MODELING OF PLANAR MECHANISMS OF THE THIRD CLASS ON ANALOG COMPUTERS

A. G. KOKIN

(Presented by the scientific seminar of the DCE, TPI)

In the analysis and design of planar mechanisms, analog computers (ACs) are finding ever broader application. Existing methods for modeling planar mechanisms have been considered mainly as applied to mechanisms of class II [3, 4, 5, 6]. However, alongside class-II mechanisms, mechanisms of class III have become widespread in practice; their modeling on ACs had not previously been considered.

Let us consider the modeling of class-III mechanisms by the method proposed by the authors in [3]. This is a method for modeling planar mechanisms by solving oblique triangles with variable side lengths on an AC. Its advantage over other methods is that, with the aid of this method, planar mechanisms of any structure and complexity can be modeled (with rotation of the driven links through \(360^\circ\), etc.). In addition, in all cases of its application this method ensures the stability of the modeling system, which is important in modeling class-III mechanisms.

Fig. 1

A planar mechanism of class III includes a three-driver group, which is attached to the main mechanism by its drivers [1]. Let us consider a class-III mechanism composed of a three-driver group and a class-I mechanism (Fig. 1). A distinctive feature of the proposed

A possible method for modeling such a mechanism on an analog computer is the geometric decomposition of it into two four-link mechanisms of class II with variable link lengths. The planar four-link mechanisms are modeled separately by the method of solving oblique triangles with variable side lengths and are then connected in a single coordinate system.

We decompose the mechanism (Fig. 1) into two four-link planar mechanisms \(OABO'\) and \(O'BDE\). \(BO'\) is the variable link of the four-link mechanism \(OABO'\).

Fig. 2

We decompose the four-link mechanism \(OABO'\) into two oblique triangles \(OAO'\) and \(ABO'\). From the triangle \(OAO'\) we determine \(AO'\) by the implicit-function method [2] from the formula

\[ AO'^2 - OA^2 - OO'^2 + 2OA \cdot OO' \cdot \sin \varphi = -\frac{AO'}{\mu}, \]

where \(\mu\) is the gain of the tracking amplifier. From the equations of projection of \(OAO'\) onto the coordinate axes we determine \(\sin \psi\), \(\cos \psi\).

\[ OA\cos\varphi - AO'\cos\psi = -\frac{\cos\psi}{\mu}, \]

\[ OA\sin\varphi + AO'\sin\psi - OO' = -\frac{\sin\psi}{\mu}. \]

We find the angle \(\gamma\) from the oblique triangle \(ABO'\), assuming that the length of the link \(BO'\) is known,

\[ BO'^2 - AB^2 - AO'^2 + 2AB \cdot AO'\cos\gamma = -\frac{\gamma}{\mu}. \]

The position of link \(AB\) in the coordinate system is determined by the angle

\[ a = 180^\circ - (\gamma + \psi). \]

To determine the length of link \(BO'\), it is necessary to know the coordinates of point \(B\) in the new coordinate system \(x'o'y'\).

\[ x'_B = OA \sin \varphi + AB \sin a - OO'; \]

\[ y'_B = OA \cos \varphi + AB \cos a. \]

\[ x'_B = OA \sin \varphi + AB[\sin \gamma \cdot \cos \psi + \cos \gamma \cdot \sin \psi] - OO'; \]

\[ y'_B = OA \cos \varphi + AB[\sin \gamma \cdot \sin \psi - \cos \gamma \cdot \cos \psi]. \]

For the four-link mechanism \(O'BDE\), from similar relations, with the coordinates of point \(B\) known, we can determine the position of the links in the coordinate system, i.e., the angles \(\varepsilon\), \(\xi\), \(\beta\). The coordinates of point \(C\) are determined by means of the angles \(\varepsilon\) and \(\xi\). The length \(O'C\) will then be equal to

\[ O'C^2 - x_B'^2 - y_B'^2 = -\frac{O'C}{\mu}. \]

Since the length \(O'C\) is in fact constant, the tracked difference is used to form the variable \(O'B\).

The block diagram for modeling a planar mechanism of the third class is given in Fig. 2. A special feature of the modeling scheme is that it consists of two similar modeling schemes for planar four-link mechanisms. This considerably facilitates unification and the setup of the problem when modeling mechanisms of this class.

In conclusion, it may be inferred that the modeling of arbitrarily complex mechanisms of classes II and III is possible provided that they are decomposed into planar four-link mechanisms. The method of modeling planar mechanisms by solving oblique triangles with variable side lengths, used in modeling planar mechanisms of class III, is a general method whose application does not depend on the class or complexity of the mechanism.

The use of analog computers for modeling planar mechanisms of classes II and III will significantly facilitate the calculation and design of mechanisms.

REFERENCES

1. I. I. Artobolevskii. Theory of Mechanisms. Fizmatgiz, 1965.

2. A. N. Lebedev. Modeling of Transcendental Equations. Sudpromgiz, 1963.

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

4. A. G. Kokin, I. G. Vintizenko. Synthesis of planar hinged mechanisms on analog computers. Izv. TPI, vol. 168.

5. V. A. Tyshkevich, E. S. Krasnoshchekii. Investigation of the kinematic properties of lever mechanisms by the method of electronic modeling. Theory of Mechanisms and Dynamics of Machines. Proceedings, Omsk Polytechnic Institute, 1967.

6. Timm. Modeling mechanisms with rigid links on analog machines. Design and Technology of Mechanical Engineering. Transactions of the American Society of Mechanical Engineers. Mir Publishers, No. 2, 1967.