UDC 621.01+519.82

DETERMINATION OF THE MAIN DIMENSIONS OF CAM MECHANISMS WITH CONSIDERATION OF PRESSURE-ANGLE CONSTRAINTS BY NONLINEAR PROGRAMMING METHODS

Yu. Ya. Kovylyin, M. Yu. Korobeinikov, Z. P. Osipova

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

The design of cam mechanisms is currently carried out with consideration of constraints on the pressure angle \(\alpha\)

\[ \alpha_2 \leqslant \alpha \leqslant \alpha_1, \tag{1} \]

where \(\alpha_1\) and \(\alpha_2\) are, respectively, the greatest and the smallest admissible values of the pressure angle. During operation of the mechanism, the current values of the pressure angle depend in a definite way on the main (free) dimensions of the mechanism. The choice of the latter is made so as to ensure satisfaction of the pressure-angle constraints and at the same time to obtain the smallest dimensions of the mechanism. Dimensions of the mechanism satisfying these requirements will be called optimal. In the general case, finding the optimal dimensions of a mechanism is thus reduced to solving a nonlinear programming problem, in which the role of constraints is played by the pressure-angle constraints.

There are well-developed graphical methods for determining the dimensions of this and other types of planar cam mechanisms. The case of a rotating disk cam is considered in [1], and the case of a translationally moving cam in [2]. Works [3] (the case of a translationally moving cam and follower) and [4] (the case of a rotating cam with a translationally moving follower) are devoted to an analytical method for determining the optimal dimensions of cam mechanisms with a translationally moving follower. In these works the main dimensions of the mechanisms were found in the form of explicit formulas. This became possible because the corresponding nonlinear programming problem had a special character admitting an explicit solution.

For mechanisms with an oscillating (rocker) follower, the available analytical solutions [5, 6, 7] appear to be very approximate and, for this reason, are most often acceptable only at the stage of preliminary calculations. Therefore, the development of analytical or effective numerical methods for the synthesis of optimal (in the indicated sense) mechanisms is an urgent problem.

In the present article, the problem is considered of determining the main dimensions \(L \equiv x_{\max}/l_{BC}\) and \(\delta\) of a planar mechanism (Fig. 1) with a translationally moving cam 1 and an oscillating (rocker) follower 2.

The determination of the indicated dimensions is carried out in accordance with the optimality conditions adopted above.

For definiteness, suppose that the graph of the function \(\psi(x)\), where \(x=x/x_{\max}\), consists of two smooth pieces \(\psi=p_i(x)\), \(i=1,2\) (Fig. 2), and let \(p_i'(x)=q_i(x)\). The segment on which the function \(p_i(x)\) is defined will be denoted by \(K_i\). When \(x\) traverses the segment \(K_i\), \(i=1,2\), the point with coordinates \(p(x),q(x)\) describes, in the rectangular coordinate system \(p,q\), the phase curve \(q=\Gamma_i(p)\) (Fig. 3). The character of the discontinuities of the phase cycle \(\Gamma=\Gamma_1\cup\Gamma_2\) is determined by the values of the one-sided derivatives \(p'(x)\) at the points \(x=0,x_0,1\). The subsequent arguments will be suitable for any piecewise continuous curves \(\Gamma\). The variable \(\operatorname{tg}\alpha\) does not depend explicitly on \(x\), but is determined by the variables \(\psi=p\), \(\psi'=q\) and by the control parameters \((L,\delta)\)

\[ \operatorname{tg}\alpha=\operatorname{ctg}(p+\delta)+q/L\cdot \sin(p+\delta),\qquad (p,q)\in\Gamma,\qquad \operatorname{tg}\alpha_2\leqslant \operatorname{tg}\alpha\leqslant \operatorname{tg}\alpha_1. \tag{2} \]

Fig. 1. Planar mechanism with a translationally moving cam and an oscillating follower

We want, for all \((p,q)\in\Gamma\), to maintain the values of \(\operatorname{tg}\alpha\) within prescribed limits by choosing the control parameters \((L,\delta)\). In doing so, it is advantageous for us that the parameter \(L>0\) take the smallest value among all possible ones. A pair \((L,\delta)\) for which the constraints (2) are satisfied will be called an admissible control. An admissible control

Fig. 2. Graph of the function \(\Psi(x)\) for a planar mechanism with a cam and follower

Fig. 3. Phase curve \(q=\Gamma_i(p)\) for a planar mechanism with a cam and follower

\((L^*,\delta^*)\) will be called an optimal control (OC) if \(L^*\) can no longer be decreased by changing \(\delta^*\), i.e., if for any \(L<L^*\) the control \((L,\delta)\) is inadmissible for every \(\delta\).

Figure (2) also shows three curves: \(\Lambda_1,\Lambda_2\), which are level lines of the function \(\operatorname{tg}\alpha\) (2), on which it assumes the limiting admissible values \(\operatorname{tg}\alpha_1,\operatorname{tg}\alpha_2\), and \(\Lambda_0\), the level line \(\operatorname{tg}\alpha=0\). To obtain the equations of these curves, one must solve equation (2) with respect to \(q\) and set \(\alpha=\alpha_i,\ i=1,2\).

\[ q=\Lambda_i(p,L,\delta)=-L\cos(p+\delta+\alpha_i)/\cos\alpha_i,\qquad i=1,2. \tag{3} \]

The position of the curves \(\Lambda_i\) is characterized by the points

\[ B=\left(\frac{\pi}{2}-\alpha_1-\delta,\,0\right);\qquad C=\left(\frac{\pi}{2}-\alpha_2-\delta,\,0\right); \]

\[ D=(\pi-\delta,\,L);\qquad E=(-\delta,\,-L). \]

The level lines of the intermediate values of \(\operatorname{tg}\alpha\) completely fill the region for which the curve \(\Lambda=\Lambda_1\cup\Lambda_2\) serves as the boundary. The pair of points \(D\) and \(E\) is common to all level lines. The amplitudes of all level lines vary in direct proportion to the parameter \(L\). In order for the control \((L,\delta)\) to be admissible, it is necessary and sufficient that the phase curve \(\Gamma\) be completely enclosed by the boundary \(\Lambda\) (Fig. 2). Hence, in particular, it follows that only those constraints (2) make sense for which \(\alpha_2>-\frac{\pi}{2}\), \(\alpha_1<\frac{\pi}{2}\), \(\alpha_1-\alpha_2>A\). Any control \((L,\delta)\) can be made admissible if \(L\) is chosen sufficiently large.

The parameter \(\delta\) may vary on the interval \(\Delta\):

\[ \frac{\pi}{2}-\alpha_1<\gamma_1\leqslant\delta\leqslant\gamma_2< \frac{\pi}{2}-\alpha_2-A. \]

The endpoints of this interval \(\gamma_1\) and \(\gamma_2\) are chosen from considerations that will be indicated below. Let us fix some \(\delta\in\Delta\) and solve equation (3) with respect to \(L\).

\[ L=\Phi_i(p,q,\delta)=-q\cos\alpha_i/\cos(p+\delta+\alpha_i),\qquad i=1,2. \tag{4} \]

Let us note that, for \(\delta\in\Delta\), the denominator in (4) does not vanish if \(0\leqslant p\leqslant A\). Relation (4), for each \(\delta\in\Delta\), determines \(L\) for which the level line \(\Lambda_i\) (3) passes through the point \((p,q)\in\Gamma_i\). If the point \((p,q)\) runs along the curve \(\Gamma=\Gamma_1\cup\Gamma_2\), then formula (4) determines, for each \(\delta\in\Delta\), a bounded closed set of values of \(L\), among which there is a greatest value \(\widetilde L\).

\[ \widetilde L(\delta)=\max_{i=1,2}\ \max_{(p,q)\in\Gamma_i}\Phi_i(p,q,\delta) =\max_{i=1,2}\ \max_{x\in K_i}\Phi_i\bigl(p_i(x),q_i(x),\delta\bigr) \tag{5} \]

Here \(p_i(x)\), \(q_i(x)=p_i'(x)\) is the parametric equation of the phase curve \(\Gamma_i\), corresponding to the piece of the function \(\psi=p_i(x)\) (Fig. 2) on the interval \(K_i\).

If, for a fixed \(\delta\in\Delta\), one chooses \(L=\widetilde L(\delta)\) (5), then the level lines \(\Lambda_i(\widetilde L,\delta)\) (3) constructed for these values have the following properties: 1) the phase curve \(\Gamma=\Gamma_1\cup\Gamma_2\) lies entirely inside the region bounded by the curve \(\Lambda=\Lambda_1\cup\Lambda_2\); 2) the boundary \(\Lambda\) touches the curve \(\Gamma\) at the point \(\widetilde p=p(\widetilde x)\), \(\widetilde q=q(\widetilde x)\), where the greatest value \(\widetilde L(\delta)\) (5) is attained. The number of points of tangency and their position are determined by the form of the curve \(L\) and by the choice of \(\delta\in\Delta\) (in Fig. 2 there is one such point on each piece \(\Gamma_i\)). Geometric properties 1)—2) are equivalent to the following: 1) the control \((\widetilde L,\delta)\) is admissible; 2) under such a control, the values of \(\operatorname{tg}\alpha\) (2) reach the boundary of the interval of admissible values one (or several) times. A control \((\widetilde L,\delta)\) possessing properties 1)—2) will be called a supporting control. A supporting control is characte-

is characterized by the fact that decreasing \(\widetilde L\) at constant \(\delta\) leads to inadmissible control. It is clear that the optimal control must be sought among the supporting controls, by examining the values \(\widetilde L(\delta)\) (5) for various \(\delta \in \Delta\) and choosing the smallest of them. The choice of the endpoints of the interval \(\Delta=[\gamma_1,\gamma_2]\) is based on the following considerations. If \(\delta=\gamma_1>\dfrac{\pi}{2}-\alpha_1\) tends to \(\dfrac{\pi}{2}-\alpha_1\), then for \(p=0,\ i=1\) the denominator in (4) tends to zero and, consequently, \(L\), for which the control \((L,\delta)\) is admissible, tends to infinity. The same occurs in (4) for \(p=A,\ i=2\), when \(\delta=\gamma_2<\dfrac{\pi}{2}-\alpha_2-A\) tends to \(\dfrac{\pi}{2}-\alpha_2-A\). Geometrically (Fig. 2) this means that as point \(B\) tends to point \(O\) (or point \(C\) to point \(A\)), the boundary \(\Lambda\) encloses the curve \(\Gamma\) only for unboundedly increasing \(L\), since the tangent to the graph \(\Gamma\) at the points \(p=0\) and \(p=A\) is always vertical. If, for some reasons, it has been established in advance that values \(L>L_0\) are not optimal or are unacceptable, then the boundaries of the interval \(\Delta=[\gamma_1,\gamma_2]\) are thereby determined; outside it there is no point in considering \(\delta\), since there \(\widetilde L(\delta)>L_0\). Taking this preliminary information into account, the problem of finding the O.C. \((L^*,\delta^*)\) is reduced to finding

\[ L^*=\widetilde L(\delta^*)=\min_{\delta\in\Delta}\{\widetilde L(\delta),\,L_0\}, \tag{6} \]

that is, to solving a minimax problem, since \(L(\delta)\) is defined in (5).

The possibilities of using analytic methods of classical analysis to find the optimum \((L^*,\delta^*)\) are limited even in the case of a smooth phase curve \(\Gamma\), since reducing problem (5)—(6) to finding a constrained extremum leads to nonlinear equations that are not explicitly solvable. Therefore, in the general case it makes sense to regard problem (5) and (6) as a nonlinear programming problem and to develop a numerical procedure (algorithm) for finding the optimum with some prescribed accuracy. Special methods have been developed for the numerical solution of minimax problems (see [8]); however, their direct application here is made difficult by the discontinuity of the derivatives of the initial function \(\Psi\). At the same time, there is no need to use these methods here, since for computing the extrema in (5) and (6) of functions of one variable there are relatively simple algorithms [9], [10].

It follows from the general results of [8] that, regardless of the presence of discontinuities in the phase curve \(\Gamma\), the positive function \(\widetilde L(\delta)\) (5) is continuous for \(\delta\in\Delta\) and is convex downward. The latter circumstance makes it possible to use, for finding the minimum (6), especially efficient methods of sequential search [9], [10].

ALGORITHM 1 (minimization of (6)). The length of the interval \(|d\delta|\) on which the value \(\delta^*\) is to be localized is chosen. According to the sequential-search method based on the use of Fibonacci numbers [9], [10], a finite number of values \(\delta_k,\ k=1,2,\ldots,n\) (beginning with \(\delta_1=\gamma_1\)) is examined in such a way that at each successful step the value \(\widetilde L(\delta_k)\) decreases. The computations are stopped when the accuracy \(|d\delta|\) has been attained for \(\delta^*\). To implement the \((k+1)\)-st successful step, it is only necessary to remember the value \(\widetilde L(\delta_k)\), the computation of which can be organized so that

ALGORITHM 2 (maximization of (5)). In (5), \(\delta=\delta_k\), prescribed by Algorithm 1, is fixed. The length of the interval \(|dx|\) is chosen on which the value \(\tilde{x}\) is to be localized that gives the maximum of the function \(\Phi_i\) (5) on the interval \(K_i\). In the general case there are several local maxima on each interval \(K_i\), and to find the global maximum one has to use the inefficient passive search [9] (for each value of \(\delta_k\), scanning anew the grid of values \(x\)). If, however (see Fig. 3), the phase curves \(\Gamma_i\) are convex (regardless of in which direction), then \(\Phi_i\) has, for \(x\in K_i\), a unique maximum, for finding which an efficient sequential search is organized.

The phase portrait suggests that \(\tilde{x}\) should be sought in a neighborhood of those values \(x\in K_i\) where \(q_i(x)\) has an extremum. Thus, the essence of computing the optimal control \((L^*,\delta^*)\) consists in finding supporting controls (Algorithm 2) and in their subsequent improvement (Algorithm 1). At the last step in Algorithm 1 we obtain the value \(\tilde{L}(\delta_k)\), for which

\[ \left|\tilde{L}(\delta_k)-L^*\right| \leq Q\,|d\delta|+R\,|dx| \leq c^{-2}\,[a\,|d\delta|+(b^2+a^2)\,|dx|]. \tag{7} \]

Here \(Q\) and \(R\) are the greatest, in absolute value, values of the derivatives of the function (4) with respect to \(\delta\in\Delta\) and \(x\in[0,1]\), respectively. These derivatives are further estimated in (7) with the aid of the auxiliary numbers

\[ c=\min_{i=1,2}\min_{p,\delta}|\cos(p+\delta+\alpha_i)| =\min\{|\cos(\gamma_1+\alpha_1)|;\ |\cos(A+\gamma_2+\alpha_2)|\} \]

(according to the remark to formula (4), \(c>0\);

\[ a_i=\max |p_i'(x)|, \]

\[ b_i=\max |p_i''(x)|,\quad i=1,2,\quad a=\max\{a_1;a_2\};\quad b=\max\{b_1,b_2\}. \]

Formula (7) makes it possible to ensure the required accuracy of computing \(L^*\) by an appropriate choice of the intervals \(|d\delta|\) and \(|dx|\) in Algorithms 1–2.

Construct, on the phase plane \(p,q\), a rectangle with vertices \(H_1(0,a_1)\), \(H_2(A,a_1)\), \(H_3(0,-a_2)\), \(H_4(A,+a_2)\). By formulas (3)—(4), three controls \((L_i,\delta_i)\), \(i=1,2,3\), are explicitly computed, for which the boundaries \(\Lambda(L_i,\delta_i)\) pass respectively through the following three pairs of vertices: \(\{H_1,H_2\}\), \(\{H_3,H_4\}\), \(\{H_1,H_4\}\). From the controls \((L_i,\delta_i)\) choose the one for which \(L_i\) is greatest, and denote it by \((L_0,\delta_0)\). The boundary \(\Lambda(L_0,\delta_0)\) entirely encloses the rectangle and touches it at the corresponding pair of vertices. Mark this pair of vertices. It is now clear that for any phase curve \(\Gamma\) that passes inside the rectangle (or along its boundary) and enters the marked pair of vertices, the control \((L_0,\delta_0)\) will be optimal. The control \((L_0,\delta_0)\) is admissible for any phase curve \(\Gamma\) contained in the rectangle and can be used in practice as the first approximation to the optimal control. Controls for which \(L>L_0\) are certainly not optimal. Therefore \(L_0\) can be used for computing the optimal control, as stated in the derivation of formula (6).

If we set

\[ l_0=\max_{i=1,2} a_i\cos\alpha_i, \]

then, for \(L<l_0\), the boundary \(\Lambda\) narrows vertically so much that, for any choice of \(\delta\), the curve will protrude beyond this boundary. Therefore, for \(L<l_0\), the control \((L,\delta)\) is inadmissible for all \(\delta\). Consequently, in any supporting control \((\tilde{L},\tilde{\delta})\) (5), the value \(\tilde{L}\geq l_0\). From this estimate and from formula (6) it follows that the optimal control \((L^*,\delta^*)\) lies within the limits \(0<l_0\leq L^*\leq L_0,\ \delta^*\in\Delta\).

The estimates of OU and the computational procedure, with obvious changes, carry over to the case when the graph of the function $\Psi$ (Fig. 2) consists of more than two smooth pieces and has more than one extremum. In conclusion, let us note that the phase portrait $\Gamma$ played a noticeable heuristic role in the mathematical formulation of the problem and in the a priori estimates. At the same time, with any numerical method for finding OU as the solution of the minimax problem (5)—(6), no graphs need be constructed; it is sufficient to have only a table of values of the function $\bar P(\chi)$ and of its derivative at a finite number of required points $\chi$.

Conclusion

The problem of nonlinear programming as applied to the optimal design of mechanisms, with allowance for constraints on some of their parameters, can in general be formulated as follows. Let a certain mechanism be characterized by three groups of parameters: 1) configuration parameters $\bar u=(p,q,\ldots)$, belonging to a rigidly specified set $F$ in the phase space $\bar u$ (in our case this is the cam profile specified by the phase curve $F$ in the plane $(p,q)$); 2) free (control) parameters $\bar\beta=(L,\delta,\ldots)$, which may be chosen from some set $B$; 3) controlled parameters $\bar\alpha=(\alpha,\ldots)$, which depend on the configuration of the mechanism and on the control parameters $\bar\beta$, i.e.,

\[ \bar\alpha=\bar\alpha(p,q,\ldots,L,\delta,\ldots)=\bar\alpha(\bar u,\bar\beta) \]

(in our case this dependence is given by formula (2)), and the parameters $\bar\alpha$ must be maintained within the limits of some set of constraints $A$.

It is required to ensure the minimum of some objective function $G(\bar\beta)=G(L,\delta,\ldots)$ subject to the constraints

\[ \bar\alpha(\bar u,\bar\beta)=\bar\alpha(p,q,\ldots,t,\delta,\ldots)\in A,\quad \bar u(p,q,\ldots)\in F,\quad \bar\beta=(L,\delta,\ldots)\in B. \]

The formulation of the problem of optimal design of mechanisms in the language of nonlinear programming will make it possible to use the great capabilities of electronic computers for its solution. In our case there are two simplifying circumstances: 1) $G(\bar\beta)=L$—the objective function coincides with the parameter $L$ being minimized; 2) the objective function $L$ can be represented as a function of the constraints of the problem, of the configuration parameters, and of the remaining free parameters, i.e. $L=\Phi(p,q,\ldots,\alpha,\ldots,\delta)$ (see formula (4)). It is precisely these two conditions that reduce the general problem posed (and the particular problem considered) to finding the minimax:

\[ L^*=\min_{\delta}\max_{\Gamma\times A}\Phi(p,q,\ldots,\alpha,\ldots,\delta), \]

where the maximum is computed over all values of the parameters $\bar u=(p,q,\ldots)\in F$ and $\bar\alpha=(\alpha,\ldots)\in A$.

REFERENCES

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

2. Yu. Ya. Kovylin. Dynamic synthesis of planar cam mechanisms with translating cams and roller followers. Izvestiya TPI, vol. 96/1. Tomsk, TSU Publishing House, 1959.

3. N. Ya. Niberg. Theory of mechanisms and machines. In: Mechanical Engineer’s Handbook. Vol. 1. Edited by N. S. Acherkan. Moscow, Mashgiz, 1954.

4. Yu. Ya. Kovylin. Determination of the main dimensions of cams without constructing phase diagrams. Present collection.

1. D. M. Lukichev, A. A. Savelova. Theory of Mechanisms. A Course of Lectures. Issue 1, 2nd ed. Moscow, MVTU Publishing House, 1968.

2. Theory of Mechanisms and Machines. Design. Edited by O. I. Kulbachny. Moscow, Vysshaya shkola, 1970.

3. Yu. Ya. Kovylin. Determination of the principal dimensions of copiers and cylindrical cams of automatic machine tools. In: Automation of Production Processes in Mechanical Engineering. Part IV. Tomsk, Tomsk State University Publishing House, 1970.

4. V. F. Demyanov, V. N. Malozemov. Introduction to Minimax. Moscow, Nauka, 1972.

5. Yu. V. Chuev, G. P. Spekhova. Technical Problems of Operations Research. Moscow, Sovetskoe radio, 1971.

6. G. Wagner. Principles of Operations Research. Vol. 2. Moscow, Mir, 1973.