CN106644188A - Method for determining uniformly distributed load of circular film under restricted condition of maximum deflection - Google Patents

Abstract

A method for determining the uniformly distributed load of a circular film under a restricted condition of maximum deflection comprises: transversely applying a uniformly distributed load q to the circular film with a thickness h, a radius a, a Young's elasticity modulus E, a Poisson's ratio v, and a fixed and clamped edge, bringing the deformed film into contact with a smooth plate, enabling the smooth plate to be parallel to the film before deformation and maintaining the distance H between the smooth plate and the film constant so as to limit the maximum deflection of the deformed circular film. 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, it is possible to determine the uniformly distributed load q applied to the circular film after circular film is in contact with the smooth plate.

Description

The determination method of circular membrane uniform load under maximum defluxion its constrained state

Technical field

The present invention relates to the circular membrane that uniform load following peripheral fixes to clamp applies under maximum defluxion its constrained state The determination method of the uniform load on circular membrane.

Background technology

Circular membrane structure has a wide range of applications in fields such as machinery, electronics, bioengineering.Capacitance pressure transducer, Development in, need to allow circular membrane arrangement works in both modes, noncontact mode：Film is not produced with smooth electrodes plate Contact；Contact mode：Film is contacted with smooth electrodes plate.Under contact mode, periphery geometrical clamp (under uniform load) under pressure The maximum defluxion of tight circular membrane, is limited all the time by the smooth electrodes plate parallel with the film before deformation.And from text Offer and look into new result and see, axial symmetry of the circular membrane that the limited rear perimeter edge of maximum defluxion is fixed to clamp under Uniform Loads becomes Shape problem, is not also resolved and solves, thus has influence on the development work of capacitance pressure transducer,.

The content of the invention

The circular membrane that the limited rear perimeter edge of Analytical Solution of the present invention maximum defluxion is fixed to clamp is under the Uniform Loads On Axisymmetric Deformation of A, and give circular membrane that uniform load following peripheral fixes to clamp on this basis in maximum defluxion The determination method of the uniform load being applied under its constrained state on circular membrane.

The determination method of circular membrane uniform load under maximum defluxion its constrained state：To thickness be h, radius be a, Young The circular membrane that the periphery that elastic modelling quantity is E, Poisson's ratio is ν is fixed to clamp is laterally applied to uniform load q, after making deformation Film contacts with a smooth plates, and makes smooth plates parallel with the film before deformation, and keeps smooth plates and become The distance between film before shape H is constant, so as to limit circular membrane maximum defluxion after deformation, then based on to this axle The standing balance analysis of symmetric deformation problem, the circular membrane is in after-applied uniform load q on film of contact smooth plates Can be expressed as with the parsing relation of the radius b of contact area

Wherein,

β=(a+b)/(2a),

α=(a²+2ab-3b²)/(4a²), c₀And c₁Value by equation

With

It is determined that, wherein,

And d₀Value by equation

It is determined that, all parameters all adopt the International System of Units.

So, as long as measuring the value of circular membrane and the radius b of smooth plates contact area, it is possible to determine the circle Film is in after-applied uniform load q on film of contact smooth plates.

Description of the drawings

Fig. 1 is the circular membrane loading organigram that the restricted periphery of maximum defluxion is fixed to clamp, wherein, 1- becomes Circular membrane after shape, 2- smooth plates, 3- clamping devices, the circular membrane before 4- deformations, and a represents circular membrane Radius, b represents the radius of film and smooth plates contact area, and r represents radial coordinate, and w (r) represents the lateral coordinates at point r, Q represents horizontal uniform load, and H represents the distance between circular membrane and smooth plates.

Specific embodiment

Technical scheme is described in further detail with reference to Fig. 1：

The round rubber film fixed to clamp to periphery is laterally applied to uniform load q, makes the film after deformation and one Individual smooth plates contact, and make smooth plates parallel with the film before deformation, and thin before keeping smooth plates and deforming The distance between film H is constant, so as to limit round rubber film maximum defluxion after deformation, wherein, smooth plates with deformation The distance between front film H=12mm, the thickness h=1mm of round rubber film, radius a=50mm, Young's modulus of lasticity E =7.84MPa, Poisson's ratio ν=0.47.Measure the radius b=20mm of film and smooth plates contact area.Using institute of the present invention The method for being given, then α=(a²+2ab-3b²)/(4a²)=0.33, β=(a+b)/(2a)=0.7, by equation

With

That is, equation

With

Wherein,

C can be obtained₀=0.3344084098, c₁=-0.04803851151.Then, by equation

That is, equation

Wherein,

d₁=-0.7048694915,

d₂=-1.042327536,

d₃=0.02688715564,

d₄=-0.5294752406,

d₅=0.1479261706,

d₆=-0.4639589262,

d₇=0.182966783,

D can be obtained₀=0.3087718725.Finally, by formula

That is, formula

Can then obtain, when the radius b of the film after deformation and smooth plates contact area reaches 20mm, be applied to this Uniform load q=0.029117MPa on round rubber film.

Claims (1)

1. under maximum defluxion its constrained state circular membrane uniform load determination method, it is characterised in that：Be to thickness h, half Footpath is a, the circular membrane that the periphery that Young's modulus of elasticity is E, Poisson's ratio is ν is fixed to clamp is laterally applied to uniform load q, Make the film after deformation contact with a smooth plates, and make smooth plates parallel with the film before deformation, and keep light The distance between film before sliding flat board and deforming H is constant, so as to limit circular membrane maximum defluxion after deformation, measures thin The value of the radius b of film and smooth plates contact area, the circular membrane is after-applied uniform on film in contact smooth plates Load q determines by below equation,

$q=\frac{{\mathrm{EhH}}^3}{a^4}/{\[\underset{n=0}{\overset{7}{\Σ}}{d}_n{(\frac{b}{a}-\mathrm{\β})}^n\]}^3,$

Wherein,

β=(a+b)/(2a),

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

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

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

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

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

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

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

α=(a²+2ab-3b²)/(4a²), c₀And c₁Value by equation

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

With

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

It is determined that, wherein,

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

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

$\begin{array}{c}{c}_4=-\frac{1}{768}\frac{1}{{\mathrm{\β}}^8{c}_0^5}(1920{\mathrm{\β}}^5{c}_0^5{c}_1+32{\mathrm{\β}}^6{c}_0^3-64{\mathrm{\α\β}}^5{c}_0^2{c}_1+24{\mathrm{\α}}^2{\mathrm{\β}}^4{c}_0{c}_1^2-160{\mathrm{\α\β}}^4{c}_0^3\\ +112{\mathrm{\α}}^2{\mathrm{\β}}^3{c}_0^2{c}_1+188{\mathrm{\α}}^2{\mathrm{\β}}^2{c}_0^3+{\mathrm{\α}}^4)\end{array},$

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

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

And d₀Value by equation

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

It is determined that, all parameters all adopt the International System of Units.