CN104792618B - A kind of determination method of big corner circular membrane maximum defluxion under uniform load - Google Patents

Abstract

The invention discloses a kind of determination method of big corner circular membrane maximum defluxion under uniform load：It is with an inside radiusClamping device, be by Young's modulus of elasticity, Poisson's ratio be, thickness beFixed to clamp without prestressed film, formed a radius beThe circular membrane structure that fixes to clamp of periphery, and a uniform load is laterally applied to circular membrane, make the corner after deformation of thin membraneOccur being more than 20^oSituation, then the Accurate Analysis based on this axial symmetry circular membrane standing balance problem, utilize load measurement value, then the maximum defluxion after deformation of thin membrane can be determinedw _m 。

Description

A kind of determination method of big corner circular membrane maximum defluxion under uniform load

Technical field

The determination method of the big corner circular membrane maximum defluxion fixed to clamp the present invention relates to Uniform Loads following peripheral.

Background technology

Circular membrane structure has application in many technical fields, the design problem such as diaphragm gauge, instrument, and thin Test problem of film/basic unit's system mechanical property etc..But so far, the circle that Uniform Loads following peripheral is fixed to clamp The analytic solutions of film standing balance problem, are obtained under the assumed condition of film corner very little, for example the applicant it Before patent of invention " a kind of method of determination uniform load lower prestress circular membrane maximum immunity value " (number of patent application for declaring： 201410238568.7) analytic solutions of the prestressing force circular membrane problem employed in, and famous Hencky solutions are (i.e. no pre- The analytic solutions of circular membrane problem during stress), these solutions are all assuming that film rotational angle theta is approximate during applying transverse load q Meet what is obtained under conditions of sin θ=tan θ, this hypothesis is commonly known as film small angle tower hypothesis.It is clear that only existing In the case of film rotational angle theta very little, can just there are approximate condition sin θ ≌ tan θ establishment, thus existing method, can only all use In the less situation of film rotational angle theta, and the larger situation of film rotational angle theta is not suitable for it, this is that is, at present for larger turn The circular membrane during situation of angle, also has no idea relatively accurately to determine the maximum defluxion w of film_m！

In fact, the rotational angle theta of film is to increase as the transverse load q applied increases and constantly.It is clear that by InTherefore no matter film rotational angle theta have how it is small (the transverse load q no matter applied have how It is small) because sin θ ≠ tan θ, film small angle tower assumes that certain calculation error will be brought.For example when θ=20 °, Sin θ=0.34202, and tan θ=0.36397, so if using sin θ=tan θ hypothesis (i.e. film small angle tower is assumed), Itself error ((tan θ-sin θ)/sin θ) of the hypothesis so used just has been over 6%, so, for circular membrane most Large deflection w_mCalculating for, small angle tower assumes that brought calculation error, just already close to 3%, is still more also existed all Such as measure other errors.In general, the error of fine measuring instrument itself it is generally desirable to be less than 3%, so, based on film The existing method that small angle tower is assumed, circular membrane maximum defluxion determination during due to cannot be used for rotational angle theta more than 20 ° of situations, therefore Also the design application of fine measuring instrument cannot be used for！

If it should be noted that to meet approximate condition sin θ ≌ tan θ, then just have to limit applied horizontal stroke Can not be excessive to load q, this is also implied that, the analytic solutions under film small angle tower assumed condition are asked for solving actual techniques Topic, has the restrictive condition that applied transverse load q can not be excessive, certainly, this restrictive condition is many technologies Application field is undesirable.In fact, being easy to show under those relatively thin, relatively soft films, transverse load larger Corner, also has as the film of a classes such as rubber, their the usual very little of Young's modulus of elasticity value (the Young's modulus of elasticity value of rubber About 7.84MPa), because this kind of film has a larger deformability, thus the transverse load of very little can just make the film be in Reveal larger corner.

But, if it is possible to influence of the film rotational angle theta to calculation error is thoroughly eliminated, i.e., thoroughly abandons film small angle tower false If so as to thoroughly eliminate the restrictive condition that transverse load q can not be excessive, providing solution for big corner film problem, this nothing It is suspected to be a very valuable job, this also exactly technical problems to be solved by the invention.

The content of the invention

The present invention, is directed to the Study of exact analytical solutions for the circular membrane problem that Uniform Loads following peripheral is fixed to clamp, Film small angle tower thoroughly is abandoned it is assumed that abandoning sin θ=tan θ assumed condition in the solution procedure of analytic solutions, is allowedSo as to completely eliminating influence of the film rotational angle theta to calculation error, and then give and be applied to Big corner circular membrane maximum defluxion w under uniform load_mDetermination method, BROAD SUMMARY is as follows：

The clamping dress that a kind of determination method of big corner circular membrane maximum defluxion under uniform load is a with an inside radius Put, be that E, Poisson's ratio are that ν, thickness are being fixed to clamp without prestressed film for h by Young's modulus of elasticity, forming a radius is The circular membrane structure that a periphery is fixed to clamp, and a uniform load q is laterally applied to circular membrane, based on this axial symmetry circle The Accurate Analysis of film standing balance problem, will obtain the maximum defluxion w after deformation of thin membrane_mParsing relation with load q isWherein function g (c) is：

And c value is determined by equation ν=2cf ' (c)/f (c)+1, function f (c) and f ' (c) are respectively in formula：

With

So, as long as accurately measuring the horizontal uniform load q applied, it is possible to determine the maximum after deformation of thin membrane Amount of deflection w_m。

Compared with prior art, the present invention has the advantages that：The vacation of film small angle tower is thoroughly abandoned due to of the invention If, therefore influence of the film rotational angle theta to calculation error is completely eliminated, thus the method for the present invention is applicable not only to film corner θ is more than 20 ° of situation, and is also applied for the situation that film rotational angle theta is less than 20 °, so that the horizontal stroke to being applied also just is not present To the restrictive condition of uniform load q sizes, it is convenient that this is provided to the design application such as diaphragm type precision instrument, instrument, and this is this The innovation of method.

It is noted that the present invention is (" a kind of just for the method without prestressed circular membrane problem, and before the applicant The method for determining uniform load lower prestress circular membrane maximum immunity value ", number of patent application：201410238568.7) although adopting With film small angle tower it is assumed that but can solve the problems, such as there is prestressed circular membrane under small angle tower situation.

Brief description of the drawings

Fig. 1 is the loading organigram for the circular membrane that Uniform Loads following peripheral is fixed to clamp, in Fig. 1,1- circles Film, 2- clamping devices, symbol therein is that a represents the inside radius of clamping device and the radius of circular membrane, and r represents radially to sit Mark, w (r) represents the lateral coordinates at point r, and q represents horizontal uniform load, w_mThe maximum defluxion of circular membrane is represented, θ represents film Corner after deformation.

Embodiment

Technical scheme is described in further detail below in conjunction with the accompanying drawings：

As shown in figure 1, with inside radius a=20mm clamping device, by Young's modulus of lasticity E=7.84MPa, Poisson Fixed to clamp than ν=0.47, thickness h=0.06mm without prestressed rubber film, form radius a=20mm periphery The circular membrane structure fixed to clamp, is laterally applied to uniform load q to circular membrane, and allows applied uniform load q from 0.01MPa 0.1MPa is slowly increased to, using the method given by the present invention, it may be determined that go out the round rubber film and carried laterally uniform Maximum defluxion w under lotus q effects_m, as shown in table 1.

In table 1, the data of rotational angle theta are the film corner values (degree) at r=10mm, in addition, using film to reflect Small angle tower assumes brought error, to embody beneficial effects of the present invention, the applicant also use before method (" one Plant the method for determining uniform load lower prestress circular membrane maximum immunity value ", number of patent application：201410238568.7, make it In prestressing force σ₀=0 is without prestressed situation), the round rubber film is given in corresponding horizontal uniform load q Maximum defluxion w under effect_m。

As can be seen from Table 1, method before as a result of film small angle tower it is assumed that small angle tower assumes the error of itself ((tan θ-sin θ)/sin θ) constantly increases with uniform load q increase, from film maximum defluxion w_mTwo methods meter Calculate result to see, as q=0.1MPa, by using film small angle tower it is assumed that the w that film rotational angle theta is brought_mCalculation error just Already exceed 16% (and calculation error that this film rotational angle theta is brought is not present in calculating process of the present invention), this numeral Have been over engineering structure calculation error 15% is permitted fair value, still more also there are other errors such as measurement！

It can see from the data in table 1, the present invention is thorough in order to thoroughly eliminate influence of the film rotational angle theta to calculation error Bottom abandons film small angle tower it is assumed that it has the technical effect that it will be evident that the present invention also completely eliminates applied transverse direction in addition Load value can not be excessive restrictive condition, this is applied to correlation technique provides convenient, and these all embody the beneficial of the present invention Effect.

Table 1

Finally illustrate, the above embodiments are merely illustrative of the technical solutions of the present invention and it is unrestricted, although applicant The present invention is described in detail with reference to preferred embodiment, it will be understood by those within the art that, to the present invention's Technical scheme is modified or equivalent substitution, without departing from the objective and scope of technical solution of the present invention, all should be covered Among scope of the presently claimed invention.

Claims (1)

1. a kind of determination method of big corner circular membrane maximum defluxion under uniform load, it is characterized in that：It is a's with an inside radius Clamping device, is that E, Poisson's ratio are that ν, thickness are being fixed to clamp without prestressed film for h by Young's modulus of elasticity, forms one The circular membrane structure that radius fixes to clamp for a periphery, and a uniform load q is laterally applied to circular membrane, make turning for film There is the situation more than 20 ° in angle θ, then the Accurate Analysis based on this axial symmetry circular membrane standing balance problem, utilizes load Measured value q, the maximum defluxion w after deformation of thin membrane is determined by below equation_m：

<mrow> <msub> <mi>w</mi> <mi>m</mi> </msub> <mo>=</mo> <msup> <mrow> <mo>(</mo> <mfrac> <mrow> <msup> <mi>a</mi> <mn>4</mn> </msup> <mi>qc</mi> </mrow> <mrow> <mn>2</mn> <mi>hE</mi> </mrow> </mfrac> <mo>)</mo> </mrow> <mrow> <mn>1</mn> <mo>/</mo> <mn>3</mn> </mrow> </msup> <mi>g</mi> <mrow> <mo>(</mo> <mi>c</mi> <mo>)</mo> </mrow> <mo>,</mo> </mrow>

Wherein function g (c) is：

And c value is by equation

ν=2cf ' (c)/f (c)+1

It is determined that, function f (c) and f ' (c) are respectively in formula：

With