Page:EB1911 - Volume 04.djvu/1006

such strips. Then to every such strip will correspond a strip of equal length x of the figures described by T₁ and T₂.

The distances of the points, T, T₁, T₂, from the axis XX may be called y, y₁, y₂. They have the values

y = l sin θ, y₁ = l cos 2θ, y₂ = −l sin 3θ,

from which

dy = l cos θ.dθ, dy₁ = - 2l sin 2θ.dθ, dy₂ = −3l cos 3θ.dθ.

The areas of the three strips are respectively

dA = xdy, dA₁ = xdy₁, dA₂ = xdy₂.

Now dy₁ can be written dy₁ = - 4l sin θ cos θdθ = −4 sin θdy; therefore

dA₁ = −4 sin θ.dA = −${\tfrac {4}{l}}$ ydA;

whence

A₁ = −${\tfrac {4}{l}}$ ∫ydA = −${\tfrac {4}{l}}$ Ay,

where A is the area of the given figure, and y the distance of its mass-centre from the axis XX. But A₁ is the area of the second figure F₁, which is proportional to the reading of W₁. Hence we may say

Ay = C₁w₁,

where C₁ is a constant depending on the dimensions of the instrument. The negative sign in the expression for A₁ is got rid of by numbering the wheel W₁ the other way round.

Again

dy₂ = −3l cos θ {4 cos² θ −3} dθ = −3 {4 cos² θ −3} dy = −3 {${\tfrac {4}{l^{2}}}$ y² −3} dy,

which gives

dA₂ = −${\tfrac {12}{l^{2}}}$y²dA + 9dA,

and

A₂ = −${\tfrac {12}{l^{2}}}$ ∫y²dA + 9A.

But the integral gives the moment of inertia I of the area A about the axis XX. As A₂ is proportional to the roll of W₂, A to that of W, we can write

I = Cw −C₂ w₂,
Ay = C₁ w₁,
A = C_c w.

If a line be drawn parallel to the axis XX at the distance y, it will pass through the mass-centre of the given figure. If this represents the section of a beam subject to bending, this line gives for a proper choice of XX the neutral fibre. The moment of inertia for it will be I + Ay². Thus the instrument gives at once all those quantities which are required for calculating the strength of the beam under bending. One chief use of this integrator is for the calculation of the displacement and stability of a ship from the drawings of a number of sections. It will be noticed that the length of the figure in the direction of XX is only limited by the length of the rail.

This integrator is also made in a simplified form without the wheel W₂. It then gives the area and first moment of any figure.

While an integrator determines the value of a definite integral, hence a mere constant, an integraph gives the value of an indefinite integral, which is a function of x. Analytically if y is a given function f(x) of x Integraphs.and

Y = $\textstyle \int _{c}^{x}$ydx or Y = ∫ydx + const.

the function Y has to be determined from the condition

dY/dx = y.

Graphically y = f(x) is either given by a curve, or the graph of the equation is drawn: y, therefore, and similarly Y, is a length. But dY/dx is in this case a mere number, and cannot equal a length y. Hence we introduce an arbitrary constant length a, the unit to which the integraph draws the curve, and write

dY/dx = y/a and aY = ∫ydx

Now for the Y-curve dY/dx = tan φ, where φ is the angle between the tangent to the curve, and the axis of x. Our condition therefore becomes

tan φ = y/a.

Fig. 21.

This φ is easily constructed for any given point on the y-curve:—From the foot B′ (fig. 21) of the ordinate y = B′B set off, as in the figure, B′D = a, then angle BDB′ = φ. Let now DB′ with a perpendicular B′B move along the axis of x, whilst B follows the y-curve, then a pen P on B′B will describe the Y-curve provided it moves at every moment in a direction parallel to BD. The object of the integraph is to draw this new curve when the tracer of the instrument is guided along the y-curve.

The first to describe such instruments was Abdank-Abakanowicz, who in 1889 published a book in which a variety of mechanisms to obtain the object in question are described. Some years later G. Coradi, in Zürich, carried out his ideas. Before this was done, C. V. Boys, without knowing of Abdank-Abakanowicz’s work, actually made an integraph which was exhibited at the Physical Society in 1881. Both make use of a sharp edge wheel. Such a wheel will not slip sideways; it will roll forwards along the line in which its plane intersects the plane of the paper, and while rolling will be able to turn gradually about its point of contact. If then the angle between its direction of rolling and the x-axis be always equal to φ, the wheel will roll along the Y-curve required. The axis of x is fixed only in direction; shifting it parallel to itself adds a constant to Y, and this gives the arbitrary constant of integration.

In fact, if Y shall vanish for x = c, or if

Y = $\textstyle \int _{c}^{x}$ydx,

then the axis of x has to be drawn through that point on the y-curve which corresponds to x = c.

Fig. 22.

In Coradi’s integraph a rectangular frame F₁F₂F₃F₄ (fig. 22) rests with four rollers R on the drawing board, and can roll freely in the direction OX, which will be called the axis of the instrument. On the front edge F₁F₂ travels a carriage AA′ supported at A′ on another rail. A bar DB can turn about D, fixed to the frame in its axis, and slide through a point B fixed in the carriage AA′. Along it a block K can slide. On the back edge F₃F₄ of the frame another carriage C travels. It holds a vertical spindle with the knife-edge wheel at the bottom. At right angles to the plane of the wheel, the spindle has an arm GH, which is kept parallel to a