阅读说明：本技术 气体压力下与刚性板接触的圆薄膜的挠度的确定方法 (Method for determining the deflection of a circular membrane in contact with a rigid plate under gas pressure ) 是由 孙俊贻 李博 何晓婷 于 2021-08-17 设计创作，主要内容包括：本发明公开了一种气体压力下与刚性板接触的圆薄膜的挠度的确定方法：对一块半径为a、厚度为h、泊松比为ν、杨氏弹性模量为E的周边固定夹紧的最初平坦的圆形薄膜的一个侧面施加气体压力q,让气体压力作用下产生轴对称变形的圆形薄膜与一块平行于该最初平坦的圆形薄膜的刚性平板形成一个半径为b的圆形光滑接触区域,其中刚性平板与最初平坦的圆形薄膜之间的距离为H,那么基于对该圆形薄膜的轴对称变形问题的静力平衡分析,利用气体压力q的测量值,就可以确定出轴对称变形后的圆形薄膜在板/膜不接触的区域上的各点的挠度。(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.)

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

Determination of b, c₀、c₁And c₂、c₃、c₄、c₅、c₆、c₇、c₈、d₁、d₂、d₃、d₄、d₅、d₆、d₇、d₈Then by the equation

Determination of d₀Is finally given by the equation

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.

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

Wherein r represents the distance from any point on the circular film after axisymmetric deformation to the symmetry axis thereof, and b < r < a,

and b, c₀、c₁And c₂、c₃、c₄、c₅、c₆、c₇、c₈Is given by the equation

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

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

To obtain d₀0.19519, the final equation

To obtain

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.