SU5788A1 - Instrument for the mechanical integration of the Poisson equation - Google Patents

Description

The differential Poisson equation for a flat section is: 2, d% ...

       where r p (s1 y). If the boundary conditions can be given in the form 0 0 on the contour of the section, then the aforementioned equation applies to a whole range of applied physics problems, for example:

1) to the determination of the shape of the deflection of a rubber film stretched over a certain contour and subjected to pressure p: then one gets:

 -T

where c depends only on the material of the film, its thickness and the steppe of pathepie;

2) to determine the velocity distribution over the cross section of a laminar fluid flow, the kinematic viscosity of which is equal to v, and the effective component of acceleration of gravity is equal to gi (where i is the so-called hydraulic slope); then receive:

2

 at

Mathematical integration of equation (1) is possible only for some of the simplest cases of a geometrically regular contour shape. The proposed device makes it possible to determine the average value of s for any contour with absolutely arbitrary 1 form.

The device (see drawing) consists of two (copper) cylinders, with which a round medal rim is clamped with a thin rubber uniformly patched on it. The upper and lower cylinders are connected by flexible tubes with two-dimensional piezometers. The inside of the cylinders is filled with water (or other liquid), which in the piezometers gives, correspondingly, a height. If the level in both piezometers is the same, i.e. li n.2 fiQ, the rubber film, which is subject to equal pressure on both sides, does not have a deflection. If now the volume of the liquid is taken from the second piezometer (/

bones, equal - L where d - diameter.

both piezometers, and pour this volume with a burette into the first piezometer, then the difference between the levels / tj - h. is obtained, which will be equal to:

Jt - ho (h - h,.. (2)

where b denotes the height in the piezometer, corresponding to the volume of the obtained rubber deflection:

.G

  / sdco with S ,,,. . (3) p /

(h - the average value of the deflection, w - sectional section).

From this it is possible that by measuring the quantities 111, 2 and l On, one can directly calculate the quantity r, „.

The set of the set to be, for example, applied to the resolution of the following problem: to find the average speed of a uniform fluid flow through the cross section of a certain shape and size. For a circular section of radius r, the mathematical solution is:

(/ (2

, „

8gI, respectively:

g, „pr-.

Clamping the rubber on both sides with suitable plastic, with a circular radius of /, and making the above mentioned piezometric measurements, find:

(A | - l, d- „ItfS

    Tm-t ,,

ft / IIff

where do you get:

2c, () -

... (four)

(j.

Thus, the equilibrium coefficient of the proposed device was found (for a given rubber film c). Now it’s necessary to determine the value of gs, for some form of section (i.e., to solve the problem), you need to make two diaphragms with cuts of this particular form, clamp the above-mentioned rubber film between them and make piezometric measurements. Then receive:

dgs

V ,,, S ,,

V (hi - A,)

lj, / h- / t2 .., L

 V2 / 4 °, V (h, -), where is the co-area of the test section, l all other values are partly known (gr, I, V, s, d), are partly obtained by measurement on the instrument (foj, h, h.

The accuracy of the proposed instrument depends on: 1) the quality of the rubber, 2) the raviomerpogti of its initial tension on the rim and 3) the thoroughness of its clamping diaphragm.

FIG. 1 of the drawing shows a longitudinal section of the proposed device according to AL FIG. 2; pas figs. 2 - device in plan (part of the upper tank is detached); in fig. 3-section BB of FIG. 1 with the clamps removed in the upper tank and in FIG. 4-piezometers with a scale, fastened iomoi) Yu rod with a stand device.

The upper and lower vessels (for example, copper bakp) 1 and 2 supply the kenas on the adjacent edges with annular thickenings 3 of that material. Between them, there is an over / kata (for example, copper) frame 4 with a rubber membrane 5 tensioned in it and 5 plates laid on both sides (for example, copper) with a cutout 5 AND of the studied contour. Compression is carried out with a brush (for example, copper) clamps 7. Each of the tanks is connected by means of pipes 8 to piezometers 9 on a common millimeter scale. The device rests on a (for example, cast iron) stand 10 with three microscopic screws 11 and two levels 12. Crane 13 is used to fill with liquid, snatch 14 is for de-airing, valve 15 is for emptying the upper tank and valve 16-for removing air from nilshego tank.

The subject of the patent.

A device for the mechanical integration of the Poisson equation as applied to a flat section of any contour, characterized in that it consists of vessels 1 and 2, between which annular thickened MP 3 is clamped to frame 4 with a rubber membrane 5 stretched on it, on which are applied on both sides plastic plates 6 with cutouts of the circuit under study, the cavities of which vessels 1 and 2, filled with liquid, are connected by means of pipes 8 with piezometric tubes 9.

E. G.