CN106644683A - Method for determining deflection of circular film under defined maximum deflection state - Google Patents

Abstract

The invention discloses a method for determining the deflection of a circular film under the defined maximum deflection state. The method comprises the following steps: horizontally applying a uniformly-distributed load q to a circular film with thickness of h, radius of a, Young's elastic modulus of E and Poisson's ratio of Nu and of which the periphery is fixed and clamped, enabling the deformed film to be in contact with a smooth plate, enabling the smooth plate to be parallel to the film before deformation, and keeping the distance H between the smooth plate and the film before deformation constant so as to limit the maximum deflection of the circular film after deformation; based on the static equilibrium analysis of the axisymmetric deformation problem, by using the measured value of the radius b of the contact area between the film and the smooth plate, determining the film deflection w (r) of all the points within the area with the radius r which is greater than b and smaller than a after the circular film is in contact with the smooth plate.

Description

method for determining circular film deflection under maximum deflection limited state

Technical Field

The invention relates to a method for determining the deflection of a circular film with the periphery fixedly clamped under uniformly distributed load in a maximum deflection limited state.

Background

The circular film structure has wide application in the fields of machinery, electronics, bioengineering and the like. In the development of the capacitive pressure sensor, the circular thin film structure needs to work in two modes, a non-contact mode: the film does not contact with the smooth electrode plate; contact mode: the film is in contact with a smooth electrode plate. In the contact mode, the maximum deflection of a circular film, which is fixedly clamped at the periphery under pressure (under uniformly distributed load), is always limited by a smooth electrode plate parallel to the film before deformation. From the study and new results, the problem of axisymmetrical deformation of the circular thin film which is fixedly clamped at the periphery after the maximum deflection is limited under the action of uniformly distributed load is not solved by analysis, so that the development work of the capacitive pressure sensor is influenced.

Disclosure of Invention

The invention solves the problem of axisymmetrical deformation of the circular film fixedly clamped at the periphery under the action of uniformly distributed loads after the maximum deflection is limited by analysis, and provides a method for determining the deflection of the circular film fixedly clamped at the periphery under the uniformly distributed loads under the state of the maximum deflection limitation on the basis.

The method for determining the deflection of the circular film in the state of limited maximum deflection comprises the following steps: transversely applying an evenly distributed load q to a circular film which is fixedly clamped at the periphery and has the thickness of H, the radius of a, the Young's modulus of elasticity of E and the Poisson ratio of v, enabling the deformed film to be in contact with a smooth flat plate, enabling the smooth flat plate to be parallel to the film before deformation, and keeping the distance H between the smooth flat plate and the film before deformation unchanged, so as to limit the maximum deflection of the circular film after deformation, and then based on the static balance analysis of the axisymmetric deformation problem, the analytical relation between the film deflection w (r) of each point in an area b < r < a and the radius b of a contact area after the circular film is in contact with the smooth flat plate can be expressed as

Wherein,

β＝(a+b)/(2a)，

α＝(a²+2ab-3b²)/(4a²)，c₀and c₁Is given by the equation

And

determining, wherein,

and d₀Is given by the equation

All parameters are determined by adopting an international system of units.

Thus, by measuring the value of the radius b of the contact area of the circular film and the smooth plate, the film deflection w (r) of each point in the area b < r < a after the circular film contacts the smooth plate can be determined.

Drawings

FIG. 1 is a schematic diagram of a maximum deflection limited perimeter clamped circular film loading configuration wherein 1-the circular film after deformation, 2-the smooth plate, 3-the clamping means, 4-the circular film before deformation, and a represents the radius of the circular film, b represents the radius of the area of contact of the film with the smooth plate, r represents the radial coordinate, w (r) represents the transverse coordinate at point r, q represents the transverse uniform load, and H represents the distance between the circular film and the smooth plate.

Detailed Description

The technical solution of the present invention is further described in detail with reference to fig. 1 below:

transversely applying an even load q to the circular rubber film fixedly clamped at the periphery, enabling the deformed film to be in contact with a smooth flat plate, enabling the smooth flat plate to be parallel to the film before deformation, and keeping the distance H between the smooth flat plate and the film before deformation constant, thereby limiting the maximum deflection of the circular rubber film after deformation, wherein the distance H between the smooth flat plate and the film before deformation is 13mm, the thickness H of the circular rubber film is 1mm, the radius a is 50mm, the Young modulus E is 7.84MPa, and the Poisson ratio v is 0.47, the radius b of the contact area of the film and the smooth flat plate is 25mm, and α is obtained by adopting the method provided by the invention (a is 25 mm)²+2ab-3b²)/(4a²) 0.3125, (a + b)/(2a) 0.75 for β, using the equation

And

namely, the equation

And

wherein,

can obtain c₀＝0.2907060055，c₁-0.03622260533. Then, from the equation

Namely, the equation

Wherein,

d₁＝-0.7166461285，

d₂＝-1.286834462，

d₃＝0.1048118431，

d₄＝-0.7783728781，

d₅＝0.07988664578，

d₆＝-0.5435878967，

d₇＝-0.1413928502，

can obtain d₀0.2610548475. Finally, the formula

I.e. the formula

It can be obtained that the film deflection (mm) at each point in the area 25mm < r < 50mm when the radius b of the area where the deformed film contacts the smooth flat plate reaches 25 mm:

Claims (1)

1. The method for determining the circular film deflection under the state of limited maximum deflection is characterized by comprising the following steps of: transversely applying an evenly distributed load q to a circular film which is fixedly clamped at the periphery and has the thickness of H, the radius of a, the Young modulus of elasticity of E and the Poisson ratio of v, enabling the deformed film to be in contact with a smooth flat plate, enabling the smooth flat plate to be parallel to the film before deformation, and keeping the distance H between the smooth flat plate and the film before deformation unchanged, thereby limiting the maximum deflection of the circular film after deformation, measuring the value of the radius b of a contact area of the film and the smooth flat plate, and determining the film deflection w (r) of each point in the area b < r < a after the circular film is in contact with the smooth flat plate by the following formula,

$w\left(r\right)=H\&lsqb;\underset{n=0}{\overset{7}{\&Sigma;}}{d}_n{(\frac{r}{a}-\mathrm{\&beta;})}^n\&rsqb;/\&lsqb;\underset{n=0}{\overset{7}{\&Sigma;}}{d}_n{(\frac{b}{a}-\mathrm{\&beta;})}^n\&rsqb;,$

wherein,

β＝(a+b)/(2a)，

$d_1=-\frac{1}{2}\frac{\mathrm{\&alpha;}}{{\mathrm{\&beta;c}}_0},$

$d_2=-\frac{1}{4}\frac{1}{{\mathrm{\&beta;}}^2{c}_0^2}(2{\mathrm{\&beta;}}^2{c}_0-{\mathrm{\&alpha;\&beta;c}}_1-{\mathrm{\&alpha;c}}_0),$

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

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

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

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

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

α＝(a²+2ab-3b²)/(4a²)，c₀and c₁Is given by the equation

$(1-v)\underset{n=0}{\overset{6}{\&Sigma;}}{c}_n{(1-\mathrm{\&beta;})}^n+\underset{n=1}{\overset{6}{\&Sigma;}}{\mathrm{nc}}_n{(1-\mathrm{\&beta;})}^{n-1}=0$

And

$\underset{n=1}{\overset{6}{\&Sigma;}}{\mathrm{nc}}_n{(\frac{b}{a}-\mathrm{\&beta;})}^{n-1}=0$

determining, wherein,

$c_2=-\frac{1}{16}\frac{1}{{\mathrm{\&beta;}}^4{c}_0^2}(24{\mathrm{\&beta;}}^3{c}_0^2{c}_1+{\mathrm{\&alpha;}}^2),$

$c_3=\frac{1}{48}\frac{1}{{\mathrm{\&beta;}}^5{c}_0^3}(96{\mathrm{\&beta;}}^3{c}_0^3{c}_1-4{\mathrm{\&alpha;\&beta;}}^2{c}_0+2{\mathrm{\&alpha;}}^2{\mathrm{\&beta;c}}_1+7{\mathrm{\&alpha;}}^2{c}_0),$

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

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

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

and d₀Is given by the equation

$\underset{n=0}{\overset{7}{\&Sigma;}}{d}_n{(1-\mathrm{\&beta;})}^n=0$

All parameters are determined by adopting an international system of units.