}^8{c}_1^6{c}_2^5+452571840{\mathrm{\β}}^8{c}_1^4{c}_2^4\\ -142403904{\mathrm{\β}}^7{c}_1^7{c}_2^4-359432640{\mathrm{\β}}^7{c}_1^5{c}_2^3+4495680{\mathrm{\β}}^6{c}_1^{10}{c}_2^4+171002304{\mathrm{\β}}^6{c}_1^8{c}_2^3\\ +173093760{\mathrm{\β}}^6{c}_1^6{c}_2^2-14640192{\mathrm{\β}}^5{c}_1^{11}{c}_2^3-112679604{\mathrm{\β}}^5{c}_1^9{c}_2^2-46131120{\mathrm{\β}}^5{c}_1^7{c}_2\\ +17513808{\mathrm{\β}}^4{c}_1^{12}{c}_2^2+38590524{\mathrm{\β}}^4{c}_1^{10}{c}_2+5205060{\mathrm{\β}}^4{c}_1^8-392216{\mathrm{\β}}^3{c}_1^{15}{c}_2^2\\ -120816{\mathrm{\β}}^3{c}_1^{13}{c}_2-5362677{\mathrm{\β}}^3{c}_1^{11}+547624{\mathrm{\β}}^2{c}_1^{16}{c}_2+1744380{\mathrm{\β}}^2{c}_1^{14}\\ -188706{\mathrm{\βc}}_1^{17}+3568c_1^{20})\end{array},$

$\begin{array}{c}{c}_{10}=\frac{1}{181440}\frac{1}{{\mathrm{\β}}^{13}{c}_1^8}(92897280{\mathrm{\β}}^{13}{c}_2^9-92897280{\mathrm{\β}}^{12}{c}_1{c}_2^8+4058449920{\mathrm{\β}}^{11}{c}_1^2{c}_2^7\\ -238970880{\mathrm{\β}}^{10}{c}_1^5{c}_2^7-10149027840{\mathrm{\β}}^{10}{c}_1^3{c}_2^6+1630158336{\mathrm{\β}}^9{c}_1^6{c}_2^6\\ +15980146560{\mathrm{\β}}^9{c}_1^4{c}_2^5-4676078592{\mathrm{\β}}^8{c}_1^7{c}_2^5-16397095680{\mathrm{\β}}^8{c}_1^5{c}_2^4\\ +131654016{\mathrm{\β}}^7{c}_1^{10}{c}_2^5+7304043456{\mathrm{\β}}^7{c}_1^8{c}_2^4+109426400{\mathrm{\β}}^7{c}_1^6{c}_2^3\\ -564117120{\mathrm{\β}}^6{c}_1^{11}{c}_2^4-6702770736{\mathrm{\β}}^6{c}_1^9{c}_2^3-4571471520{\mathrm{\β}}^6{c}_1^7{c}_2^2\\ +949114656{\mathrm{\β}}^5{c}_1^{12}{c}_2^3+3610117836{\mathrm{\β}}^5{c}_1^{10}{c}_2^2+1083264840{\mathrm{\β}}^5{c}_1^8{c}_2\\ -18404544{\mathrm{\β}}^4{c}_1^{15}{c}_2^3-783565920{\mathrm{\β}}^4{c}_1^{13}{c}_2^2-1055710800{\mathrm{\β}}^4{c}_1^{11}{c}_2\\ -110837160{\mathrm{\β}}^4{c}_1^9+41460120{\mathrm{\β}}^3{c}_1^{16}{c}_2^2+317333772{\mathrm{\β}}^3{c}_1^{14}{c}_2\\ +129215439{\mathrm{\β}}^3{c}_1^{12}-30678936{\mathrm{\β}}^2{c}_1^{17}{c}_2-50420286{\mathrm{\β}}^2{c}_1^{15}\\ +484984{\mathrm{\βc}}_1^{20}{c}_2+7457610{\mathrm{\βc}}_1^{18}-316628c_1^{21})\end{array},$

And β=(a+b)/2a, c₁And c₂Value by equation

$\mathrm{\ν}=2-\frac{b}{a}\[\underset{i=2}{\overset{10}{\Σ}}i(i-1)c_i{(\frac{b}{a}-\mathrm{\β})}^{i-2}\]/\[\underset{j=1}{\overset{10}{\Σ}}{\mathrm{jc}}_j{(\frac{b}{a}-\mathrm{\β})}^{j-1}\]$

With

$\mathrm{\ν}=2-\frac{b}{a}\[\underset{i=2}{\overset{10}{\Σ}}i(i-1)c_i{(1-\mathrm{\β})}^{i-2}\]/\[\underset{j=1}{\overset{10}{\Σ}}{\mathrm{jc}}_j{(1-\mathrm{\β})}^{j-1}\]$

Determine, c₀Value by equation

$c_0=-\underset{j=1}{\overset{10}{\Σ}}{c}_j{(1-\mathrm{\β})}^j$

Determine.

So, as long as accurately measuring the value of the uniform load q applied, it is possible to determine the elasticity energy U after the deformation of this annular membrane. Wherein, all parameters all adopt the International System of Units.

Accompanying drawing explanation

Fig. 1 is the loading organigram of the annular membrane of the center band rigid plate that uniform load following peripheral fixes to clamp, wherein, 1-annular membrane, 2-rigid plate, 3-clamping device, and a represents the inside radius of clamping device and the outer radius of annular membrane, b represents the radius of rigid plate and the inside radius of annular membrane, and r represents radial coordinate, and w (r) represents the lateral coordinates at some r place, q represents horizontal uniform load, w_mRepresent the maximum defluxion of thin film.

Detailed description of the invention

Below in conjunction with accompanying drawing, technical scheme is described in further detail:

As shown in Figure 1, clamping device with an inside radius a=20mm, by thickness h=0.06mm, Young's modulus of lasticity E=7.84MPa, Poisson's ratio ν=0.47, center band radius b=5mm the rubber film of rigid plate fix to clamp, form an outer radius a=20mm, the axial symmetry annular membrane of center band rigid plate that the periphery of inside radius b=5mm fixes to clamp, it is laterally applied to uniform load q, and records q=0.01MPa. Adopt the method given by the present invention, pass through equation

$\mathrm{\ν}=2-\frac{b}{a}\[\underset{i=2}{\overset{10}{\Σ}}i(i-1)c_i{(\frac{b}{a}-\mathrm{\β})}^{i-2}\]/\[\underset{j=1}{\overset{10}{\Σ}}{\mathrm{jc}}_j{(\frac{b}{a}-\mathrm{\β})}^{j-1}\]$

With

$\mathrm{\ν}=2-\frac{b}{a}\[\underset{i=2}{\overset{10}{\Σ}}i(i-1)c_i{(1-\mathrm{\β})}^{i-2}\]/\[\underset{j=1}{\overset{10}{\Σ}}{\mathrm{jc}}_j{(1-\mathrm{\β})}^{j-1}\]$

Then can obtain c₁=-0.7026607165, c₂=-0.6927450924, wherein,

β=(a+b)/2a=0.625,

$c_3=\frac{1}{6}\frac{1}{{\mathrm{\β}}^3{c}_1}(8{\mathrm{\β}}^3{c}_2^2-10{\mathrm{\β}}^2{c}_1{c}_2+3{\mathrm{\βc}}_1^2-c_1^5),$

$c_4=\frac{1}{24}\frac{1}{{\mathrm{\β}}^4{c}_1^2}(48{\mathrm{\β}}^4{c}_2^3-120{\mathrm{\β}}^3{c}_1{c}_2^2+90{\mathrm{\β}}^2{c}_1^2{c}_2-16{\mathrm{\βc}}_1^5{c}_2-21{\mathrm{\βc}}_1^3+8c_1^6),$

$\begin{array}{c}{c}_5=\frac{1}{120}\frac{1}{{\mathrm{\β}}^6{c}_1^3}(384{\mathrm{\β}}^6{c}_2^4-1440{\mathrm{\β}}^5{c}_1{c}_2^3+1896{\mathrm{\β}}^4{c}_1^2{c}_2^2-232{\mathrm{\β}}^3{c}_1^5{c}_2^2-1032{\mathrm{\β}}^3{c}_1^3{c}_2\\ +312{\mathrm{\β}}^2{c}_1^6{c}_2+198{\mathrm{\β}}^2{c}_1^4-101{\mathrm{\βc}}_1^7+8c_1^{10})\end{array},$

$\begin{array}{c}{c}_6=\frac{1}{720}\frac{1}{{\mathrm{\β}}^7{c}_1^4}(3840{\mathrm{\β}}^7{c}_2^5-19200{\mathrm{\β}}^6{c}_1{c}_2^4+36720{\mathrm{\β}}^5{c}_1^2{c}_2^3-3552{\mathrm{\β}}^4{c}_1^5{c}_2^3\\ -33360{\mathrm{\β}}^4{c}_1^3{c}_2^2+8296{\mathrm{\β}}^3{c}_1^6{c}_2^2+14340{\mathrm{\β}}^3{c}_1^4{c}_2-6208{\mathrm{\β}}^2{c}_1^7{c}_2\\ -2340{\mathrm{\β}}^2{c}_1^5+344{\mathrm{\βc}}_1^{10}{c}_2+1489{\mathrm{\βc}}_1^8-204c_1^{11})\end{array},$

$\begin{array}{c}{c}_7=\frac{1}{2520}\frac{1}{{\mathrm{\β}}^9{c}_1^5}(23040{\mathrm{\β}}^9{c}_2^6-144000{\mathrm{\β}}^8{c}_1{c}_2^5+362880{\mathrm{\β}}^7{c}_1^2{c}_2^4-29664{\mathrm{\β}}^6{c}_1^5{c}_2^4\\ -469800{\mathrm{\β}}^6{c}_1^3{c}_2^3+100944{\mathrm{\β}}^5{c}_1^6{c}_2^3+328140{\mathrm{\β}}^5{c}_1^4{c}_2^2-124644{\mathrm{\β}}^4{c}_1^7{c}_2^2\\ -116910{\mathrm{\β}}^4{c}_1^5{c}_2+5416{\mathrm{\β}}^3{c}_1^{10}{c}_2^2+66120{\mathrm{\β}}^3{c}_1^8{c}_2+16605{\mathrm{\β}}^3{c}_1^6\\ -7468{\mathrm{\β}}^2{c}_1^{11}{c}_2-1278{\mathrm{\β}}^2{c}_1^9+2524{\mathrm{\βc}}_1^{12}-86c_1^{15})\end{array},$

$\begin{array}{c}{c}_8=\frac{1}{20160}\frac{1}{{\mathrm{\β}}^{10}{c}_1^6}(322560{\mathrm{\β}}^{10}{c}_2^7-2419200{\mathrm{\β}}^9{c}_1{c}_2^6+7580160{\mathrm{\β}}^8{c}_1^2{c}_2^5-543744{\mathrm{\β}}^7{c}_1^5{c}_2^5\\ -12821760{\mathrm{\β}}^7{c}_1^3{c}_2^4+2455488{\mathrm{\β}}^6{c}_1^6{c}_2^4+12602520{\mathrm{\β}}^6{c}_1^4{c}_2^3-4317408{\mathrm{\β}}^5{c}_1^7{c}_2^3\\ -7175700{\mathrm{\β}}^5{c}_1^5{c}_2^2+156816{\mathrm{\β}}^4{c}_1^{10}{c}_2^3+3689808{\mathrm{\β}}^4{c}_1^8{c}_2^2+2186730{\mathrm{\β}}^4{c}_1^6{c}_2\\ -357560{\mathrm{\β}}^3{c}_1^{11}{c}_2^2-1531056{\mathrm{\β}}^3{c}_1^9{c}_2-274995{\mathrm{\β}}^3{c}_1^7+265844{\mathrm{\β}}^2{c}_1^{12}{c}_2\\ +246591{\mathrm{\β}}^2{c}_1^{10}-7136{\mathrm{\βc}}_1^{15}{c}_2-64454{\mathrm{\βc}}_1^{13}+4508c_1^{16})\end{array},$

$\begin{array}{c}{c}_9=\frac{1}{181440}\frac{1}{{\mathrm{\β}}^{12}{c}_1^7}(5160960{\mathrm{\β}}^{12}{c}_2^8-45158400{\mathrm{\β}}^{11}{c}_1{c}_2^7+169344000{\mathrm{\β}}^{10}{c}_1^2{c}_2^6\\ -10916352{\mathrm{\β}}^9{c}_1^5{c}_2^6-354654720{\mathrm{\β}}^9{c}_1^3{c}_2^5+61770240{\mathrm{\β}}^8{c}_1^6{c}_2^5+452571840{\mathrm{\β}}^8{c}_1^4{c}_2^4\\ -142403904{\mathrm{\β}}^7{c}_1^7{c}_2^4-359432640{\mathrm{\β}}^7{c}_1^5{c}_2^3+4495680{\mathrm{\β}}^6{c}_1^{10}{c}_2^4+171002304{\mathrm{\β}}^6{c}_1^8{c}_2^3\\ +173093760{\mathrm{\β}}^6{c}_1^6{c}_2^2-14640192{\mathrm{\β}}^5{c}_1^{11}{c}_2^3-112679604{\mathrm{\β}}^5{c}_1^9{c}_2^2-46131120{\mathrm{\β}}^5{c}_1^7{c}_2\\ +17513808{\mathrm{\β}}^4{c}_1^{12}{c}_2^2+38590524{\mathrm{\β}}^4{c}_1^{10}{c}_2+5205060{\mathrm{\β}}^4{c}_1^8-392216{\mathrm{\β}}^3{c}_1^{15}{c}_2^2\\ -120816{\mathrm{\β}}^3{c}_1^{13}{c}_2-5362677{\mathrm{\β}}^3{c}_1^{11}+547624{\mathrm{\β}}^2{c}_1^{16}{c}_2+1744380{\mathrm{\β}}^2{c}_1^{14}\\ -188706{\mathrm{\βc}}_1^{17}+3568c_1^{20})\end{array},$

$\begin{array}{c}{c}_{10}=\frac{1}{181440}\frac{1}{{\mathrm{\β}}^{13}{c}_1^8}(92897280{\mathrm{\β}}^{13}{c}_2^9-92897280{\mathrm{\β}}^{12}{c}_1{c}_2^8+4058449920{\mathrm{\β}}^{11}{c}_1^2{c}_2^7\\ -238970880{\mathrm{\β}}^{10}{c}_1^5{c}_2^7-10149027840{\mathrm{\β}}^{10}{c}_1^3{c}_2^6+1630158336{\mathrm{\β}}^9{c}_1^6{c}_2^6\\ +15980146560{\mathrm{\β}}^9{c}_1^4{c}_2^5-4676078592{\mathrm{\β}}^8{c}_1^7{c}_2^5-16397095680{\mathrm{\β}}^8{c}_1^5{c}_2^4\\ +131654016{\mathrm{\β}}^7{c}_1^{10}{c}_2^5+7304043456{\mathrm{\β}}^7{c}_1^8{c}_2^4+109426400{\mathrm{\β}}^7{c}_1^6{c}_2^3\\ -564117120{\mathrm{\β}}^6{c}_1^{11}{c}_2^4-6702770736{\mathrm{\β}}^6{c}_1^9{c}_2^3-4571471520{\mathrm{\β}}^6{c}_1^7{c}_2^2\\ +949114656{\mathrm{\β}}^5{c}_1^{12}{c}_2^3+3610117836{\mathrm{\β}}^5{c}_1^{10}{c}_2^2+1083264840{\mathrm{\β}}^5{c}_1^8{c}_2\\ -18404544{\mathrm{\β}}^4{c}_1^{15}{c}_2^3-783565920{\mathrm{\β}}^4{c}_1^{13}{c}_2^2-1055710800{\mathrm{\β}}^4{c}_1^{11}{c}_2\\ -110837160{\mathrm{\β}}^4{c}_1^9+41460120{\mathrm{\β}}^3{c}_1^{16}{c}_2^2+317333772{\mathrm{\β}}^3{c}_1^{14}{c}_2\\ +129215439{\mathrm{\β}}^3{c}_1^{12}-30678936{\mathrm{\β}}^2{c}_1^{17}{c}_2-50420286{\mathrm{\β}}^2{c}_1^{15}\\ +484984{\mathrm{\βc}}_1^{20}{c}_2+7457610{\mathrm{\βc}}_1^{18}-316628c_1^{21})\end{array}.$

Then, then by equationC can be obtained₀=0.3710002899. Finally, by equation

$U=\mathrm{\π}{\left(\frac{a^4{q}^4}{hE}\right)}^{1/3}\underset{n=0}{\overset{10}{\Σ}}\[\frac{4a^2{c}_n}{n+2}{\left(\frac{a-b}{2a}\right)}^{n+2}+\frac{2a(a+b)c_n}{n+1}{\left(\frac{a-b}{2a}\right)}^{n+1}+b^2{c}_n{\left(\frac{b-a}{2a}\right)}^n\]$

Then can obtain elasticity energy U=50.2088 × 10 of this annular membrane^-3J。

Claims (1)

1. the defining method of the annular membrane elasticity energy of center band rigid plate under uniform load, it is characterized in that: adopting inside radius is the clamping device of a, the annular membrane of the rigid plate of thickness to be h, Young's modulus of elasticity be E, Poisson's ratio to be ν, center band radius be b is fixed to clamp, form outer radius to be a, inside radius be the axial symmetry annular membrane of the center band rigid plate that the periphery of b fixes to clamp, it is laterally applied to a uniform load q, and record the value of the uniform load q applied, below equation determine the elasticity energy U of this annular membrane:

$U=\mathrm{\π}{\left(\frac{a^4{q}^4}{hE}\right)}^{1/3}\underset{n=0}{\overset{10}{\mathrm{\Σ}}}\[\frac{4a^2{c}_n}{n+2}{\left(\frac{a-b}{2a}\right)}^{n+2}+\frac{2a\left(a+b\right)c_n}{n+1}{\left(\frac{a-b}{2a}\right)}^{n+1}+b^2{c}_n{\left(\frac{b-a}{2a}\right)}^n\],$

Wherein,

$c_3=\frac{1}{6}\frac{1}{{\mathrm{\β}}^3{c}_1}(8{\mathrm{\β}}^3{c}_2^2-10{\mathrm{\β}}^2{c}_1{c}_2+3{\mathrm{\βc}}_1^2-c_1^5),$

$c_4=\frac{1}{24}\frac{1}{{\mathrm{\β}}^4{c}_1^2}(48{\mathrm{\β}}^4{c}_2^3-120{\mathrm{\β}}^3{c}_1{c}_2^2+90{\mathrm{\β}}^2{c}_1^2{c}_2-16{\mathrm{\βc}}_1^5{c}_2-21{\mathrm{\βc}}_1^3+8c_1^6),$

$\begin{array}{c}{c}_5=\frac{1}{120}\frac{1}{{\mathrm{\β}}^6{c}_1^3}(384{\mathrm{\β}}^6{c}_2^4-1440{\mathrm{\β}}^5{c}_1{c}_2^3+1896{\mathrm{\β}}^4{c}_1^2{c}_2^2-232{\mathrm{\β}}^3{c}_1^5{c}_2^2-1032{\mathrm{\β}}^3{c}_1^3{c}_2\\ +312{\mathrm{\β}}^2{c}_1^6{c}_2+198{\mathrm{\β}}^2{c}_1^4-101{\mathrm{\βc}}_1^7+8c_1^{10})\end{array},$

$\begin{array}{c}{c}_6=\frac{1}{720}\frac{1}{{\mathrm{\β}}^7{c}_1^4}(3840{\mathrm{\β}}^7{c}_2^5-19200{\mathrm{\β}}^6{c}_1{c}_2^4+36720{\mathrm{\β}}^5{c}_1^2{c}_2^3-3552{\mathrm{\β}}^4{c}_1^5{c}_2^3\\ -33360{\mathrm{\β}}^4{c}_1^3{c}_2^2+8296{\mathrm{\β}}^3{c}_1^6{c}_2^2+14340{\mathrm{\β}}^3{c}_1^4{c}_2-6208{\mathrm{\β}}^2{c}_1^7{c}_2\\ -2340{\mathrm{\β}}^2{c}_1^5+344{\mathrm{\βc}}_1^{10}{c}_2+1489{\mathrm{\βc}}_1^8-204c_1^{11})\end{array},$

$\begin{array}{c}{c}_7=\frac{1}{2520}\frac{1}{{\mathrm{\β}}^9{c}_1^5}(23040{\mathrm{\β}}^9{c}_2^6-144000{\mathrm{\β}}^8{c}_1{c}_2^5+362880{\mathrm{\β}}^7{c}_1^2{c}_2^4-39664{\mathrm{\β}}^6{c}_1^5{c}_2^4\\ -469800{\mathrm{\β}}^6{c}_1^3{c}_2^3+100944{\mathrm{\β}}^5{c}_1^6{c}_2^3+328140{\mathrm{\β}}^5{c}_1^4{c}_2^2-124644{\mathrm{\β}}^4{c}_1^7{c}_2^2\\ -116910{\mathrm{\β}}^4{c}_1^5{c}_2+5416{\mathrm{\β}}^3{c}_1^{10}{c}_2^2+66120{\mathrm{\β}}^3{c}_1^8{c}_2+16605{\mathrm{\β}}^3{c}_1^6\\ -7468{\mathrm{\β}}^2{c}_1^{11}{c}_2-12708{\mathrm{\β}}^2{c}_1^9+2524{\mathrm{\βc}}_1^{12}-86c_1^{15})\end{array},$

$\begin{array}{c}{c}_8=\frac{1}{20160}\frac{1}{{\mathrm{\β}}^{10}{c}_1^6}(322560{\mathrm{\β}}^{10}{c}_2^7-2419200{\mathrm{\β}}^9{c}_1{c}_2^6+7580160{\mathrm{\β}}^8{c}_1^2{c}_2^5-543744{\mathrm{\β}}^7{c}_1^5{c}_2^5\\ -12821760{\mathrm{\β}}^7{c}_1^3{c}_2^4+2455488{\mathrm{\β}}^6{c}_1^6{c}_2^4+12602520{\mathrm{\β}}^6{c}_1^4{c}_2^3-4317408{\mathrm{\β}}^5{c}_1^7{c}_2^3\\ -7175700{\mathrm{\β}}^5{c}_1^5{c}_2^2+156816{\mathrm{\β}}^4{c}_1^{10}{c}_2^3+3689808{\mathrm{\β}}^4{c}_1^8{c}_2^2+2186730{\mathrm{\β}}^4{c}_1^6{c}_2\\ -357560{\mathrm{\β}}^3{c}_1^{11}{c}_2^2-1531056{\mathrm{\β}}^3{c}_1^9{c}_2-274995{\mathrm{\β}}^3{c}_1^7+265844{\mathrm{\β}}^2{c}_1^{12}{c}_2\\ +246591{\mathrm{\β}}^2{c}_1^{10}-7136{\mathrm{\βc}}_1^{15}{c}_2-64454{\mathrm{\βc}}_1^{13}+4508c_1^{16})\end{array},$

$\begin{array}{c}{c}_9=\frac{1}{181440}\frac{1}{{\mathrm{\β}}^{12}{c}_1^7}(5160960{\mathrm{\β}}^{12}{c}_2^8-45158400{\mathrm{\β}}^{11}{c}_1{c}_2^7+169344000{\mathrm{\β}}^{10}{c}_1^2{c}_2^6\\ -10916352{\mathrm{\β}}^9{c}_1^5{c}_2^6-354654720{\mathrm{\β}}^9{c}_1^3{c}_2^5+61770240{\mathrm{\β}}^8{c}_1^6{c}_2^5+452571840{\mathrm{\β}}^8{c}_1^4{c}_2^4\\ -142403904{\mathrm{\β}}^7{c}_1^7{c}_2^4-359432640{\mathrm{\β}}^7{c}_1^5{c}_2^3+4495680{\mathrm{\β}}^6{c}_1^{10}{c}_2^4+171002304{\mathrm{\β}}^6{c}_1^8{c}_2^3\\ +173093760{\mathrm{\β}}^6{c}_1^6{c}_2^2-14640192{\mathrm{\β}}^5{c}_1^{11}{c}_2^3-112679604{\mathrm{\β}}^5{c}_1^9{c}_2^2-46131120{\mathrm{\β}}^5{c}_1^7{c}_2\\ +17513808{\mathrm{\β}}^4{c}_1^{12}{c}_2^2+38590524{\mathrm{\β}}^4{c}_1^{10}{c}_2+5205060{\mathrm{\β}}^4{c}_1^8-392216{\mathrm{\β}}^3{c}_1^{15}{c}_2^2\\ -9120816{\mathrm{\β}}^3{c}_1^{13}{c}_2-5362677{\mathrm{\β}}^3{c}_1^{11}+547624{\mathrm{\β}}^2{c}_1^{16}{c}_2+1744380{\mathrm{\β}}^2{c}_1^{14}\\ -188706{\mathrm{\βc}}_1^{17}+3568c_1^{20})\end{array},$

$\begin{array}{c}{c}_{10}=\frac{1}{181440}\frac{1}{{\mathrm{\β}}^{13}{c}_1^8}(92897280{\mathrm{\β}}^{13}{c}_2^9-92897280{\mathrm{\β}}^{12}{c}_1{c}_2^8+4058449920{\mathrm{\β}}^{11}{c}_1^2{c}_2^7\\ -238970880{\mathrm{\β}}^{10}{c}_1^5{c}_2^7-10149027840{\mathrm{\β}}^{10}{c}_1^3{c}_2^6+1630158336{\mathrm{\β}}^9{c}_1^6{c}_2^6\\ +15980146560{\mathrm{\β}}^9{c}_1^4{c}_2^5-4676078592{\mathrm{\β}}^8{c}_1^7{c}_2^5-16397095680{\mathrm{\β}}^8{c}_1^5{c}_2^4\\ +131654016{\mathrm{\β}}^7{c}_1^{10}{c}_2^5+7304043456{\mathrm{\β}}^7{c}_1^8{c}_2^4+10942646400{\mathrm{\β}}^7{c}_1^6{c}_2^3\\ -564117120{\mathrm{\β}}^6{c}_1^{11}{c}_2^4-6702770736{\mathrm{\β}}^6{c}_1^9{c}_2^3-4571471520{\mathrm{\β}}^6{c}_1^7{c}_2^2\\ +949114656{\mathrm{\β}}^5{c}_1^{12}{c}_2^3+3610117836{\mathrm{\β}}^5{c}_1^{10}{c}_2^2+1083264840{\mathrm{\β}}^5{c}_1^8{c}_2\\ -18404544{\mathrm{\β}}^4{c}_1^{15}{c}_2^3-783565920{\mathrm{\β}}^4{c}_1^{13}{c}_2^2-1055710800{\mathrm{\β}}^4{c}_1^{11}{c}_2\\ -110837160{\mathrm{\β}}^4{c}_1^9+41460120{\mathrm{\β}}^3{c}_1^{16}{c}_2^2+317333772{\mathrm{\β}}^3{c}_1^{14}{c}_2\\ +129215439{\mathrm{\β}}^3{c}_1^{12}-30678936{\mathrm{\β}}^2{c}_1^{17}{c}_2-50420286{\mathrm{\β}}^2{c}_1^{15}\\ +484984{\mathrm{\βc}}_1^{20}{c}_2+7457610{\mathrm{\βc}}_1^{18}-316628c_1^{21})\end{array},$

And β=(a+b)/2a, c₁And c₂Value by equation

$\mathrm{\ν}=2-\frac{b}{a}\[\underset{i=2}{\overset{10}{\mathrm{\Σ}}}i\left(i-1\right)c_i{\left(\frac{b}{a}-\mathrm{\β}\right)}^{i-2}\]/\[\underset{j=1}{\overset{10}{\mathrm{\Σ}}}{\mathrm{jc}}_j{\left(\frac{b}{a}-\mathrm{\β}\right)}^{j-1}\]$

With

$\mathrm{\ν}=2-\frac{b}{a}\[\underset{i=2}{\overset{10}{\mathrm{\Σ}}}i\left(i-1\right)c_i{\left(1-\mathrm{\β}\right)}^{i-2}\]/\[\underset{j=1}{\overset{10}{\mathrm{\Σ}}}{\mathrm{jc}}_j{\left(1-\mathrm{\β}\right)}^{j-1}\]$

Determine, c₀Value by equation

$c_0=-\underset{j=1}{\overset{10}{\Σ}}{c}_j{(1-\mathrm{\β})}^j$

Determining, all parameters all adopt the International System of Units.