SYNTHESIS OF A GRAB CRANK-SLIDER MECHANISM

N. P. Tyrsa

(Presented by the scientific seminar of the Department of Applied Mechanics)

The grab mechanism in a motion-picture camera serves for the intermittent transport of film. Questions of designing grab mechanisms have been studied by V. Lichtenheld [3] and S. M. Provornov [2], who proposed graphical methods for synthesizing grab four-bar linkages. The analytical determination of the parameters of the kinematic scheme of grab mechanisms by means of general methods of mechanism synthesis is a rather cumbersome and difficult problem. However, it can be substantially simplified and brought to an engineering solution if a number of propositions of kinematic geometry and Chebyshev’s theory of best approximation are used.

The curve 1 or 2 (Fig. 1), along which the transporting point of the grab tooth moves, must, during the entry stroke, ensure conditions under which engagement of the tooth with the edge of the perforation hole takes place without any large joint path between them. This must eliminate the danger of tearing the bridge of the film under the action of the driving force. During the working stroke it is necessary that transport be carried out over a certain part of the rotation of the driving link of the mechanism, and that the trajectory of the leading point of the tooth correspond to the shape of the film channel. With a rectilinear film channel, one should obtain a sufficiently good approximation of the transporting segment of the grab curve to a straight line. Owing to this, both the wear of the working face of the tooth and of the edge of the perforated opening is reduced, as is the danger of taking the film out of a plane state of tension. During the stroke in which the tooth leaves engagement with the film, it is required that, in the final phase, the tooth not transport the film.

The fulfillment of a number of requirements imposed on the grab mechanism as a whole makes it necessary to arrange the limited number of parameters of the kinematic scheme of the mechanism in such a way as to satisfy the stated conditions as fully as possible. Below, analytical and graphical methods are considered for designing a crank-slider grab mechanism which, in our opinion, ensure the fulfillment of a number of the requirements listed.

In the synthesis of the grab mechanism we shall regard as given:
a) the angle of rotation $\varphi_{14}$ of the driving link and the segment $C_1C_4$ on the line $f—f$ of the film channel, corresponding to the transport stroke, with

$C_1C_4 = L_0 = 1$;

b) the position of the slider center $B_1$ with coordinates $X_{B_1}$, $Y_{B_1}$, and the direction of the line of displacement $b—b$ of its center—the angle $\nu$ (Fig. 2).

Fig. 1. \(AB\) — stroke of entry of the grab tooth; \(BC \cdot H_{\mathrm{gr}}\) — stroke of transporting the film; \(CD\) — stroke of exit of the tooth from the perforation.

Fig. 2

It is required to determine the positions of the centers of the moving and fixed hinges of the driving link of the mechanism.

1. To ensure high-quality transportation, let us require that the trajectory of the leading point of the grab tooth have four common points with the line of the segment \(C_1C_4\). The positions of the points \(C_1\), \(C_2\), \(C_3\), and \(C_4\) are determined by the formula

\[ L_i = 0.5L_0\left(1-\frac{\operatorname{Cos}\dfrac{2i-1}{2n}}{\operatorname{Cos}\dfrac{\pi}{8}}\right), \tag{1} \]

where \(i\) is the number of the interpolation node, \(i = 1, 2, 3, 4\);

\(n\) is the number of simple interpolation nodes, \(n = 4\);

\(L_0\) is the length of the prescribed approximation segment, \(L_0 = 1\);

\(L_i\) is the length of the segment of the prescribed approximation portion, measured from the beginning of the approximation portion to the \(i\)-th node.

1. The positions of the moving plane \(S_1\), \(S_2\), \(S_3\), and \(S_4\) of the mechanism are completely specified by the positions \(C_1B_1\), \(C_2B_2\), \(C_3B_3\), and \(C_4B_4\) of the connecting-rod segment \(C_1B_1\). Let us find the poles of the finite rotations \(P_{12}\), \(P_{13}\), \(P_{14}\), \(P_{23}\), \(P_{24}\), \(P_{34}\).

The auxiliary geometric loci \(M^{124}\) and \(M^{134}\), subtended respectively by the chords \(P_{12}P_{24}\) and \(P_{13}P_{34}\) at the central angle \(\varphi_{14}\), intersect at the points \(M'\) and \(M''\) (Fig. 3).

Fig. 3

It is obvious that

\[ \left. \begin{aligned} \angle \overline{P_{12}M'P_{24}} &= \angle \overline{P_{13}M'P_{34}} = \frac{1}{2}\varphi_{14};\\ \angle \overline{P_{12}M''P_{24}} &= \angle \overline{P_{13}M''P_{34}} = \frac{1}{2}\varphi_{14} \end{aligned} \right\} \tag{2} \]

To the center \(M'\) in the fixed plane \(\Sigma\) there corresponds, in \(S_1\), the circular point \(A'_1\); to the center \(M''\), the point \(A''_1\) (not shown in Fig. 3). On the basis of the validity of the expressions

\[ \left. \begin{aligned} \angle \overline{P_{13}M'P_{34}} &= \angle \overline{A'_1M'P_{14}};\\ \angle \overline{P_{13}M''P_{34}} &= \angle \overline{A''_1M''P_{14}} \end{aligned} \right\} \tag{3} \]

one may say that

\[ \angle \overline{A'_1M'A'_4}=\angle \overline{A''_1M''A''_4}=\varphi_{14} \tag{4} \]

Thus, the points \(M'\) and \(M''\) are points of the center curve \(M^{1234}\), whose circular points \(A'_1\) and \(A''_1\), from position \(S_1\) to position \(S_4\), rotate through the prescribed angle \(\varphi_{14}\). Placing, for example, at the point \(M'\) the center \(O\) of the fixed hinge of the crank, and at the point \(A'\) the center \(A_1\) of the moving hinge of the crank, we obtain the crank-slider mechanism \(OA_1B_1\), in which the connecting-rod point \(C_1\) passes through the segment \(C_1C_4\), while the crank \(OA_1\) turns through the angle \(\varphi_{14}\).

3. In the coordinate system connected with the frame of the mechanism, the coordinates of the fixed center \(O\) of the crank \(OA_1\) are calculated by the formulas:

\[ \left. \begin{aligned} Y_o&=-\frac{p_2-\sqrt{p_2^2-4p_1p_3}}{2p_1};\\ X_o&=\frac{Y_o q_1+q_2}{q_3}, \end{aligned} \right\} \tag{5} \]

where the coefficients \(p_1, p_2, p_3\) are found from the expressions

\[ \left. \begin{aligned} p_1&=\tan\frac{\varphi_{14}}{2}\left(q_1^2+q_3^2\right);\\[4pt] p_2&=\tan\frac{\varphi_{14}}{2}\left(2q_1q_2-m_1q_1q_3-m_2q_3^2\right)+q_1q_3n_2-q_3^2n_1;\\[4pt] p_3&=\tan\frac{\varphi_{14}}{2}\left(q_2^2-m_1q_2q_3\right)+n_2q_2q_3+q_3^2q_4. \end{aligned} \right\} \tag{6} \]

Here the following notation is used:

\[ \left. \begin{aligned} q_1&=\tan\frac{\varphi_{14}}{2}(m_4-m_2)-n_1+n_3;\\[4pt] q_2&=\tan\frac{\varphi_{14}}{2}(l_1-l_2)+l_3-l_4;\\[4pt] q_3&=\tan\frac{\varphi_{14}}{2}(m_1-m_3)+n_4-n_2;\\[4pt] q_4&=\tan\frac{\varphi_{14}}{2}l_1+l_3. \end{aligned} \right\} \tag{7} \]

\[ \left. \begin{aligned} m_1&=X_{12}+X_{24};\\ m_2&=Y_{12}+Y_{24};\\ m_3&=X_{13}+X_{34};\\ m_4&=Y_{13}+Y_{34}. \end{aligned} \right\} \tag{8} \]

\[ \left. \begin{aligned} n_1&=X_{12}-X_{24};\\ n_2&=Y_{12}-Y_{24};\\ n_3&=X_{13}-X_{34};\\ n_4&=Y_{13}-Y_{34}. \end{aligned} \right\} \tag{9} \]

\[ \left. \begin{aligned} l_1&=X_{12}\cdot X_{24}+Y_{12}\cdot Y_{24};\\ l_3&=X_{12}\cdot Y_{24}-Y_{12}\cdot X_{24};\\ l_2&=X_{13}\cdot X_{34}+Y_{13}\cdot Y_{34};\\ l_4&=X_{13}\cdot Y_{34}-Y_{13}\cdot X_{34}. \end{aligned} \right\} \tag{10} \]

The coordinates of the center of the moving joint \(A_1\) are found using the method of Professor S. A. Cherkudinov [1].

\[ \left. \begin{aligned} X_{A_1}&=\frac{Y_{13}-Y_{12}+X_{12}\tg\delta_{A_1 12}-X_{13}\tg\delta_{A_1 13}} {\tg\delta_{A_1 12}-\tg\delta_{A_1 13}};\\ Y_{A_1}&=Y_{12}+\tg\delta_{A_1 12}(X_{A_1}-X_{12}), \end{aligned} \right\} \tag{11} \]

where

\[ \left. \begin{aligned} \tg\delta_{A_1 12} &= \frac{ Y_o-Y_{12}-(X_o-X_{12})\tg\frac{\Theta_{12}}{2} }{ X_o-X_{12}+(Y_o-Y_{12})\tg\frac{\Theta_{12}}{2} };\\[6pt] \tg\delta_{A_1 13} &= \frac{ Y_o-Y_{13}-(X_o-X_{13})\tg\frac{\Theta_{13}}{2} }{ X_o-X_{13}+(Y_o-Y_{13})\tg\frac{\Theta_{13}}{2} }. \end{aligned} \right\} \tag{12} \]

Here \(\delta_{A_1 12}\) and \(\delta_{A_1 13}\) are the angles of inclination to the abscissa axis of the straight lines \(P_{12}A_2\) and \(P_{13}A_1\); \(\frac{\Theta_{12}}{2}\) and \(\frac{\Theta_{13}}{2}\) are the angles at \(P_{13}\) and \(P_{13}\) in the pole triangle \(P_{12}P_{13}P_{23}\) (Fig. 3).

The values of the trigonometric functions of the angles \(\Theta_{13}\) and \(\Theta_{12}\) can be found as follows:

\[ \left. \begin{aligned} \tg\frac{\Theta_{12}}{2} &= \frac{(X_{13}-X_{12})(Y_{12}-Y_{23})-(X_{12}-X_{23})(Y_{13}-Y_{12})} {(X_{13}-X_{12})(X_{12}-X_{23})+(Y_{12}-Y_{23})(Y_{13}-Y_{12})};\\[6pt] \tg\frac{\Theta_{13}}{2} &= \frac{(X_{13}-X_{12})(Y_{13}-Y_{23})-(X_{13}-X_{23})(Y_{13}-Y_{12})} {(X_{13}-X_{23})(X_{13}-X_{12})+(Y_{13}-Y_{23})(Y_{13}-Y_{12})}. \end{aligned} \right\} \tag{13} \]

From the known coordinates of the joints \(O\) and \(A_1\), it is not difficult to calculate the parameters of the kinematic scheme of the grab crank-slider mechanism.

1. In Fig. 2 the constructions are shown that are necessary for finding the centers \(O\) and \(A_1\) of the grab mechanism with a negative angle \(\varphi_{14}\). With accuracy sufficient for practice, it is assumed that \(C_2C_3=0.4\,C_1C_4\). The point \(O\) of intersection of the circles \(M^{124}\) and \(M^{134}\) gives the center of the fixed joint of the crank. To find the center of the joint \(A_1\), it is necessary:

a) to rotate the line \(OP_{14}\) about \(O\) through the angle \(\frac{\varphi_{14}}{2}\) counterclockwise (for \(\varphi_{14}<0\), clockwise) and draw the ray \(Oa\);

b) to rotate the line \(OP_{12}\) about \(P_{12}\) through the angle \(\frac{\Theta_{12}}{2}\) clockwise.

arrows; the point \(A_1\) of intersection with the ray \(Oa'\) gives the position of the center of the movable hinge of the crank \(OA_1\).

1. By varying the parameter \(\varphi_{14}\), the position of the center, and the direction of the line of displacement \(b-b\) of the slider \(B_1\), one can obtain families of mechanisms and thereby solve questions connected with the installability of the mechanism in the motion-picture camera, the required transport coefficient of the film, and the conditions for the entry and exit of the tooth from the perforation hole.

REFERENCES

1. S. A. Cherkudinov. Synthesis of Plane Hinged-Lever Mechanisms. Publishing House of the Academy of Sciences of the USSR, Moscow, 1959.

2. A. M. Melik-Stepanyan, S. M. Provornov. Parts and Mechanisms of Motion-Picture Apparatus. Iskusstvo Publishing House, Moscow, 1959.

3. V. Lichtenheldt. Synthesis of Mechanisms. Nauka Publishing House, Moscow, 1964.