SU67116A1 - Photoelectric method of differentiation of functions - Google Patents

Description

The approximate value of the derivative of any function can be obtained on the basis of the following considerations.

Let it be a function

y- / W (1)

If this function is plotted as a graph and then shifted along the x axis by some value D X, then the segments, and parallel to the y axis, enclosed between the graphs, i.e., y f (x) and ij - f (x + x ), represent the increments of two adjacent ordinates; //, and //, i of curve (I), separated by x from each other (Fig. 1 of the drawing).

By postponing these differences on an axed scale

(Dx in this case const) and X, we obtain with a known approximation the value of the derivative of this function y f (x).

Practically, these considerations can be implemented in a manner that is composed of n) of the present invention.

The graph of the differentiable function is plotted on the film with a flat pen held in an unchanged position relative to the paper plane, and then the graph is moved in front of the slot transverse to the direction of movement of the graph, located between the light source and the photocell, in order to change the photocell illumination depending on the thickness of the photocell. line graph, which is almost proportional to the magnitude of the first derivative.

The invention is explained in more detail by the further description and drawing, in FIG. 1 - 7 of which diagrams and parts of apparatus are shown that implement the proposed method,. as well as explanatory diagrams.

On a transparent plate or

The film A (Fig. 2) draws a graph y f (x) with a flat pen, the larger dimension of which must always move parallel to the x axis.

If we take an element of the graph in an enlarged scale (Fig. 3), then it is easy to see that the vertical segment enclosed between the upper and lower boundaries of the contour

iZ X-f ,,. i) X-fAv, (2) where) is the thickness of the pen.

The skip graph in front of a narrow section B of a constant cross section made in an opaque plate C and an illuminated-beam of light of constant intensity, for each moment will have an expression for the luminous flux (Fig. 4):

7 - / about 1 (about Q - Do),

where / is the light intensity, 5o is the area of the slit, AS is the area of the graph element, cut out by the slit.

Taking this element for a rectangle with dimensions o (width, at the slit) and L /, we obtain for the flux F perceived by the photocell F, the value

F I (Sg - a A I).

Knowing of equality (2), we get:

 (So-OL-GD / U; / a (I-X-D2 /) 1 (; (G-Lou).

Accepting the photocell characteristic of a linear, i.e. counting

i kiE / CiA-gF IcF,

where- (is the current given by the photocell, B is the illumination, k, and k is conMt, we get i kla for the photocell current, (,

that with / const gives r with (G - d y).

With this method, the current given by the photocell F is nronoriopalea I-D /, where /, generally speaking, it is much more than D1 /, which may affect the accuracy of the device.

To obtain proportionality S; it is possible to turn on the second photocell according to the positin transformation scheme or the liquid one (Fig. B) to take a film covered with an opaque layer (for example, carbon black); in this case, the pen will play the role of a flat spatula, removing the opaque layer, and thus the image of the curve will be represented by a transparent line on an opaque background.

Regardless of the way the curve is applied, the indicator is calibrated so that the scale zero corresponds to the value of the luminous flux area or (Z-X) or 52 oA, i.e. when the curve-cut element is parallel to the X-axis (Ar / 0).

In the device shown in FIG. 2, marked D - optical system, focusing, and the light flux F on the cathode of the photocell F.

The described method allows to obtain an approximate value of the derivative of this function in absolute value. By some complication, it is possible to achieve algebraic values.

According to the proposed method, it is possible to realize the automatic differentiation of any function continuously drawn by the recorder.

Let the transparent plate or film move with a constant velocity y, (Fig. 6).

The feather moves in the perpendicular direction, and, moving it, thanks to the transmission system, is empty)) will be the foreground logging y, whose derivative is subject to determination.

The pen will obviously draw a graph on the plate, the equation of which will be:

y f (O.

There is a correlation between the portable and the absolute and the speeds of the pen at any time:

V r, tg with /. (3)

where a is the angle formed by the tangent to the curve of the graph at a given point with the axis / or, which is the same, the vector of relative velocity and, with the same axis.

Consider in FIG. 3 enlarged graph element

.

The segment 4 / enclosed between the upper and lower boundaries of the curve is equal to

ii A -) - atga

OR, substituted here from (3), we get:

, ay

av AZ X4--: X4--

v,

A - thickness a

pen,

where is the width, with const,

target speed.

Dependence (4) can be used to determine and can be done using the instrument shown in FIG. 7 and generally similar to the device in FIG. 2

The transparent film A, coiling off the drum M, is wound on the drum W rotating at a constant speed.

Perpendicular to the direction of movement of the film, the flat feather P moves, and its displacements are proportional to the value of y, the derivative of which needs to be found.

A film with a graph drawn on it at f (t) passes in front of a narrow slit B, illuminated by a light source of constant intensity.

If you repeat the above calculations, we get that current.

given by the photocell F, will be equal to

i Ci (/. - C.I.).

a - him

where c | and the Constituents.

-

To obtain a more accurate result dt, you can apply the above methods for obtaining proportionality S f V.

Subject invention

A photoelectric method of differentiation of functions, characterized in that the graph of the differentiable function is applied to a film of a flat pen held in an unchanged position relative to the paper plane, and then the graph is moved in front of the slot transverse to the direction of movement of the graph, between the light source and the photocell, with the aim changes in the photoelement of MeiiTa depending on the component of the thickness of the LINE graph, which is almost proportional to the value of the first derivative.

9 tflh

r

t

FIG. five

FIG. b

TG-T

yf (t)