SYNTHESIS OF PLANAR MECHANISMS ON ANALOG COMPUTERS

A. G. Kokin, I. G. Vintizenko

(Presented by the scientific seminar of the Computing Laboratory of TPI)

In solving technical problems, especially in the field of mechanics and automatic control, it is often necessary to solve transcendental equations on analog machines.

The model must ensure the realization of the solution of the equation

\[ f(z, x_i)=0. \tag{1} \]

Here the root \(z=z(x_i)\) is a function of the independent variables \(x_i\).

In the general case, one can conceive of a model for solving a system of \(m\) transcendental equations of the form

\[ f_j(z_1, z_2,\ldots z_j,\ldots z_m, x_i)=0 \tag{2} \]

with respect to the unknowns \(z_j\).

Mathematical models for solving transcendental equations and their systems are widely used.

First of all, they can be used in creating artificial reversibility of the simplest computing-and-solving devices as elements of automatic-control systems. In addition, they may be applied to the modeling of planar mechanisms whose motion is described by transcendental equations.

In finding the real roots of transcendental equations and their systems on mathematical models, the root-selection method is used. Its essence consists in selecting such a root by trial substitution of its value into the given equation that the equation becomes an identity.

A mathematical model for solving a transcendental equation with one unknown by the root-selection method can be used for modeling the simplest crank-and-connecting-rod mechanism—a three-link mechanism.

It is an ordinary tracking system consisting of a computing device reproducing the function \(\varepsilon=f(z,x_i)\) and a feedback line, which in the general case includes a sensing element, an amplifier, and an actuator [4].

The operation of the tracking system in selecting the root of an equation is determined primarily by the sign of the quantity \(\varepsilon=f(z,x_i)\). The equilibrium state of the system corresponds to the condition \(\varepsilon=0\). If the equilibrium state is disturbed, a signal appears in the feedback line,

the sign of which depends on the sign of \(\varepsilon\). When \(\varepsilon=0\) there is no signal; when \(\varepsilon>0\) the signal should be regarded as positive, and when \(\varepsilon<0\) as negative.

When a signal is present \((\varepsilon \ne 0)\), the following system realizes a change of the variable \(z\) to the value \(z+\Delta z\). In order that the device should again be in equilibrium, the increment \(\Delta z\) must satisfy the equation

\[ f'_z \cdot \Delta z \simeq -\varepsilon,\quad \text{where } f'_z=\frac{\partial f}{\partial z}. \tag{3} \]

The sign of the increment

\[ \Delta z \simeq -\frac{\varepsilon}{f'_z} \tag{4} \]

is determined by the signs of \(\varepsilon\) and \(f'_z\). When \(f'_z>0\), the system must introduce into the computing device an increment \(\Delta z\) with a sign opposite to the sign of \(\varepsilon\), and when \(f'_z<0\), the signs of \(\Delta z\) and \(\varepsilon\) must be identical.

In general form, the stability condition of the mathematical model may be written as

\[ \operatorname{sign}(\Delta z)=-\operatorname{sign}(\varepsilon)\cdot \operatorname{sign}(f'_z). \tag{5} \]

It follows from equation (4) that \(|f'_z|\) is the degree of stability of the system. On this basis we arrive at the conclusion that, from the various possible variants of the mathematical model being designed for solving the given equation, one should choose the one for which the value of \(|f'_z|\) is greatest.

For a three-link mechanism, such a quantitative characteristic of stability is the quantity

\[ |b\cos\psi|, \]

where \(b\) is the length of the connecting rod;
\(\psi\) is the angle formed by the connecting rod and the \(x\)-axis.

In concrete cases, the behavior of the mathematical model depends primarily on the state in which it was at the moment the power was switched on. In practice it is necessary that this state, determined by the value \(z=z_0\), correspond to the stability region. Otherwise it is necessary to act on the system forcibly in order to bring it into the stability region.

A mathematical model for solving a system of two equations with two unknowns by the root-selection method can be used for modeling a four-link mechanism.

In such a model, two variants of feedback lines may be distinguished: parallel feedback lines and crossed ones.

For a model with parallel feedback lines, the condition of principal stability will be:

\[ \left\{ \begin{aligned} \operatorname{sign}(\Delta z_1) &= -\operatorname{sign}(\varepsilon_1)\cdot \operatorname{sign}(f'_{11}),\\ \operatorname{sign}(\Delta z_2) &= \operatorname{sign}(\varepsilon_2)\cdot \operatorname{sign}(f'_{22}); \end{aligned} \right. \tag{6} \]

\[ \varepsilon_1=f_1(z_1,z_2,x_i),\quad \varepsilon_2=f_2(z_1,z_2,x_i), \]

\[ f'_{11}=\frac{\partial f_1}{\partial z_1},\quad f'_{22}=\frac{\partial f_2}{\partial z_2}. \]

When these conditions are satisfied, the character of stability will be different depending on the magnitudes \(|f'_{11}|\) and \(|f'_{22}|\).

Therefore the quantity \(|f'_{11}\cdot f'_{22}|\) may be regarded as a quantitative characteristic of stability. For a model with crossed feedback lines the condition of principal stability is:

\[ \begin{cases} \operatorname{sign}(\Delta z_1)=-\operatorname{sign}(\varepsilon_2)\operatorname{sign}(f'_{21}),\\ \operatorname{sign}(\Delta z_2)=-\operatorname{sign}(\varepsilon_1)\operatorname{sign}(f'_{12}). \end{cases} \tag{7} \]

The stability characteristic is the quantity \(|f'_{12}\cdot f'_{21}|\);

\[ f'_{12}=\frac{\partial f_1}{\partial z_2},\qquad f'_{21}=\frac{\partial f_2}{\partial z_1}. \]

For a four-bar mechanism described by the system of equations

\[ \begin{cases} a\cos\varphi+b\cos\gamma-c\cos\psi=0,\\ a\sin\varphi+b\sin\gamma-c\sin\psi-d=0; \end{cases} \]

\(a\) — length of the crank,
\(b\) — length of the connecting rod,
\(c\) — length of the rocker;

the values \(|c\cos\psi\cdot b\sin\gamma|\) and \(|b\cos\gamma\cdot c\sin\psi|\) may serve as a stability characteristic.

The application of mathematical models for modeling planar mechanisms is a new problem. There are several methods for modeling such mechanisms: a method using function converters, a method with an arcsin transformation, a method using a square-root extraction circuit, and a method using the relation of the sides in a right triangle. The disadvantages of all these methods are the low accuracy of modeling due to the use of function converters and the small angle of rotation of the rocker at which the circuit is still stable, which limits the range of mechanisms that can be modeled on an analog machine.

In order to be able to model planar mechanisms with high accuracy and large \(\psi\), it is proposed to combine the method of implicit functions with the method of integration with respect to a variable that is not independent.

The new method for modeling the kinematics of planar hinged mechanisms consists in creating flexible feedbacks in the mathematical model and replacing holonomic constraints by nonholonomic ones [2, 3]. A block diagram for modeling the kinematics of a four-bar mechanism by this method is given in Fig. 1.

The resulting circuit is a mathematical model with parallel crossed feedback lines, with integrators in the feedback loops. It uses a circuit for obtaining the functions \(\sin\) and \(\cos\) by integration with respect to a dependent variable, using multiplier blocks and integrators [7].

The error of the method is considerably smaller than the error of the computing elements. Hence, the error of the solution will be determined entirely by the computing elements, especially the multipliers. The error of the multiplier blocks of the EMU-10 is \(0.3\%\), which is much less than the error of function converters (1% and more). The total error of the solution on the EMU-10 analog computer with recording of the solution on the “DRP-2” does not exceed 2%. The method makes it possible not only to find the trajectories of points of mechanisms, but also to determine their velocities and accelerations.

Figure 2 shows the mechanism and the trajectories of its individual points.

Fig. 1. Block diagram for modeling a four-link articulated mechanism

Fig. 2. Trajectories of individual points of the mechanism, obtained on the EMU-10 analog computer

Mechanism data:

\[ a = 20\ \text{cm}, \quad b = 67\ \text{cm}, \quad c = 50\ \text{cm}, \]

\[ d = 60\ \text{cm}, \quad \kappa = \kappa' = 20\ \text{cm}. \]

For the synthesis of mechanisms, the automatic support optimizer of the SPS machine “EMU-10” may be used.

The new method we have considered for modeling planar articulated mechanisms makes it possible to model mechanisms with high accuracy. In addition, this method, when switches are introduced into the feedback lines that change the stability conditions, makes it possible to obtain a rotation angle of the rocker of a four-bar linkage within the range of \(360^\circ\). The use of an automatic optimizer makes it possible to obtain circuits with prescribed degrees of stability and to optimize not only the parameters of mechanisms, but also the trajectories of their points, the velocities of individual points of the mechanisms, and accelerations.

REFERENCES

1. M. L. Bykhovskii. Ultrastability in electronic computing devices in the realization of nonlinear equations in implicit form. IFAC, Academy of Sciences of the USSR, 1961.

2. A. A. Voronov. Fundamentals of the Theory of Automatic Control. Energiya Publishing House, 1965.

3. M. B. Ignat’ev. Holonomic Automatic Systems. Publishing House of the Academy of Sciences of the USSR, 1963.

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

5. E. P. Popov. Automatic Regulation and Control. Fizmatgiz, 1962.

6. Smolov et al. Continuous-Action Computing Machines. Vysshaya Shkola Publishing House, 1964.

7. Agler H. Elektronische Analogrechner. Verlag der Wissenschaften, Berlin, 1962.

8. Crossley F. Die Nachbildung eines mechanischen Kurbelgetriebes mittels eines electronschen Analogrechners. Feinwerk Technik (FWT), formerly Feinmechanik und Präzision, 1963, Nos. 4, 5, 6, pp. 218–222.