CN105890970A - 均布载荷下中心带刚性板的环形薄膜最大挠度的确定方法 - Google Patents

Abstract

本发明公开了均布载荷下中心带刚性板的环形薄膜最大挠度的确定方法：采用内半径为a的夹紧装置，将厚度为h、杨氏弹性模量为E、泊松比为ν、中心带半径为b的刚性板的环形薄膜固定夹紧，形成一个外半径为a、内半径为b的周边固定夹紧的中心带刚性板的轴对称环形薄膜，对其横向施加一个均布载荷q，基于这个轴对称变形问题的静力平衡分析，并利用均布载荷q的测量值，则可确定出该环形薄膜的最大挠度w_m。

Description

均布载荷下中心带刚性板的环形薄膜最大挠度的确定方法

技术领域

本发明涉及均布载荷下中心带刚性板的环形薄膜最大挠度的确定方法。

背景技术

均布载荷下中心带刚性板的环形薄膜轴对称变形问题的解析解，对传感器以及仪器、仪表的研制具有重要意义。由于薄膜是柔性材料，在均布载荷作用下通常呈现出较大的挠度，因而其变形问题具有较强的非线性，这些非线性问题通常难以解析求解。本发明致力于均布载荷下中心带刚性板的环形薄膜轴对称变形问题的解析研究，获得了该问题的解析解，并在此基础上给出了均布载荷下中心带刚性板的环形薄膜最大挠度的确定方法。

发明内容

均布载荷下中心带刚性板的环形薄膜最大挠度的确定方法：采用内半径为a的夹紧装置，将厚度为h、杨氏弹性模量为E、泊松比为ν、中心带半径为b的刚性板的环形薄膜固定夹紧，形成一个外半径为a、内半径为b的周边固定夹紧的中心带刚性板的轴对称环形薄膜，对其横向施加一个均布载荷q，基于这个轴对称变形问题的静力平衡分析，就可以得到该环形薄膜的最大挠度w_m与均布载荷q的解析关系为

$w_m={\left(\frac{a^{4}q}{hE}\right)}^{1/3}\underset{n=0}{\overset{10}{\Σ}}{c}_n{\left(\frac{b-a}{2a}\right)}^n,$

其中，

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

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

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

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

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

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

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

$\begin{array}{c}{c}_{10}=\frac{1}{181440}\frac{1}{{\mathrm{\β}}^{13}{c}_1^8}(928970{\mathrm{\β}}^{13}{c}_2^9-92897280{\mathrm{\β}}^{12}{c}_1{c}_2^8+4058449920{\mathrm{\β}}^{11}{c}_1^2{c}_2^7\\ -238970880{\mathrm{\β}}^{10}{c}_1^5{c}_2^7-10149027840{\mathrm{\β}}^{10}{c}_1^3{c}_2^6+1630158336{\mathrm{\β}}^9{c}_1^6{c}_2^6\\ +15980146560{\mathrm{\β}}^9{c}_1^4{c}_2^5-4676078592{\mathrm{\β}}^8{c}_1^7{c}_2^5-16397095680{\mathrm{\β}}^8{c}_1^5{c}_2^4\\ +131654016{\mathrm{\β}}^7{c}_1^{10}{c}_2^5+7304043456{\mathrm{\β}}^7{c}_1^8{c}_2^4+10942646400{\mathrm{\β}}^7{c}_1^6{c}_2^3\\ -564117120{\mathrm{\β}}^6{c}_1^{11}{c}_2^4-6702770736{\mathrm{\β}}^6{c}_1^9{c}_2^3-4571471520{\mathrm{\β}}^6{c}_1^7{c}_2^2\\ +949114656{\mathrm{\β}}^5{c}_1^{12}{c}_2^3+3610117836{\mathrm{\β}}^5{c}_1^{10}{c}_2^2+1083264840{\mathrm{\β}}^5{c}_1^8{c}_2\\ -18404544{\mathrm{\β}}^4{c}_1^{15}{c}_2^3-78356920{\mathrm{\β}}^4{c}_1^{13}{c}_2^2-1055710800{\mathrm{\β}}^4{c}_1^{11}{c}_2\\ -110837160{\mathrm{\β}}^4{c}_1^9+41460120{\mathrm{\β}}^3{c}_1^{16}{c}_2^2+317333772{\mathrm{\β}}^3{c}_1^{14}{c}_2\\ +129215439{\mathrm{\β}}^3{c}_1^{12}-30678936{\mathrm{\β}}^2{c}_1^{17}{c}_2-50420286{\mathrm{\β}}^2{c}_1^{15}\\ +484984{\mathrm{\βc}}_1^{20}{c}_2+7457610{\mathrm{\βc}}_1^{18}-316628c_1^{21})\end{array},$

而β＝(a+b)/2a，c₁和c₂的值由方程

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

和

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

确定,c₀的值由方程

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

确定。

这样，只要准确测量出所施加的均布载荷q的值，就可以确定出该环形薄膜变形后的最大挠度w_m。其中，所有参量皆采用国际单位制。

附图说明

图1为均布载荷下周边固定夹紧的中心带刚性板的环形薄膜的加载构造示意图，其中，1－环形薄膜，2－刚性板，3－夹紧装置，而a表示夹紧装置的内半径和环形薄膜的外半径，b表示刚性板的半径和环形薄膜的内半径，r表示径向坐标，w(r)表示点r处的横向坐标，q表示横向均布载荷，w_m表示薄膜的最大挠度。

具体实施方式

下面结合附图对本发明的技术方案作进一步的详细说明：

如图1所示，用一个内半径a＝20mm的夹紧装置，将厚度h＝0.06mm、杨氏弹性模量E＝7.84MPa、泊松比ν＝0.47、中心带半径b＝5mm的刚性板的橡胶薄膜固定夹紧，形成一个外半径a＝20mm、内半径b＝5mm的周边固定夹紧的中心带刚性板的轴对称环形薄膜，对其横向施加一个均布载荷q，并测得q＝0.01MPa。采用本发明所给出的方法，通过方程

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

和

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

则可以得到c₁＝-0.7026607165，c₂＝-0.6927450924，其中，

β＝(a+b)/2a＝0.625，

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

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

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

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

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

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

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

$\begin{array}{c}{c}_{10}=\frac{1}{181440}\frac{1}{{\mathrm{\β}}^{13}{c}_1^8}(928970{\mathrm{\β}}^{13}{c}_2^9-92897280{\mathrm{\β}}^{12}{c}_1{c}_2^8+4058449920{\mathrm{\β}}^{11}{c}_1^2{c}_2^7\\ -238970880{\mathrm{\β}}^{10}{c}_1^5{c}_2^7-10149027840{\mathrm{\β}}^{10}{c}_1^3{c}_2^6+1630158336{\mathrm{\β}}^9{c}_1^6{c}_2^6\\ +15980146560{\mathrm{\β}}^9{c}_1^4{c}_2^5-4676078592{\mathrm{\β}}^8{c}_1^7{c}_2^5-16397095680{\mathrm{\β}}^8{c}_1^5{c}_2^4\\ +131654016{\mathrm{\β}}^7{c}_1^{10}{c}_2^5+7304043456{\mathrm{\β}}^7{c}_1^8{c}_2^4+10942646400{\mathrm{\β}}^7{c}_1^6{c}_2^3\\ -564117120{\mathrm{\β}}^6{c}_1^{11}{c}_2^4-6702770736{\mathrm{\β}}^6{c}_1^9{c}_2^3-4571471520{\mathrm{\β}}^6{c}_1^7{c}_2^2\\ +949114656{\mathrm{\β}}^5{c}_1^{12}{c}_2^3+3610117836{\mathrm{\β}}^5{c}_1^{10}{c}_2^2+1083264840{\mathrm{\β}}^5{c}_1^8{c}_2\\ -18404544{\mathrm{\β}}^4{c}_1^{15}{c}_2^3-78356920{\mathrm{\β}}^4{c}_1^{13}{c}_2^2-1055710800{\mathrm{\β}}^4{c}_1^{11}{c}_2\\ -110837160{\mathrm{\β}}^4{c}_1^9+41460120{\mathrm{\β}}^3{c}_1^{16}{c}_2^2+317333772{\mathrm{\β}}^3{c}_1^{14}{c}_2\\ +129215439{\mathrm{\β}}^3{c}_1^{12}-30678936{\mathrm{\β}}^2{c}_1^{17}{c}_2-50420286{\mathrm{\β}}^2{c}_1^{15}\\ +484984{\mathrm{\βc}}_1^{20}{c}_2+7457610{\mathrm{\βc}}_1^{18}-316628c_1^{21})\end{array},$

然后，再由方程可以得到c₀＝0.3710002899。最后，由方程

$w_m={\left(\frac{a^{4}q}{hE}\right)}^{1/3}\underset{n=0}{\overset{10}{\Σ}}{c}_n{\left(\frac{b-a}{2a}\right)}^n$

则可以得到该环形薄膜的最大挠度w_m＝8.14mm。

Claims (1)

1.均布载荷下中心带刚性板的环形薄膜最大挠度的确定方法，其特征在于：采用内半径为a的夹紧装置，将厚度为h、杨氏弹性模量为E、泊松比为ν、中心带半径为b的刚性板的环形薄膜固定夹紧，形成一个外半径为a、内半径为b的周边固定夹紧的中心带刚性板的轴对称环形薄膜，对其横向施加一个均布载荷q，并测得所施加的均布载荷q的值，由以下公式确定该环形薄膜的最大挠度w_m：

$w_m={\left(\frac{a^{4}q}{hE}\right)}^{1/3}\underset{n=0}{\overset{10}{\Σ}}{c}_n{\left(\frac{b-a}{2a}\right)}^n,$

其中，

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

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

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

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

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

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

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

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

而β＝(a+b)/2a，c₁和c₂的值由方程

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

和

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

确定,c₀的值由方程

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

确定，所有参量皆采用国际单位制。