APPLICATION OF THE METHOD OF OBLIQUE TRIANGLES TO THE STUDY ON AN ANALOG COMPUTER OF THE KINEMATICS OF PLANAR CLASS IV MECHANISMS

A. G. KOKIN

(Presented by the scientific seminar of the computing department of the Research Institute of Electronic Modeling)

The application of methods for modeling the kinematics of planar mechanisms with the use of analog computing equipment makes it possible not only to reduce the labor required for studying the kinematics of various types of planar mechanisms and to increase the accuracy and speed of computations, but also to carry out kinematic analysis of mechanisms of classes III and IV [1], [2]. One, and so far the only, machine method suitable for modeling the kinematics of planar mechanisms of classes III and IV is the method of oblique triangles [3]. However, the application of the method in modeling the kinematics of planar mechanisms of classes III and IV encounters certain difficulties. Modeling on an analog computer the kinematics of planar mechanisms of classes III and IV does not differ essentially from modeling the kinematics of planar mechanisms of class II [4], [5]. The principle remains the same: the mechanism is geometrically decomposed into oblique triangles, the solution of which is found on the analog computer. The distinctive feature of modeling the kinematics of planar mechanisms of classes III and IV is that such mechanisms have several possible variants of geometric decomposition into oblique triangles, and it is necessary to choose an optimal variant such that there is sufficient data for their successive solution. In decomposition into oblique triangles, groups are clearly separated. The solution begins with the initial triangle, which includes the driving link. The difficulty lies in selecting the initial triangle of the group, since all triangles in the group are determined equally, and in solving each of the triangles it is necessary to know the solution of the others, including in the assignment of this initial triangle. In addition to being used for solving each triangle of the group separately, the implicit method is applied in solving the entire group [6].

Let us consider several variants of modeling the kinematics of a planar class IV mechanism, the application of the method of oblique triangles, and the implicit method for solving the triangles of a group. Figure 1 shows a class IV mechanism and its geometric decomposition into oblique triangles. The motion is specified by the driving link \(OA\), by the angle \(\varphi\). The resulting system consists of 22 interconnected oblique triangles. These are \(OAB\), \(OAD\), \(ABD\), \(BCD\), \(AHD\), \(AKH\), \(AEK\), \(AEH\), \(ABE\), \(BEL\), \(BFL\), \(BEF\), \(BCF\), \(CFM\), \(MCG\), \(FCG\), \(CGD\), \(GND\), \(DHN\), \(GHD\), \(EFH\), \(FGH\). The solution begins with the triangles \(OAB\) and \(OAD\). After determining the values of \(AB\), \(AD\) by the law of cosines using the implicit method and their position in the coordinate system, i.e., the angles \(\psi_1\), \(\psi_2\), we continue the solution by considering the triangle \(AHD\) or any other triangle of the group. For its

For the determination we specify the angle \(\alpha\) and the length of the side \(AH\). The angle \(\alpha_1\) is found by an implicit method, using the law of cosines, from the expression:

\[ AH^2 - AK^2 - KH^2 + 2AK \cdot KH \cos \alpha_1 = -\frac{\alpha_1}{\mu}, \tag{1} \]

\(\mu\) is the gain coefficient of the tracking amplifier, \(\mu > 1 \cdot 10^6\).

The angle \(\alpha_2\) is equal to:

\[ \alpha_2 = \angle HKE - \alpha_1 \tag{2} \]

The side \(AE\) is determined from triangle \(AEK\):

\[ AE^2 - EK^2 - AK^2 + 2EK \cdot AK \cos \alpha_2 = -\frac{AE}{\mu}. \tag{3} \]

The angle \(\alpha_3\) is found from triangle \(AEH\):

\[ EH^2 - AE^2 - AH^2 + 2AE \cdot AH \cos \alpha_3 = -\frac{\alpha_3}{\mu}. \tag{4} \]

Fig. 1. First variant of the geometric decomposition of a planar mechanism of class \(IV\) into oblique triangles.

The angle \(BAD\) is determined from triangle \(ABD\):

\[ BD^2 - AB^2 - AD^2 + 2AB \cdot AD \cos \angle BAD = -\frac{\angle BAD}{\mu}. \tag{5} \]

The unknown angle \(\alpha_4\) will be equal to:

\[ \alpha_4 = \angle BAD - \alpha_3 - \alpha, \tag{6} \]

From triangle \(ABE\) we determine the length of the side \(BE\) and the angle \(\alpha_5\):

\[ BE^2 - AB^2 - AE^2 + 2AB \cdot AE \cdot \cos \alpha_4 = -\frac{BE}{\mu}, \]

\[ AE^2 - AB^2 - BE^2 + 2AB \cdot BE \cos \alpha_5 = -\frac{\alpha_5}{\mu}. \tag{7} \]

Then the process of determining the sides and angles of the following oblique triangles continues until the length of the side \(HD\) and the angle \(\alpha_{19}\) have been determined. Then, proceeding from the specified value of the length of the side

$AH$ and the obtained values of the length $HD$ and the angle $\alpha_{19}$, we find from triangle $AHD$ the length of the side $AD^*$. Since the value $AD$ is known, the difference between the obtained and the given value of $AD$ is used to form the variable

Fig. 2. Block diagram for modeling the kinematics of a planar mechanism of class $IV$.

the length of the side $AH$; a system for determining the side $AH$ by the implicit method was obtained:

\[ AD^* - AD = -\frac{\Delta AH}{\mu}. \tag{8} \]

The change in the magnitude of the angle \(\alpha\) is determined from the condition that the sum of the interior angles of the quadrilateral \(ABCD\) is \(360^\circ\).

\[ < BAD + ABC + <BCD + <CDA - 360^\circ = -\frac{\Delta \alpha}{\mu}. \tag{9} \]

Figure 2 gives a block diagram for modeling the kinematics of a planar class IV mechanism. The block diagram consists of single-type models, interconnected, for solving transcendental or algebraic equa-

Fig. 3. Second variant of the geometric decomposition of a planar class IV mechanism into oblique triangles.

tions with one unknown. The number of such models will be determined by the number of triangles, which in the present case is equal to 22. The number of analog blocks required, differing in purpose, in the scheme will be determined by the following figures: adders — 76, tracking amplifiers — 31, multiplication blocks — 37, functional \(\cos\) blocks — 26. Such a scheme is comparatively difficult to assemble on universal analog computing machines, since there may arise the problem of a shortage of analog blocks of various purposes. The system can be composed of separate unified blocks (adders, multiplication blocks, \(\sin\), \(\cos\) blocks)

in conjunction with the available analog computer. Such standardized units are manufactured industrially.

The problem of a shortage of analog equipment can be partially solved if other variants of the geometric decomposition of a planar class IV mechanism are considered. In this case, two four-bar linkages expressed implicitly and connected with one another may be obtained [2]. It is possible to isolate, in a class IV group, an implicit four-bar linkage, as shown in Fig. 3. The entire calculation of the planar mechanism in this case is based on the solution of the four-bar linkage \(ABCD\), for which we regard the angle \(\alpha\) and the length of the side \(BC\) as prescribed. The value \(AD\) is determined from the oblique triangle \(OAD\). The magnitude \(BC\) and its position in the coordinate system determine the position of the quadrilateral \(EFGH\), and of the points \(M\) and \(N\). Knowing the values of the lengths of the links \(PM\), \(NQ\), and the coordinates of the points \(P\) and \(Q\), we determine, by the implicit method, the change in the length of the side \(BC\) and of the angle \(\alpha\), which, in turn, is implicitly related to the angle

Fig. 4. Geometric decomposition of a planar four-bar mechanism.

\(\alpha_1\) through the lengths of the links of the four-bar linkage \(ABCD\). Instead of the four-bar linkage \(ABCD\), one may consider the four-bar linkage \(ABMP\) or \(ABNQ\); the approach to solving the mechanism remains exactly the same. For modeling the kinematics of a planar class IV mechanism in this case, we have the oblique triangles \(OAD\), \(ABD\), \(BCD\), \(BEC\), \(FEH\), \(FGH\). Having solved these oblique triangles and found the coordinates of the points \(N\) and \(M\), we determine the lengths of the links \(PM\) and \(NQ\). Comparing the obtained values of \(PM\) and \(NQ\) with the prescribed constant values, we, as it were, “attach” the mechanism in the coordinate system to the fixed pivots \(P\) and \(Q\). In this process, the obtained \(PM\) and \(NQ\) become rigid constant constraints, while the difference tracked by the follower amplifiers goes to the formation of the variable values \(BC\) and \(\alpha\). The circuit obtained in this way is a system for finding the unknowns \(BC\) and \(\alpha\) by an implicit method. In this variant of the geometric decomposition of the mechanism into oblique triangles and their solution on an analog computer, 43 adders, 15 follower amplifiers, 17 multipliers, and 21 sine–cosine functional units are used. As can be seen, this method, in comparison with those considered above, gives a saving in the amount of analog equipment employed.

Let us consider in more detail the application of the method of implicit functions to modeling the kinematics of planar mechanisms. In addition to the application of the method of implicit functions for solving individual oblique triangles, it is used in a more general form both in modeling the kinematics of planar class III mechanisms [1] and in the examples considered above of modeling the kinematics of planar class IV mechanisms. Let us consider its application using as an example the solution of a planar four-bar hinged mechanism, decomposing the mechanism into oblique triangles as shown in Fig. 4. None of the triangles is determined; only the sides of the triangles \(OA\), \(AB\), \(BC\), \(OC\) and the angle \(\varphi\) are known. Taking the angle \(\alpha\) as prescribed, we find the side \(OB\) of the triangle \(OAB\) by the cosine theorem:

\[ AB^2 - OA^2 - OB^2 + 2OA \cdot OB \cdot \cos\alpha = -\frac{OB}{\mu}. \tag{10} \]

The angle \(\alpha_1\) is equal to:

\[ \alpha_1=\varphi-\alpha \tag{11} \]

From triangle \(OBC\) we find one of the known sides, for example \(BC\):

\[ BC^2-OB^2-OC^2+2OB\cdot OC\cdot \cos\alpha_1=-\frac{BC}{\mu}. \tag{12} \]

When the angle \(\varphi\) in triangle \(OBC\) changes, the angle \(\alpha_1\) will change. When \(\alpha_1\) changes, the magnitude of link \(BC\) will change. But in fact \(BC\) is a—

Fig. 5. Block diagram for solving a four-link mechanism.

constant quantity; therefore it is necessary to make the obtained variable length \(BC\) rigid by comparing it with the specified constant value \(BC\), switching in a tracking amplifier that makes the difference between the obtained and specified values of \(BC\) close to zero. The tracked difference is fed in order to change the angle \(\alpha\). This change will continue until, according to expression (10), the variable length of side \(OB\) satisfies triangle \(OBC\). Such a tracking system makes it possible to determine the angle \(\alpha\) by an implicit method. The block diagram for solving the triangles is shown—

Fig. 6. Curves of variation of the sides and angles of a planar four-link hinged mechanism, obtained on an analog computer.
\(BC\), \(\alpha\), \(OB\) are the values of the side lengths and angles for a quadrilateral with the following data: \(OA=30\), \(AB=64\), \(BC=50\), \(OC=60\). \(BC'\), \(OB'\) are the values of the side lengths of the triangles for a specified constant \(\alpha'\).

in Fig. 5. Figure 6 shows the curves of variation of the sides and angles of a planar four-link hinged mechanism, obtained on the EMU-10 analog computer.

Having determined the sides and angles of the triangles of the four-link mechanism, we can easily find the position of any points of the mechanism links.

The application of the method of oblique triangles in combination with the method of implicit functions for modeling the kinematics of planar mechanisms makes it possible to carry out the kinematic analysis of class IV mechanisms; moreover, the difficulties that arise in this case, consisting in an increase in the analog-computer equipment and in the use of the implicit method for finding functions specified implicitly, are overcome by rational construction of analog circuits and by considering various variants of the geometric decomposition of mechanisms, which is most essential for complex mechanisms of classes III and IV. The application of the implicit method in solving a class IV group is similar to its use in modeling the kinematics of planar mechanisms of classes II and III [1], [5].

The analysis carried out shows that the investigation on an analog computer of the kinematics of planar class IV mechanisms by means of the method of oblique triangles does not introduce fundamental difficulties into the solution of the problem. This makes it possible, given a sufficient amount of analog equipment, to solve complex kinematic problems.

REFERENCES

1. A. G. Kokin. Modeling of planar class III mechanisms on analog computing machines. Izv. TPI, vol. 187 (in press).

2. A. G. Kokin. Modeling of planar class IV mechanisms on analog computing machines. Izv. TPI, vol. 203 (in press).

3. I. G. Vintizenko, A. G. Kokin. A new method for modeling the kinematics of planar mechanisms by solving oblique triangles. Report at the All-Union Seminar on the Theory of Machines and Mechanisms. NII MASH, Academy of Sciences of the USSR, 1969, October 28.

4. I. G. Vintizenko, A. G. Kokin. Method of analog modeling of planar mechanisms (with rotation of the driving link through 360°). Izv. TPI, vol. 187 (in press).

5. Yu. Ya. Kovlin, I. G. Vintizenko, A. G. Kokin. Structural modeling of planar mechanisms on analog computing machines. Report at the All-Union Seminar on the Theory of Machines and Mechanisms. NII MASH, Academy of Sciences of the USSR, 1969, October 28.

6. A. N. Lebedev. Modeling of transcendental equations. Sudpromgiz, 1963.