MECHANICS

Academician I. I. Artobolevskii

ON A MECHANICAL TRANSFORMER

The present paper sets forth the theory of a motion transformer of a new type. The transformer is a kinematic chain shown in Fig. 1. Link 2 rotates about the center \(A\) and is a two-sided crank lever with a rigid angle \(kAm\) equal to \(90^\circ\). The sides \(Ak\) and \(Am\) of lever 2 slide in sliders 5 and 7, which form rotational pairs at points \(C\) and \(D\) with sliders 4 and 6. Sliders 4 and 6 slide along the crossbar \(t — t\), belonging to link 3, which moves translationally in the fixed guides \(B\).

Fig. 1                Fig. 2

It is not difficult to see that this kinematic chain has two degrees of freedom. Consequently, if point \(C\) is guided along some curve \(q — q\), defined by the equation \(\Psi(\xi,\eta)=0\), then point \(D\) will move along the curve \(p — p\), defined by the equation \(\Phi(x,y)=0\). Thus, the indicated kinematic chain transforms the curve \(\Psi(\xi,\eta)=0\) into the curve \(\Phi(x,y)=0\).

Let us consider the relation between the coordinates of the curves \(\Psi(\xi,\eta)=0\) and \(\Phi(x,y)=0\).

From Fig. 1 it follows directly that

\[ x=\xi, \tag{1} \]

\[ y=\frac{\xi^2}{\eta}. \tag{2} \]

It is not difficult to see that the chain has the property of reversibility of motion transformation, i.e., if point \(D\) is guided along the curve \(p — p\), then point \(C\) will describe the curve \(q — q\).

Let the curve \(\Psi(\xi,\eta)=0\) be the straight line \(q-q\) (Fig. 2), whose equation is

\[ \eta=\xi \operatorname{tg}\alpha+n, \tag{3} \]

where \(\alpha\) is the angle formed by the axis \(Cq\) of the fixed guide \(E\), in which link 4 moves, with the axis \(Ax\). Eliminating the coordinates \(\xi\) and \(\eta\) from equations (1), (2), and (3), we obtain the equation of the curve \(\Phi(x,y)=0\):

\[ x^2-\operatorname{tg}\alpha \cdot xy-ny=0, \tag{4} \]

i.e., the equation of an equilateral hyperbola. Thus, the mechanism shown in Fig. 2 is a hyperbolograph.

Fig. 3

If the straight line \(q-q\) is parallel to the axis \(Ax\), then the six-link mechanism shown in Fig. 3 is obtained. We take the coordinate \(\eta\) equal to \(\eta=2p\), where \(p\) is a certain constant. From Fig. 3 and equations (1) and (2) it follows that

\[ x^2=2py, \tag{5} \]

i.e., the curve \(p-p\) will be a parabola with parameter equal to \(p\). It is not difficult to see that this case leads us to Antonov’s parabolagraph, the theory of which was given by N. B. Delone (1).

Let the curve \(\Psi(\xi,\eta)=0\) be a conic section given by an equation of the general form

\[ A\xi^2+2B\xi\eta+C\eta^2+2D\xi+2E\eta+F=0. \tag{6} \]

Eliminating the coordinates \(\xi\) and \(\eta\) from equations (1), (2), and (6), we obtain the equation of the curve \(\Phi(x,y)=0\) in the form

\[ Ax^2y^2+2Bx^3y+Cx^4+2Dxy^2+ +2Ex^2y+Fy^2=0. \tag{7} \]

This is a curve of the 4th order; consequently, the kinematic chain shown in Fig. 1 transforms curves of the 2nd order into curves of the 4th order.

If the conic section \(\Psi(\xi,\eta)=0\) is an ellipse with center at point \(A\) (Fig. 4) and semiaxes \(a\) and \(b\),

\[ b^2\xi^2+a^2\eta^2-a^2b^2=0, \tag{8} \]

then the equation of the transformed curve \(\Phi(x,y)=0\) has the form

\[ y^2=\frac{a^2}{b^2}\frac{x^4}{a^2-x^2}. \tag{9} \]

Fig. 4

The mechanism for transforming the ellipse (8) into the curve (9) is shown in Fig. 4. Point \(C\) of the transformer belongs to link 8, which enters into revolute pairs with sliders 9 and 10, sliding along the fixed ...

axes \(Ax\) and \(Ay\). In order that the point \(C\) of link 8 move along an ellipse with semiaxes \(a\) and \(b\), it must divide the segment \(EF\) in the ratio \(EC/CF = a/b\).

If the point \(C\) (Fig. 4) moves along the circle

\[ \xi^2+\eta^2-r^2=0, \tag{10} \]

then equation (7) assumes the form

\[ {}^2y=\frac{x^4}{r^2-x^2}. \tag{11} \]

This is the “kappa curve.” The mechanism for generating the “kappa curve” must have a crank \(AC\) (Fig. 4) of constant length \(r\). To obtain this mechanism in the mechanism shown in Fig. 4, the links 5, 8, 9, and 10, shown by dashed lines, must be omitted.

If the curve \(\Psi(\xi,\eta)=0\) is a parabola with vertex at the origin,

\[ \eta^2=2p\xi, \tag{12} \]

then the equation of the curve \(\Phi(x,y)=0\) will be

\[ y^2=\frac{1}{2p}x^3. \tag{13} \]

This is the equation of a semicubical parabola. To obtain, by means of the converter under consideration (Fig. 1), a mechanism for generating a semicubical parabola, one may use the Antonov parabolograph considered above (Fig. 3). The point \(D\) of this mechanism traces the parabola \(p—p\), given by equation (5). Let us attach to the mechanism a two-slider group, shown by dashed lines and consisting of sliders 6 and 7, which enter the revolute pair \(E\). Slider 6 must slide along a guide belonging to link 5, and slider 7 along the guide \(k—k\), belonging to link 2. The guide \(Dl\) forms a right angle with the straight line \(Ct\). The point \(E\) of the mechanism will describe the curve \(s—s\), whose equation is

\[ x'^2=\frac{1}{2p}y'^3, \tag{14} \]

i.e., a semicubical parabola.

Thus, using the proposed mechanical converter, by adjusting its links one can obtain mechanisms for generating a number of curves of different orders.

Institute of Machine Science
Academy of Sciences of the USSR

Received
5 IX 1956

REFERENCES

1. N. B. Delone, Transmission of rotation and mechanical tracing of curves by hinge-lever mechanisms, St. Petersburg, 1894.