METHOD OF ANALOG MODELING OF PLANAR ARTICULATED MECHANISMS (WITH ROTATION OF THE DRIVEN LINK THROUGH 360°)

I. G. Vintizenko, A. G. Kokin

(Presented by the scientific seminar of the Computing Laboratory)

At the present time, significant successes have been achieved in the analysis and synthesis of mechanisms whose link points describe various trajectories. The study of these questions is of important applied significance, since in the practice of machine design there very often arises the requirement that individual points of actuating mechanisms move along prescribed trajectories.

For the design and calculation of planar mechanisms, analog computing machines (ACM) have recently been used. Of great interest is the modeling of four-link mechanisms with an angle of rotation of the driven link through 360°, since such a mechanism is the basis for designing arbitrarily complex articulated mechanisms. There are several works devoted to modeling the kinematics of such mechanisms [2], [4], [5].

In work [5], the method of direct solution of transcendental equations obtained in projecting the mechanism onto the coordinate axes was used. However, as was shown in Bykhovskii’s work [3], the solution on an ACM of transcendental equations describing stable systems is often unstable. Therefore, in the later work [4], which uses the same methods as in work [5], stable modeling of the driven link of a four-link mechanism was observed only within the range 0—180°. Various methods for eliminating instability were proposed. In Bykhovskii’s work [3], a method was proposed for realizing an ultrastable structure; however, because instability arises before the system approaches the switching point, this method proves not entirely convenient. To eliminate instability of the system it was proposed (as in [3]) to use switches in the feedback lines, which would change the stability conditions [4], but such a method, when crossing the stability boundaries, does not give an exact picture of the rotation of the driven link, and therefore exact analysis and calculation of mechanisms is impossible. The authors of the present article proposed a method for eliminating instability by introducing holonomic constraints into the system [4]. In this way, the stability problem in the range 0—360° was solved. However, an indeterminacy arose, associated with the introduction of an additional degree of freedom, consisting in the fact that the same sum of the projections of the mechanism links onto the coordinate axes corresponded to a large number of quadrilaterals; therefore the model easily transi-

led from the reproduction of the trajectories of points of one mechanism to the trajectories of points of another mechanism, although in the initial part of the first revolution the initial conditions determined the behavior of the mechanism correctly.

A new method is proposed for modeling four-link mechanisms with an angle of rotation of the driven link of \(360^\circ\). Its essence is that a mechanism of any desired complexity can be decomposed into four-link mechanisms of the slider-crank type with a rotating slider; to obtain the angles and coordinates of any points of these mechanisms it is sufficient to solve a triangle with variable side lengths. A feature of the model, not yet used widely enough, is the possibility of modeling, in a relatively simple way, articulated mechanisms with changing link lengths.

Let us consider the application of the method to the four-link mechanism \(ABCD\) (Fig. 1). Such a mechanism is reduced to a four-link mechanism with a rotating slider \(ABD\) (Fig. 1) if, to the slider-crank mechanism, one adds a triangle with sides \(BC\) and \(CD\) (with hinged joints at points \(B\), \(C\), and \(D\)) and with the variable side \(BD\). Its length \(e\) is determined in triangle \(ABD\) from the relation

\[ e^2=a^2+d^2-2ad\cdot \sin\varphi, \]

\[ -e^2+a^2+d^2-2ad\cdot \sin\varphi=-\frac{e}{\mu},\quad \mu>10^6,\quad \varphi=0. \]

From triangle \(ABD\) and its projections on the coordinate axes we obtain

\[ a\cos\varphi=e\cos\psi, \]

\[ a\sin\varphi-d=-e\sin\psi. \]

Fig. 1. Transformation diagram of a planar four-link mechanism

From triangle \(BCD\) it follows that

\[ c^2=e^2+b^2-2eb\cdot \cos\gamma, \]

\[ -c^2+e^2+b^2-2eb\cdot \cos\gamma=-\frac{\gamma}{\mu},\quad \mu>10^6. \]

The coordinates of points located on the connecting rod and on rods rigidly connected with it and directed at right angles to the connecting rod are determined by

\[ Y_K=e\cos\psi+b[\sin\psi\cdot \sin\gamma-\cos\psi\cdot \cos\gamma]- \]

\[ -k[\cos\psi\cdot \sin\gamma+\sin\psi\cdot \cos\gamma], \]

\[ X_K=-e\sin\psi+b[\cos\psi\cdot \sin\gamma+\sin\psi\cdot \cos\gamma]+ \]

\[ +k[\sin\psi\cdot \sin\gamma-\cos\psi\cdot \cos\gamma]. \]

The modeling diagram is given in Fig. 2. It includes a diagram for modeling the driving link, circuits for forming, by the method of implicit functions, \(\sin\psi\), \(\cos\psi\); the variable length \(e\), as well as \(\sin\gamma\), \(\cos\gamma\). The circuit uses 8 multiplication blocks, 2 functional blocks, and 4 tracking amplifiers. The modeling was carried out on an analog

on the EMU-10 computing machine using the nonlinear blocks of the SBN and SBPD sections of the MN-14 analog computer.

The modeling error of the mechanisms will be determined mainly by the errors of the function and multiplier blocks. It

Fig. 2. Block diagram for modeling a planar four-link mechanism with rotation of the driven link through 360 degrees

is approximately 1.2%. The trajectories of the mechanism points were recorded on two-coordinate recording instruments (DRP) and digital voltmeters. Figure 3 shows the trajectories of the mechanism points obtained on the machine.

Fig. 3. Trajectories of individual points of the four-link mechanism obtained on the EMU-10 analog computer

Thus, the transition from modeling trapezoidal contours with four constant link lengths to triangular contours with variable link lengths makes it possible to model continuously and stably, in all four quadrants, the kinematics of complex articulated mechanisms.

REFERENCES

1. I. I. Artobolevskii. Synthesis of Planar Mechanisms. Fizmatgiz, 1959.

2. Crossley F. R. Nachbildung eines mechanischen Kurbelgetriebes mittels eines elektronischen Analogrechners. Feinwerk Technik (FWT), früher “Feinmechanik und Präzesion,” 67 I. H. 4. 1963.

3. M. L. Bykhovskii. Ultrastability in electronic computing devices in the implementation of nonlinear equations in implicit form. Proceedings of the 1st International IFAC Congress. Statistical Methods of Research. Theory of Structures, Modeling, Terminology, Education. Publishing House of the Academy of Sciences of the USSR, Moscow, 1961.

4. A. G. Kokin, I. G. Vintizenko. Synthesis of planar mechanisms on analog computing machines. Collection of Abstracts and Annotations of Reports of the Second All-Union Conference on AVT. Moscow, 1966.

5. I. G. Vintizenko, Yu. A. Kovylin. Modeling of planar articulated mechanisms on an AVM. Report at the Second All-Union Conference on the Theory of Mechanisms and Machines. Moscow, 1960.