STUDY OF THE MOTION OF MECHANISMS BY THE LINEARIZATION METHOD FOR FORCE MOMENTS DEPENDING ON THE POSITION AND VELOCITY OF THE REDUCTION LINK

Yu. Ya. Kovylin

The development of modern instrument-making and machine-building is associated with the widespread use of complex systems that include electrical machines and hydraulic and pneumatic devices. The links of the mechanisms forming the basis of these systems may be subjected to forces depending not only on their displacements, but also on their velocities. On the other hand, the requirements for the speed and accuracy with which the executive links of such systems perform the prescribed operations are continuously increasing.

All this dictates the need to develop effective engineering methods for calculating the motion of mechanisms with allowance for the dependence of the forces acting on their links on displacements and velocities. This circumstance, in particular, was pointed out in the resolution of the Fifth All-Union Conference on the basic problems of the theory of machines and mechanisms (Sukhumi, 1967).

We shall consider mechanisms with one degree of freedom. With respect to such mechanisms, the indicated problem has been solved by a number of authors. Of the proposed methods, the most widespread have been the graphical method of Prof. V. A. Zinoviev [1] and the numerical method of Prof. G. G. Baranov [2]. These methods implement simple algorithms. However, their application in practical calculations is not always sufficiently effective: the first method has limited accuracy and is developed for the case when the prescribed dependence of the force moment \(M(\varphi,\omega)\), reduced to the reduction link (\(\varphi\) is the angle of rotation of the reduction link, \(\omega\) is its angular velocity), can be represented as the sum of two functions, one of which depends only on the coordinate \(\varphi\), and the other on its derivative \(\omega\); the second method is in principle free of the listed limitations, but, in order to achieve the accuracy required in technical calculations, the investigated angle of rotation of the reduction link must be traversed with a very small step \(\Delta\varphi\), since within the limits of one such step the reduced moment of forces \(M(\varphi,\omega)\) is regarded as constant, while the reduced moment of inertia \(I(\varphi)\) is assumed to vary linearly. Consequently, practical application of Prof. G. G. Baranov’s method is possible only when using an electronic digital computer.

The simple numerical linearization method presented below can be applied with equal success both in manual and in machine calculations of the motion of mechanisms.

Let the moment of the forces reduced to the reduction link be given by the function \(M = M(\varphi,\omega)\), and let the mass moment of inertia of the mechanism reduced to the same link be the function \(I = I(\varphi)\).

It is required to determine the law of motion of the reduction link if \(\omega=\omega_0\) at \(\varphi=\varphi_0\).

We divide the entire angle of rotation of the reduction link intended for investigation into small intervals

\[ \Delta\varphi_{01}=\varphi_1-\varphi_0,\quad \Delta\varphi_{12}=\varphi_2-\varphi_1,\ldots,\quad \Delta\varphi_{ik}=\varphi_k-\varphi_i,\ldots \]

For the displacement of the mechanism from position \(i\) to position \(k\), the equation is valid:

\[ \frac{I_k\omega_k^2}{2}-\frac{I_i\omega_i^2}{2} = \int_{\varphi_i}^{\varphi_k} M\,d\varphi . \tag{1} \]

Let the value \(\omega_i\) at \(\varphi=\varphi_i\) already be known. Next, represent

\[ \omega_k=\omega_i+\Delta\omega_{ik}, \tag{2} \]

where \(\Delta\omega_{ik}\) is the increment of velocity on the interval \(\Delta\varphi_{ik}\).

If the interval \(\Delta\varphi_{ik}\) is sufficiently small, then the velocity increment will also be small. On this basis we have

\[ \omega_k^2=(\omega_i+\Delta\omega_{ik})^2 \simeq \omega_i^2+2\omega_i\Delta\omega_{ik}. \tag{3} \]

On the same basis we may assume that, on the interval \(\Delta\varphi_{ik}\), the reduced moment of the forces varies in accordance with the linearized dependence

\[ M_* = M(\varphi,\omega_i)+ \left(\frac{\partial M}{\partial \omega}\right)_{\omega=\omega_i} \Delta\omega \quad (\varphi_i \leq \varphi \leq \varphi_k), \]

since the real dependences \(M(\varphi,\omega)\) are practically always differentiable with respect to \(\omega\).

Finally, the third assumption consists in the fact that, on the interval \(\Delta\varphi_{ik}\) under consideration, the component of the moment of the forces equal to

\[ \left(\frac{\partial M}{\partial\omega}\right)_{\omega=\omega_i}\Delta\omega \]

can be represented by a linear function of the angle \(\varphi\). This makes it possible to write

\[ \int_{\varphi_i}^{\varphi_k} M\,d\varphi \simeq \Delta\varphi_{ik} \left( M'_{ik}+\frac{1}{2}C_{ik}\Delta\omega_{ik} \right). \tag{4} \]

Here

\[ M'_{ik} = \frac{1}{\Delta\varphi_{ik}} \int_{\varphi_i}^{\varphi_k} M(\varphi,\omega_i)\,d\varphi, \]

or approximately

\[ M'_{ik}\simeq \frac{1}{2}\left[M(\varphi_i,\omega_i)+M(\varphi_k,\omega_i)\right]; \]

\[ C_{ik}= \left(\frac{\partial M}{\partial\omega}\right)_{\substack{\varphi=\varphi_k\\ \omega=\omega_i}} . \]

Substituting (3) and (4) into (1), after elementary transformations we finally obtain:

\[ \Delta\omega_{ik}\simeq \frac{2M'_{ik}\Delta\varphi_{ik}+(I_i-I_k)\omega_i^2} {2I_k\omega_i-C_{ik}\Delta\varphi_{ik}} . \tag{5} \]

Applying formulas (5) and (2) successively to all the designated intervals \(\Delta\varphi_{ik}\), we find the function \(\omega(\varphi)\). To determine the dependences of velocity on time and of the angle of rotation on time, one may use known methods [3].

The method described can be applied when the functions \(M(\varphi,\omega)\) and \(I(\varphi)\) are specified analytically, in tabular form, or graphically. Experience in applying it to the analysis of the motion of mechanisms of the most varied purpose shows that good results are obtained already with \(12 \div 16\) sections per revolution of the reduction link. In this case, all characteristic details of the law of motion can be revealed. An important advantage of the adopted computational scheme is that formula (5) determines not the velocity \(\omega_k\), but only its small part \(\Delta\omega_{ik}\), which makes it possible, in computations, to restrict oneself to a minimum number of significant figures. In practice, a twenty-five-centimeter slide rule is usually sufficient even to reproduce confidently the curve \(\Delta\omega(\varphi)\) in the regime of steady motion.

REFERENCES

1. V. A. Zinoviev. Course in the Theory of Mechanisms and Machines. Moscow, Fizmatgiz, 1960.
2. G. G. Baranov. Course in the Theory of Mechanisms and Machines. Moscow, “Mashinostroenie,” 1967.
3. I. I. Artobolevsky. Theory of Mechanisms. Moscow, “Nauka,” 1967.