UDC 621.835

DETERMINATION OF THE MAIN DIMENSIONS OF CAMS WITHOUT CONSTRUCTING PHASE DIAGRAMS

Yu. Ya. Kovylin

(Presented by the scientific-methodological seminar of the Department of Applied Mechanics)

As is known, the problem of determining the main dimensions of cam mechanisms with allowance for constraints on the pressure angle (or transmission angle) has, in principle, an exact general graphical solution by means of phase diagrams, also called diagrams of arms [1–4].

A number of proposals are also known for the analytical solution of this problem. As applied to the mechanism most widespread in engineering, shown in Fig. 1, one proceeds from the known [1–3] dependence of the pressure angle \(\alpha\) on the main dimensions of the mechanism \(\Delta\) and \(S_0\) and on the prescribed law of motion of the follower \(S = S(\varphi)\)

\[ \tg \alpha = \frac{\dfrac{ds}{d\varphi} - \Delta}{S_0 + s} \tag{1} \]

taking into account the constraints

\[ \alpha_{\max} \geq \alpha \geq \alpha_{\min}, \tag{2} \]

where \(\alpha_{\max}\) and \(\alpha_{\min}\) are the limiting permissible values of the pressure angle \(\alpha\), ensuring operation of the mechanism without jamming and with a sufficiently high efficiency.

Fig. 1. For calculating the main dimensions of cams

At the same time, in works [5, 6, etc.] it is assumed that the extreme pressure angles occur in those positions of the cam in which the transmission ratio \(ds/d\varphi\) of the mechanism (the analogue of the follower velocity) takes its greatest and smallest values. This assumption (as applied to the mechanism under consideration) by no means always ensures results of acceptable accuracy: the limiting absolute error of the dimension \(S_0\) is estimated by a quantity commensurable with the follower stroke \(s_{\max}\), and it always entails an expansion of the range of variation of the pressure angle in comparison with that specified in (2).

In the monograph [7], to determine the mentioned “dangerous” positions of the mechanism, the equation

\[ \frac{d}{d\varphi}\operatorname{tg}\alpha(\varphi)=0. \tag{3} \]

is used.

However, in the general case this equation turns out to be transcendental, and its solution is carried out graphically. Only for a certain limited number of the simplest laws of motion of the follower can the solution of the problem be brought to completion in analytic form.

In view of the foregoing, the algorithms listed above are practically rather ineffective. We shall show a simpler and, at the same time, more universal method for finding the main dimensions of the mechanism.

Taking into account the specified constraints (2), we write equation (1) in the following two forms:

\[ \operatorname{tg}\alpha-\operatorname{tg}\alpha_{\max} = \frac{ \dfrac{ds}{d\varphi}-\Delta-(S_0+s)\operatorname{tg}\alpha_{\max} }{ S_0+s }; \tag{4} \]

\[ \operatorname{tg}\alpha-\operatorname{tg}\alpha_{\min} = \frac{ \dfrac{ds}{d\varphi}-\Delta-(S_0+s)\operatorname{tg}\alpha_{\min} }{ S_0+s }. \tag{5} \]

From (4) and (5) it follows that

\[ \Delta+S_0\operatorname{tg}\alpha_{\max} = \left( \frac{ds}{d\varphi}-s\operatorname{tg}\alpha_{\max} \right)_{\max}; \tag{6} \]

\[ \Delta+S_0\operatorname{tg}\alpha_{\min} = \left( \frac{ds}{d\varphi}-s\operatorname{tg}\alpha_{\min} \right)_{\min}. \tag{7} \]

Since the dependence \(s(\varphi)\) and the angles \(\alpha_{\max}\) and \(\alpha_{\min}\) are assumed here to be given, finding the quantities on the right-hand sides of (6) and (7) presents no special difficulties. In this case, from the system (6), (7) we immediately find the least value of the dimension \(S_0\) (optimal in the sense of the constraints (2)):

\[ S_0= \frac{ \left( \dfrac{ds}{d\varphi}-s\operatorname{tg}\alpha_{\max} \right)_{\max} - \left( \dfrac{ds}{d\varphi}-s\operatorname{tg}\alpha_{\min} \right)_{\min} }{ \operatorname{tg}\alpha_{\max}-\operatorname{tg}\alpha_{\min} }, \tag{8} \]

after which the corresponding value of \(\Delta\) is most simply found from equality (6) or (7).

If necessary (for example, proceeding from design considerations, strength conditions, etc.), one may, without changing \(\Delta\), increase \(S_0\) by any amount relative to the calculated value. In this case the angle \(\alpha\) will not leave the limits established by condition (2) (it is assumed that \(\alpha_{\max}>0\), \(\alpha_{\min}<0\)).

If, in the synthesis of the mechanism, the offset \(\Delta\) is specified, then equations (6) and (7) become independent. In this case the minimum dimension \(S_0\) must be no less than the larger of the two values of this quantity determined successively from (6) and (7).

In conclusion, we note that the method presented can in many practically important cases be implemented in analytic form. In the remaining cases—in a convenient numerical form, as well as in a simple graphical form.

References

1. I. I. Artobolevskii. Theory of Mechanisms. Moscow, Nauka, 1967.

2. N. I. Levitskii. Methods for Calculating Cam Mechanisms. Moscow, Mashgiz, 1950.

3. L. N. Reshetov. Cam Mechanisms. Moscow, Mashgiz, 1953.

4. Yu. Ya. Kovylin. Dynamic synthesis of planar cam mechanisms with translational cams and roller followers. Proceedings of TPI, vol. 96. Tomsk, TPI Publishing House, 1959.

5. D. M. Lukichev, A. A. Savelova. Theory of Mechanisms. Hinged-Lever and Cam Mechanisms. Issue 1, 2nd ed. Moscow, MVTU Publishing House, 1968.

6. Theory of Mechanisms and Machines. Design. Edited by O. I. Kulbachnyi. Moscow, Vysshaya Shkola, 1970.

7. N. N. Popov. Calculation and Design of Cam Mechanisms. Moscow, Mashinostroenie, 1965.