最大挠度受限制状态下圆形薄膜挠度的确定方法与流程

技术特征：

1.最大挠度受限制状态下圆形薄膜挠度的确定方法，其特征在于：对厚度为h、半径为a、杨氏弹性模量为E、泊松比为ν的周边固定夹紧的圆形薄膜横向施加一个均布载荷q，使变形后的薄膜与一个光滑平板相接触，并让光滑平板与变形前的薄膜相平行，且保持光滑平板与变形前的薄膜之间的距离H不变，从而限制圆形薄膜在变形后的最大挠度，测得薄膜与光滑平板接触区域的半径b的值，该圆形薄膜接触光滑平板后在区域b＜r＜a内各点的薄膜挠度w(r)由以下公式确定，

$w\left(r\right)=H\[\underset{n=0}{\overset{7}{\Σ}}{d}_n{(\frac{r}{a}-\mathrm{\β})}^n\]/\[\underset{n=0}{\overset{7}{\Σ}}{d}_n{(\frac{b}{a}-\mathrm{\β})}^n\],$

其中，

β＝(a+b)/(2a)，

$d_1=-\frac{1}{2}\frac{\mathrm{\α}}{{\mathrm{\βc}}_0},$

$d_2=-\frac{1}{4}\frac{1}{{\mathrm{\β}}^2{c}_0^2}(2{\mathrm{\β}}^2{c}_0-{\mathrm{\α\βc}}_1-{\mathrm{\αc}}_0),$

$d_3=\frac{1}{96}\frac{1}{{\mathrm{\β}}^5{c}_0^4}(32{\mathrm{\β}}^5{c}_0^2{c}_1-16{\mathrm{\α\β}}^4{c}_0{c}_1^2+16{\mathrm{\β}}^4{c}_0^3-40{\mathrm{\α\β}}^3{c}_0^2{c}_1-16{\mathrm{\α\β}}^2{c}_0^3-{\mathrm{\α}}^3),$

$\begin{array}{c}{d}_4=\frac{1}{192}\frac{1}{{\mathrm{\β}}^6{c}_0^5}(48{\mathrm{\β}}^6{c}_0^2{c}_1^2-24{\mathrm{\α\β}}^5{c}_0{x}_1^3+96{\mathrm{\β}}^5{c}_0^3{c}_1-96{\mathrm{\α\β}}^6{c}_0^5{c}_1^2+24{\mathrm{\β}}^4{c}_0^4\\ -108{\mathrm{\α\β}}^3{c}_0^3{c}_1-24{\mathrm{\α\β}}^2{c}_0^4+5{\mathrm{\α}}^2{\mathrm{\β}}^2{c}_0-4{\mathrm{\α}}^3{\mathrm{\βc}}_1-5{\mathrm{\α}}^3{c}_0)\end{array},$

$\begin{array}{c}{d}_5=-\frac{1}{19020}\frac{1}{{\mathrm{\β}}^9{c}_0^7}(384{\mathrm{\β}}^9{c}_0^3{c}_1^3-192{\mathrm{\α\β}}^8{c}_0^2{c}_1^4+1344{\mathrm{\β}}^8{c}_0^4{c}_1^2-1056{\mathrm{\α\β}}^7{c}_0^3{c}_1^3+1248{\mathrm{\β}}^7{c}_0^5{c}_1\\ -1968{\mathrm{\α\β}}^6{c}_0^4{c}_1^2+192{\mathrm{\β}}^6{c}_0^6-1344{\mathrm{\α\β}}^5{c}_0^5{c}_1-192{\mathrm{\α\β}}^4{c}_0^6-40{\mathrm{\α\β}}^6{c}_0^3+112{\mathrm{\α}}^2{\mathrm{\β}}^5{c}_0^2{c}_1\\ -58{\mathrm{\α}}^3{\mathrm{\β}}^4{c}_0{c}_1^2+124{\mathrm{\α}}^2{\mathrm{\β}}^4{c}_0^3-152{\mathrm{\α}}^3{\mathrm{\β}}^3{c}_0^2{c}_1-87{\mathrm{\α}}^3{\mathrm{\β}}^2{c}_0^3-{\mathrm{\α}}^5)\end{array},$

$\begin{array}{c}{d}_6=-\frac{1}{23040}\frac{1}{{\mathrm{\β}}^{10}{c}_0^8}(3840{\mathrm{\β}}^{10}{c}_0^3{c}_1^4-1920{\mathrm{\α\β}}^9{c}_0^2{c}_1^5+19200{\mathrm{\β}}^9{c}_0^4{c}_1^3-13440{\mathrm{\α\β}}^8{c}_0^3{c}_1^4\\ +31680{\mathrm{\β}}^8{c}_0^5{c}_1^2-35040{\mathrm{\α\β}}^7{c}_0^4{c}_1^3+18240{\mathrm{\β}}^7{c}_0^6{c}_1-40800{\mathrm{\α\β}}^6{c}_0^5{c}_1^2+1920{\mathrm{\β}}^6{c}_0^7\\ -19200{\mathrm{\α\β}}^5{c}_0^6{c}_1+160{\mathrm{\β}}^8{c}_0^4-1216{\mathrm{\α\β}}^7{c}_0^3{c}_1+2104{\mathrm{\α}}^2{\mathrm{\β}}^6{c}_0^2{c}_1^2-888{\mathrm{\α}}^3{\mathrm{\β}}^5{c}_0{c}_1^3\\ -1920{\mathrm{\α\β}}^4{c}_0^7-1232{\mathrm{\α\β}}^6{c}_0^4+5056{\mathrm{\α}}^2{\mathrm{\β}}^5{c}_0^3{c}_1-3588{\mathrm{\α}}^3{\mathrm{\β}}^4{c}_0^2{c}_1^2+2596{\mathrm{\α}}^2{\mathrm{\β}}^4{c}_0^4\\ -4416{\mathrm{\α}}^3{\mathrm{\β}}^3{c}_0^3{c}_1-1554{\mathrm{\α}}^3{\mathrm{\β}}^2{c}_0^4+48{\mathrm{\α}}^4{\mathrm{\β}}^2{c}_0-43{\mathrm{\α}}^5{\mathrm{\βc}}_1-56{\mathrm{\α}}^5{c}_0)\end{array},$

$\begin{array}{c}{d}_7=\frac{1}{1290240}\frac{1}{{\mathrm{\β}}^{13}{c}_0^{10}}(184320{\mathrm{\β}}^{13}{c}_0^4{c}_1^5-92160{\mathrm{\α\β}}^{12}{c}_0^3{c}_1^6+1198080{\mathrm{\β}}^{12}{c}_0^5{c}_1^4\\ -783360{\mathrm{\α\β}}^{11}{c}_0^4{c}_1^5+2856960{\mathrm{\β}}^{11}{c}_0^6{c}_1^3-2626560{\mathrm{\α\β}}^{10}{c}_0^5{c}_1^4+2972160{\mathrm{\α\β}}^{10}{c}_0^7{c}_1^2\\ -4343040{\mathrm{\α\β}}^9{c}_0^6{c}_1^3+1198080{\mathrm{\β}}^9{c}_0^8{c}_1-3571200{\mathrm{\α\β}}^8{c}_0^7{c}_1^2+24576{\mathrm{\β}}^{11}{c}_0^5{c}_1\\ -115968{\mathrm{\α\β}}^{10}{c}_0^4{c}_1^2+158976{\mathrm{\α}}^2{\mathrm{\β}}^9{c}_0^3{c}_1^3-59328{\mathrm{\α}}^3{\mathrm{\β}}^8{c}_0^2{c}_1^4+92160{\mathrm{\β}}^8{c}_0^9\\ -1244160{\mathrm{\α\β}}^7{c}_0^8{c}_1+22272{\mathrm{\β}}^{10}{c}_0^6-259584{\mathrm{\α\β}}^9{c}_0^5{c}_1+600384{\mathrm{\α}}^2{\mathrm{\β}}^8{c}_0^4{c}_1^2\\ -325440{\mathrm{\α}}^3{\mathrm{\β}}^7{c}_0^3{c}_1^3-92160{\mathrm{\α\β}}^6{c}_0^9-122496{\mathrm{\α\β}}^8{c}_0^6+681984{\mathrm{\α}}^2{\mathrm{\β}}^7{c}_0^5{c}_1\\ -626208{\mathrm{\α}}^3{\mathrm{\β}}^6{c}_0^4{c}_1^2+217440{\mathrm{\α}}^2{\mathrm{\β}}^6{c}_0^6-484608{\mathrm{\α}}^3{\mathrm{\β}}^5{c}_0^5{c}_1-118656{\mathrm{\α}}^3{\mathrm{\β}}^4{c}_0^6\\ -4288{\mathrm{\α}}^3{\mathrm{\β}}^6{c}_0^3+10336{\mathrm{\α}}^4{\mathrm{\β}}^5{c}_0^2{c}_1-5416{\mathrm{\α}}^5{\mathrm{\β}}^4{c}_0{c}_1^2+12608{\mathrm{\α}}^4{\mathrm{\β}}^4{c}_0^3\\ -14440{\mathrm{\α}}^5{\mathrm{\β}}^3{c}_0^2{c}_1-8896{\mathrm{\α}}^5{\mathrm{\β}}^2{c}_0^3-43{\mathrm{\α}}^7)\end{array},$

α＝(a²+2ab-3b²)/(4a²)，c₀和c₁的值由方程

$(1-v)\underset{n=0}{\overset{6}{\Σ}}{c}_n{(1-\mathrm{\β})}^n+\underset{n=1}{\overset{6}{\Σ}}{\mathrm{nc}}_n{(1-\mathrm{\β})}^{n-1}=0$

和

$\underset{n=1}{\overset{6}{\Σ}}{\mathrm{nc}}_n{(\frac{b}{a}-\mathrm{\β})}^{n-1}=0$

确定，其中，

$c_2=-\frac{1}{16}\frac{1}{{\mathrm{\β}}^4{c}_0^2}(24{\mathrm{\β}}^3{c}_0^2{c}_1+{\mathrm{\α}}^2),$

$c_3=\frac{1}{48}\frac{1}{{\mathrm{\β}}^5{c}_0^3}(96{\mathrm{\β}}^3{c}_0^3{c}_1-4{\mathrm{\α\β}}^2{c}_0+2{\mathrm{\α}}^2{\mathrm{\βc}}_1+7{\mathrm{\α}}^2{c}_0),$

$\begin{array}{c}{c}_4=-\frac{1}{768}\frac{1}{{\mathrm{\β}}^8{c}_0^5}(1920{\mathrm{\β}}^5{c}_0^5{c}_1+32{\mathrm{\β}}^6{c}_0^3-64{\mathrm{\α\β}}^5{c}_0^2{c}_1+24{\mathrm{\α}}^2{\mathrm{\β}}^4{c}_0{c}_1^2-160{\mathrm{\α\β}}^4{c}_0^3\\ +122{\mathrm{\α}}^2{\mathrm{\β}}^3{c}_0^2{c}_1+188{\mathrm{\α}}^2{\mathrm{\β}}^2{c}_0^3+{\mathrm{\α}}^4)\end{array},$

$\begin{array}{c}{c}_5=\frac{1}{3840}\frac{1}{{\mathrm{\β}}^9{c}_0^6}(11520{\mathrm{\β}}^5{c}_0^6{c}_1+192{\mathrm{\β}}^7{c}_0^3{c}_1-288{\mathrm{\α\β}}^6{c}_0^2{c}_1^2+96{\mathrm{\α}}^2{\mathrm{\β}}^5{c}_0{c}_1^3+384{\mathrm{\β}}^6{c}_0^4\\ -1152{\mathrm{\α\β}}^5{c}_0^3{c}_1+576{\mathrm{\α}}^2{\mathrm{\β}}^4{c}_0^2{c}_1^2-1392{\mathrm{\α\β}}^4{c}_0^4+1272{\mathrm{\α}}^2{\mathrm{\β}}^3{c}_0^3{c}_1+1368{\mathrm{\α}}^2{\mathrm{\β}}^2{c}_0^4\\ -16{\mathrm{\α}}^3{\mathrm{\β}}^2{c}_0+11{\mathrm{\α}}^4{\mathrm{\βc}}_1+22{\mathrm{\α}}^4{c}_0)\end{array},$

$\begin{array}{c}{c}_6=-\frac{1}{184320}\frac{1}{{\mathrm{\β}}^{12}{c}_0^8}(9216{\mathrm{\β}}^{10}{c}_0^4{c}_1^2-12288{\mathrm{\α\β}}^9{c}_0^3{c}_1^3+3840{\mathrm{\α}}^2{\mathrm{\β}}^8{c}_0^2{c}_1^4+645120{\mathrm{\β}}^7{c}_0^8{c}_1\\ +32256{\mathrm{\β}}^9{c}_0^5{c}_1-66816{\mathrm{\α\β}}^8{c}_0^4{c}_1^2+28416{\mathrm{\α}}^2{\mathrm{\β}}^7{c}_0^3{c}_1^3+31488{\mathrm{\β}}^8{c}_0^6-127488{\mathrm{\α\β}}^7{c}_0^5{c}_1\\ +81216{\mathrm{\α}}^2{\mathrm{\β}}^6{c}_0^4{c}_1^2-99456{\mathrm{\α\β}}^6{c}_0^6+113472{\mathrm{\α}}^2{\mathrm{\β}}^5{c}_0^5{c}_1+88128{\mathrm{\α}}^2{\mathrm{\β}}^4{c}_0^6+960{\mathrm{\α}}^2{\mathrm{\β}}^6{c}_0^3\\ -1920{\mathrm{\α}}^3{\mathrm{\β}}^5{c}_0^2{c}_1+816{\mathrm{\α}}^4{\mathrm{\β}}^4{c}_0{c}_1^2-3456{\mathrm{\α}}^3{\mathrm{\β}}^4{c}_0^3+3000{\mathrm{\α}}^4{\mathrm{\β}}^3{c}_0^2{c}_1+2856{\mathrm{\α\β}}^2{c}_0^3+11{\mathrm{\α}}^6)\end{array},$

而d₀的值由方程

$\underset{n=0}{\overset{7}{\Σ}}{d}_n{(1-\mathrm{\β})}^n=0$

确定，所有参量皆采用国际单位制。