MODELING OF PLANAR CENTROID MECHANISMS ON ANALOG COMPUTERS

I. G. Vintizenko, A. G. Kokin

(Presented by the scientific seminar of the computing division of the Research Institute of AEM)

At the present time, analog computers have become widely used for the synthesis of planar mechanisms. In particular, they can be applied to modeling planar centroid mechanisms.

Fig. 1. Diagram of a centroid mechanism and trajectories of some of its points, obtained on an analog machine.

The principle of modeling planar centroid mechanisms on an analog computer consists in obtaining the trajectories described by a point on a circle, outside it or inside it, rolling without slipping along another circle, outside or inside it. The equation of motion of such a point is written in parametric form as

\[ x = (R \pm r)\cos \varphi \mp \lambda r \cos\left(\frac{R \pm r}{\pm r}\right)\varphi, \tag{1} \]

\[ y = (R \pm r)\sin \varphi \mp \lambda r \sin\left(\frac{R \pm r}{\pm r}\right)\varphi, \]

where \(R\) is the radius of the fixed circle,

\(r\) is the radius of the moving circle,

\(\lambda\) is a parameter (for a point outside the circumference \(\lambda>1\), inside it \(\lambda<1\)).

The solution of these equations can be found on an analog computer by means of circuits generating the functions \(\sin\varphi\), \(\cos\varphi\), \(\cos\varphi\) and of summators that add these functions

Fig. 2. Block diagram for modeling a centroid mechanism.

for each wheel according to formula (1). We shall consider the modeling of planar centroid mechanisms using as an example a planetary centroid mechanism with the data (Fig. 1) [1]:

\[ R_1=40,\qquad R_2=R_3=R_4=10,\qquad R_5=2,\qquad R_6=8 \]

Fig. 3.

Diagram of a generator of harmonic oscillations
with stable amplitude.

The block diagram for modeling the planetary mechanism is shown in Fig. 2. Its basis is a generator comprising two integrators and one inverter. It reproduces the solution of the differential equation

\[ \ddot{x}+\omega^2 x=0 \tag{2} \]

with frequency \(\omega\), generating \(\sin\omega t\), \(\cos\omega t\).

To implement a driving generator with stable amplitude there exist various methods. One of them is the solution of the Van der Pol differential equation [2].

\[ \ddot{x} - \varepsilon \omega_0 \left(1 - x^2 - \frac{\dot{x}^2}{\omega_0^2}\right)\dot{x} + \omega_0^2 x = 0 . \tag{3} \]

The block diagram of a generator of this type is given in Fig. 3. The number of generators modeling the centroid mechanism must be equal to the number of wheels of the mechanism. The ratio of the diameters of the wheels of the mechanism is inversely proportional to the ratio of the frequencies of the driving generators.

Figure 1 shows the trajectories of points of the centroid mechanism obtained on an EMU-10 analog computer with the curves recorded on a DRP-2 two-coordinate recording instrument. The error is no more than 1–2% (instrumental error).

References

1. I. I. Artobolevsky, N. I. Levitsky, S. A. Cherkudinov, Synthesis of Planar Mechanisms. Fizmatgiz, 1959.

2. Von G. Meyer—Brotz. Anwendungen analoger Rechenelemente in der Tiefstfrequenz—Meßtechnik. FREQUENZ, Band 16, No. 1, 1962.