CN113701679A - Method for determining the deflection of a circular membrane in contact with a rigid plate under gas pressure - Google Patents

Method for determining the deflection of a circular membrane in contact with a rigid plate under gas pressure Download PDF

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CN113701679A
CN113701679A CN202110944940.6A CN202110944940A CN113701679A CN 113701679 A CN113701679 A CN 113701679A CN 202110944940 A CN202110944940 A CN 202110944940A CN 113701679 A CN113701679 A CN 113701679A
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gas pressure
circular
film
circular film
deflection
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孙俊贻
李博
何晓婷
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Chongqing University
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    • G—PHYSICS
    • G01—MEASURING; TESTING
    • G01B—MEASURING LENGTH, THICKNESS OR SIMILAR LINEAR DIMENSIONS; MEASURING ANGLES; MEASURING AREAS; MEASURING IRREGULARITIES OF SURFACES OR CONTOURS
    • G01B13/00—Measuring arrangements characterised by the use of fluids
    • G01B13/24—Measuring arrangements characterised by the use of fluids for measuring the deformation in a solid

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Abstract

The invention discloses a method for determining the deflection of a circular film in contact with a rigid plate under gas pressure, which comprises the following steps: applying gas pressure q to one side surface of an initial flat circular film fixedly clamped at the periphery with the radius of a, the thickness of H, the Poisson ratio of v and the Young modulus of elasticity of E, and enabling the circular film which is subjected to axisymmetric deformation under the action of the gas pressure and a rigid flat plate which is parallel to the initial flat circular film to form a circular smooth contact area with the radius of b, wherein the distance between the rigid flat plate and the initial flat circular film is H, so that the deflection of each point of the circular film which is subjected to axisymmetric deformation on a plate/film non-contact area can be determined by utilizing the measured value of the gas pressure q based on the static balance analysis of the axisymmetric deformation problem of the circular film.

Description

Method for determining the deflection of a circular membrane in contact with a rigid plate under gas pressure
Technical Field
The invention relates to a method for determining the deflection of a circular film which is smoothly contacted with a rigid flat plate under the action of gas pressure.
Background
If a smooth-surfaced rigid plate is arranged parallel to the initially flat circular membrane and at a distance H from it on one side of the initially flat circular membrane, which is clamped firmly on the periphery, and a gas pressure q is applied to the circular membrane on the other side, the circular membrane initially deforms and deflects axially symmetrically on the side on which the rigid plate is located, and then, with increasing applied gas pressure q, the circular membrane gradually comes into contact with the smooth-surfaced rigid plate and gradually forms a circular smooth contact area with a radius b. Obviously, by adjusting the distance H between the rigid plate and the initially flat circular membrane, it is possible to ensure that the circular membrane under the action of the gas pressure q is always within the elastic deformation range during the contact with the rigid plate having a smooth surface.
The above physical phenomena that the circular film is firstly axially symmetrically deformed and deflected under the action of gas pressure q and then contacts a rigid flat plate with a smooth surface are used for developing a capacitive pressure sensor in a Micro Electro Mechanical System (MEMS). The situation before the plate/membrane contact corresponds to the so-called standard mode, while the situation after the plate/membrane contact corresponds to the so-called contact mode. The normal mode corresponds to a pressure sensor of the so-called non-parallel plate capacitor type, while the contact mode corresponds to a pressure sensor of the so-called central partial parallel plate capacitor type (i.e. one parallel plate capacitor in the plate/membrane contact area and one non-parallel plate capacitor outside the plate/membrane contact area). Thus, it is a prerequisite to design a non-parallel plate capacitor type pressure sensor and a partial center parallel plate capacitor type pressure sensor by giving an analytical solution of the mechanical behavior of the circular thin film under the action of the gas pressure q, regardless of the standard mode or the contact mode.
The invention is dedicated to the design research of a pressure sensor in a contact mode, namely, the design problem of a central local parallel plate capacitor type pressure sensor is researched, and the axial symmetry deformation and deflection problems of a circular film which is smoothly contacted with a rigid flat plate under the action of gas pressure are solved through analysis. In the establishment of the so-called out-of-plane equilibrium equation, the commonly used condition for limiting the size of the film corners is abandoned (i.e., the size of the film corners is not limited in the establishment of the out-of-plane equilibrium equation); the effect of the horizontal component of the gas pressure exerted on the membrane is taken into account when establishing the so-called in-plane equilibrium equation. This enables the analytical solution to be obtained for the problem of axisymmetric deformation and deflection to be applied to the case where the membrane is subjected to a large membrane rotation angle under the action of gas pressure, and new results have been made, which indicates that there is no such analytical solution in the prior art. The invention provides a method for determining the deflection of each point of a circular film which is in smooth contact with a rigid flat plate under the action of gas pressure on a plate/film non-contact area based on the obtained analytic solution of the axisymmetric deformation and deflection problems of the circular film which is in smooth contact with the rigid flat plate under the action of gas pressure (the deflection of each point of the film on the plate/film contact area is always equal to the distance H between the rigid flat plate and the initially flat circular film).
Disclosure of Invention
The method for determining the deflection of a circular membrane in contact with a rigid plate under gas pressure comprises the following steps: applying gas pressure q to one side surface of an initial flat circular film with the radius of a, the thickness of H, the Poisson ratio of v and the Young modulus of elasticity of E, and enabling the circular film which is subjected to axial symmetric deformation under the action of the gas pressure and a rigid flat plate which is parallel to the initial flat circular film to form a circular smooth contact area with the radius of b, wherein the distance between the rigid flat plate and the initial flat circular film is H, then based on the static balance analysis of the problem of axial symmetric deformation of the circular film, the analytical relation between the deflection w (r) of each point of the circular film on the area where the plate/film is not in contact and the applied gas pressure q is obtained
Figure BDA0003216321780000021
Wherein r represents the distance from any point on the circular film after axisymmetric deformation to the symmetry axis thereof, and b < r < a,
Figure BDA0003216321780000022
Figure BDA0003216321780000023
Figure BDA0003216321780000024
Figure BDA0003216321780000025
Figure BDA0003216321780000026
Figure BDA0003216321780000027
Figure BDA0003216321780000031
Figure BDA0003216321780000032
Figure BDA0003216321780000041
Figure BDA0003216321780000042
Figure BDA0003216321780000043
Figure BDA0003216321780000044
and b, c0、c1And c2、c3、c4、c5、c6、c7、c8Is given by the equation
Figure BDA0003216321780000045
Figure BDA0003216321780000046
Figure BDA0003216321780000047
Figure BDA0003216321780000048
Figure BDA0003216321780000049
Figure BDA0003216321780000051
Figure BDA0003216321780000052
Figure BDA0003216321780000053
Figure BDA0003216321780000054
Figure BDA0003216321780000055
Figure BDA0003216321780000056
Figure BDA0003216321780000057
Figure BDA0003216321780000058
Figure BDA0003216321780000061
Figure BDA0003216321780000062
Figure BDA0003216321780000063
Figure BDA0003216321780000071
Figure BDA0003216321780000072
Figure BDA0003216321780000073
Figure BDA0003216321780000074
Figure BDA0003216321780000075
And (4) determining.
Thus, by measuring the value of the gas pressure q, it is possible to determine the deflection w (r) of each point of the circular film after axisymmetric deformation in the area where the plate/film is not in contact (b < r < a) (the deflection of each point of the film in the area where the plate/film is in contact is always equal to the deflection of the rigid plate and the initially flat plateDistance H) between circular films, wherein r, a, b, H, H, w (r) are all in millimeters (mm) and E, q are all in Newton per square millimeter (N/mm)2) And v, c0、c1、c2、c3、c4、c5、c6、c7、c8、d0、d1、d2、d3、d4、d5、d6、d7、d8Q, e, β are dimensionless quantities.
Drawings
FIG. 1 is a schematic diagram of the problem of axisymmetric deformation and deflection of a circular thin film in smooth contact with a rigid flat plate under gas pressure, wherein 1 is the circular thin film after axisymmetric deformation, 2 is the rigid flat plate, 3 is a clamping device for the circular thin film, 4 is a geometric middle plane of the initially flat circular thin film, 5 is a fixed support, a is a radius of the circular thin film and an inner radius of the clamping device, b is a radius of a circular smooth contact area formed between the circular thin film after axisymmetric deformation and the rigid flat plate, o is a coordinate origin (centroid located at the geometric middle plane of the initially flat circular thin film), r is a radial coordinate (representing a distance from any point on the circular thin film after axisymmetric deformation to its symmetry axis), w is a transverse coordinate (representing deflection of the circular thin film after axisymmetric deformation), and q is a gas pressure acting on the circular thin film, h represents the distance between the initially flat circular membrane and the rigid flat plate.
Detailed Description
The technical scheme of the invention is further explained by combining the specific cases as follows:
as shown in fig. 1, for a block with a radius a of 10mm, a thickness h of 1mm, a poisson ratio v of 0.47, and a young's modulus E of 7.84N/mm2The gas pressure q is applied to one side surface of the initially flat circular membrane fixedly clamped at the periphery, so that the circular membrane which is axially symmetrically deformed under the action of the gas pressure and a rigid flat plate parallel to the initially flat circular membrane form a circular smooth contact area with the radius b, wherein the distance between the rigid flat plate and the initially flat circular membrane is H-2.5mm, measured gas pressure q 0.1N/mm2Then the method given by the invention is adopted, and the equation is expressed
Figure BDA0003216321780000081
Figure BDA0003216321780000082
Figure BDA0003216321780000083
Figure BDA0003216321780000084
Figure BDA0003216321780000085
Figure BDA0003216321780000086
Figure BDA0003216321780000087
Figure BDA0003216321780000088
Figure BDA0003216321780000091
Figure BDA0003216321780000092
Figure BDA0003216321780000093
Figure BDA0003216321780000094
Figure BDA0003216321780000095
Figure BDA0003216321780000096
Figure BDA0003216321780000101
Figure BDA0003216321780000102
Figure BDA0003216321780000103
Figure BDA0003216321780000111
Figure BDA0003216321780000112
Figure BDA0003216321780000113
Figure BDA0003216321780000114
B is 2.90362mm、c0＝0.11149、c1-0.04715 and c2＝-0.08187、c3＝-0.05657、c4＝-0.08930、c5＝-0.14466、c6＝-0.23246、c7＝-0.45082、c8＝-0.84485、d1＝-0.30795、d2＝-0.46531、d3＝-0.26695、d4＝-0.54876、d5＝-0.67733、d6＝-1.45870、d7＝-2.35960、d8-5.05579, then expressed by the equation
Figure BDA0003216321780000115
To obtain d00.19519, the final equation
Figure BDA0003216321780000116
To obtain
Figure BDA0003216321780000117
Therefore, at a gas pressure q of 0.1N/mm2The deflection w (r) of each point of the circular film on the area where the plate/film is not contacted is determined by the equation
w(r)＝-50.5579(0.1r-0.645181)8-23.596(0.1r-0.645181)7-14.587(0.1r-0.645181)6-6.7733(0.1r-0.645181)5-5.48757(0.1r-0.645181)4-2.66946(0.1r-0.645181)3-4.65307(0.1r-0.645181)2-0.307951r+3.93873
And determining, wherein r represents the distance from any point on the circular film after the axisymmetric deformation to the symmetry axis thereof, and 2.90362mm < r < 10 mm.

Claims (1)

1. The method for determining the deflection of the circular film in contact with the rigid plate under the gas pressure is characterized in that: applying gas pressure q to one side of an initial flat circular film fixedly clamped at the periphery with the radius of a, the thickness of H, the Poisson ratio of v and the Young modulus of elasticity of E, and enabling the circular film which is axially symmetrically deformed under the action of the gas pressure and a rigid flat plate which is parallel to the initial flat circular film to form a circular smooth contact area with the radius of b, wherein the distance between the rigid flat plate and the initial flat circular film is H, then based on the static balance analysis of the axial symmetry deformation problem of the circular film, utilizing the measured value of the gas pressure q, and using an equation to calculate the value of the static balance analysis
Figure FDA0003216321770000011
Figure FDA0003216321770000012
Figure FDA0003216321770000013
Figure FDA0003216321770000014
Figure FDA0003216321770000015
Figure FDA0003216321770000016
Figure FDA0003216321770000017
Figure FDA0003216321770000018
Figure FDA0003216321770000019
Figure FDA00032163217700000110
Figure FDA0003216321770000021
Figure FDA0003216321770000022
Figure FDA0003216321770000023
Figure FDA0003216321770000024
Figure FDA0003216321770000031
Figure FDA0003216321770000032
Figure FDA0003216321770000033
Figure FDA0003216321770000041
Figure FDA0003216321770000042
Figure FDA0003216321770000043
Figure FDA0003216321770000044
Determination of b, c0、c1And c2、c3、c4、c5、c6、c7、c8、d1、d2、d3、d4、d5、d6、d7、d8Then by the equation
Figure FDA0003216321770000045
Determination of d0Is finally given by the equation
Figure FDA0003216321770000046
Determining the deflection w (r) of each point of the axisymmetrically deformed circular film on the non-contact area of the plate/film, wherein r represents the distance from any point on the axisymmetrically deformed circular film to the symmetry axis thereof, and b < r < a, the units of r, a, b, H, H and w (r) are all millimeters (mm), and the units of E, q are all Newton per square millimeter (N/mm)2) And v, c0、c1、c2、c3、c4、c5、c6、c7、c8、d0、d1、d2、d3、d4、d5、d6、d7、d8Q, e, β are dimensionless quantities.
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Cited By (1)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
CN115307789A (en) * 2022-07-08 2022-11-08 重庆大学 Method for determining capacitance of contact type circular conductive film variable capacitor

Cited By (1)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
CN115307789A (en) * 2022-07-08 2022-11-08 重庆大学 Method for determining capacitance of contact type circular conductive film variable capacitor

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