Profiling Cylindrical Cams Using Elliptical Arcs

(for a Given Follower Acceleration and Pressure Angle)

E. I. Toropygin

(Presented by the joint seminar of the Departments of Applied Mechanics and Descriptive Geometry and Graphics)

In our article [1] we dealt with questions concerning the profiling of cylindrical cams with an axial pointed follower for a given acceleration and maximum displacement of the follower. In the present article we set ourselves the aim of explaining the method and deriving calculation formulas for profiling the same cams for a known acceleration and value of the cam transmission angle, which for these mechanisms is equal to the follower pressure angle.

As in the preceding case, among the specified quantities we shall take the coefficient of the acceleration ratio \(\gamma\) and the circumferential velocity of the cam \(U\).

We shall begin the determination of the unknown quantities for constructing the cam profile with the section of accelerated motion of the follower, and then pass to the section of its retarded motion.

For constructing the cam profile it will be necessary to determine such quantities as \(h_1, h_2; a_1, a_2; A_1, A_2; B_1, B_2\).

Let us introduce the following notation:

All these unknown quantities are interconnected, and a change in any one of them necessarily leads to a corresponding change in the others. To solve the problem it will be necessary to specify one of them and proceed to determine the rest. Depending on this, several variants will be considered. But first we shall examine those special cases of profiling a cam by arcs of ellipses that occur when the parameters of the profiling ellipses are changed accordingly, in order to know within what limits one should prescribe particular unknowns when solving the problem by the corresponding variant.

It is known that [1] for an elliptical cam

\[ f_1=\frac{A_1B_1U^2}{\left(\sqrt{B_1^2-x^2}\right)^3}. \tag{1} \]

A corresponding change in the parameters of the profiling ellipse leads in one case to a parabola, and in another to a circle. Thus, if the parameters of the ellipse \(A_1\) and \(B_1\) are increased in a certain way to \(\infty\), then in the limit we can obtain a parabola. Decreasing the parameter \(A_1\) to the value at which \(A_1=B_1\) transforms the ellipse into a circle.

The following property of the parabola is known [2] (Fig. 1):

\[ a_{1n}=2k_1. \]

Fig. 1. Theoretical cam profile formed by a parabola.

On this basis,

\[ a_{1n}=\frac{2h_{1n}}{\operatorname{tg}\Theta}. \tag{2} \]

From [1] we have

\[ a_{1n}^{2}=\frac{2h_{1n}U^{2}}{j_{1}}. \tag{3} \]

Solving (3) and (2) simultaneously, we obtain the values of the cam displacement \(a_{1n}\) and of the follower displacement \(h_{1n}\) in the case when the profiling ellipse, for \(A_{1}=B_{1}=\infty\), degenerates into a parabola:

\[ a_{1n}=\frac{U^{2}}{j_{1}}\operatorname{tg}\Theta,\qquad h_{1n}=\frac{(U\operatorname{tg}\Theta)^{2}}{2j_{1}}. \tag{4} \]

Thus, expressions (4) are the design formulas for determining the dimensions of the parabola, which is one limiting case of variation of the profiling ellipse.

Fig. 2. Theoretical cam profile formed along an arc of a circle.

Let us proceed to the derivation of design formulas for determining the parameters of the circle, which is the other limiting case of variation of the profiling ellipse.

From triangle \(O_{1}CM\), for \(x=a_{10}\) (Fig. 2), we find

\[ a_{10}=R_{1}\sin\Theta . \tag{5} \]

If in equation (1), instead of \(A_{1}=B_{1}\), we substitute the value \(R_{1}\) and solve it together with equation (5), then we obtain

\[ R_{1}=\frac{U^{2}}{j_{1}\cos^{2}\Theta}, \]

\[ a_{10}=\frac{U^{2}\operatorname{tg}\Theta}{j_{1}\cos^{2}\Theta}. \tag{6} \]

From triangle \(OCM\) we have

\[ h_{10}=a_{10}\operatorname{tg}\mu . \tag{7} \]

Considering triangles \(OO_{1}M\) and \(OCM\), and taking into account that \(\angle O_{1}OM=\angle OMO_{1}\), we obtain

\[ \mu=\frac{\Theta}{2}. \]

Substituting the obtained value of \(\mu\) into (7), we find

\[ h_{10}=a_{10}\operatorname{tg}\frac{\Theta}{2}, \]

or, on the basis of (6),

\[ h_{10}=\frac{U^{2}\operatorname{tg}^{2}\Theta}{j_{1}\cos\Theta(1+\cos\Theta)}. \tag{8} \]

We shall now find the relationships that make it possible to profile cylindrical and flat translating cams along arcs of ellipses.

First variant. The solution of the problem is based on the preliminary selection of the value \(a_1\). The following expressions are known [1]:

\[ A_1=\frac{j_1a_1h_1-U^2\tan\Theta\,h_1+U^2a_1\tan^2\Theta}{j_1a_1-U^2\tan\Theta}, \]

\[ A_1=\frac{(a_1\tan\Theta-h_1)h_1}{a_1\tan\Theta-2h_1}. \tag{9} \]

Here \(A_1\) is laid off along the \(OY\) axis, and \(B_1\) along the \(OX\) axis. Solving equations (9) simultaneously, we obtain the following design formula:

\[ h_1= \frac{ U^2a_1\tan^2\Theta \left[ \sqrt{\frac{j_1a_1}{U^2\tan\Theta}}-1 \right] }{ j_1a_1-U^2\tan\Theta }. \tag{10} \]

We determine the value \(A_1\) using (9), and \(B_1\) using the known expression [1],

\[ B_1=A_1\sqrt{\frac{a_1}{(A_1-h_1)\tan\Theta}}. \tag{11} \]

Second variant. The solution of the problem is based on the choice of the value \(h_1\). To determine \(a_1\), we use the following relation [1]:

\[ a_1= \frac{ (j_1h_1+2U^2\tan^2\Theta)h_1+ \sqrt{\left[(j_1h_1+2U^2\tan^2\Theta)h_1\right]^2-4U^4h_1^2\tan^4\Theta} }{ 2U^2\tan^3\Theta }. \tag{12} \]

As for the parameters \(A_1\) and \(B_1\), their values are determined in accordance with expressions (9) and (11).

Third variant. We prescribe the value \(A_1\). From equations (9) we have

\[ a_1=\frac{(A_1-h_1)U^2\tan\Theta}{A_1j_1-h_1j_1-U^2\tan^2\Theta}, \qquad a_1=\frac{(2A_1-h_1)h_1}{(A_1-h_1)\tan\Theta}. \tag{13} \]

Solving equations (13) simultaneously, we obtain the cubic equation

\[ h_1^3-3A_1h_1^2+2A_1^2h_1-\frac{A_1^2U^2\tan^2\Theta}{j_1}=0. \tag{14} \]

Equation (14) can be solved graphically.

As a result of the preliminary determination of the limits of variation of the parameters of the profiling ellipse, the approximate value of the follower displacement \(h_1\) is already known, which substantially facilitates determination of the sought root.

Next, by formula (12) we find \(a_1\), after which, in accordance with (9) and (11), we determine the remaining necessary quantities.

As for obtaining the quantities on the section of decelerated motion of the follower, for this purpose we use the known expression [1]:

\[ \frac{j_1}{j_2} = \frac{a_{2n}}{a_{1n}} = \frac{h_{2n}}{h_{1n}} = \frac{R_2}{R_1} = \frac{a_{20}}{a_{10}} = \frac{h_{20}}{h_{10}} = \frac{A_2}{A_1} = \frac{B_2}{B_1} = \frac{a_2}{a_1} = \frac{h_2}{h_1}. \tag{15} \]

Example 1

Given: \(j_1=100\ \text{cm/sec}^2;\ U=10\ \text{cm/sec};\ \Theta=50^\circ.\)

Calculate the profile of a cylindrical cam over the portion of accelerated motion of the follower using arcs of ellipses, parabolas, and circles.

Let us solve the problem according to the first variant.

From (4), (6), and (8) we have: \(a_{1n}=1,19\ \text{cm}\), \(h_{1n}=0,71\ \text{cm}\), \(a_{10}=2,9\ \text{cm}\), \(R_1=3,77\ \text{cm}\), \(h_{10}=1,35\ \text{cm}\).

Set \(a_1=1,4\ \text{cm}\).

In accordance with (10), we obtain \(h_1=0,85\ \text{cm}\).

From equations (9) and (11), successively, we have \(A_1=9,5\ \text{cm}\), \(B_1=3,63\ \text{cm}\).

Example 2

Given \(j_1=600\ \text{cm}/\text{sec}^3\); \(U=50\ \text{cm}/\text{sec}\); \(\Theta=45^\circ\); \(\nu=2\).

Calculate and draw the profiles of a cylindrical cam with an axial pointed follower using arcs of parabolas, ellipses, and circles, and the corresponding acceleration curves.

Determine the necessary quantities on the portion of accelerated motion of the follower.

From (4), for one of the limits of variation of the parameters of the profiling ellipse, we obtain

\[ a_{1n}=4,17\ \text{cm}; \qquad h_{1n}=2,10\ \text{cm}. \]

From (6), we find the value of the radius of the circle in which the profiling ellipse is inscribed,

\[ R_1=11,8\ \text{cm}. \]

The quantity \(R_1\), when solving the problem according to the third variant, will serve as a certain guide for choosing the value of \(A_1\).

From (6) and (8) we find \(a_{10}=8,35\ \text{cm}\), \(h_{10}=3,45\ \text{cm}\).

We shall carry out the further solution of the problem according to the third variant. Set the value \(A_1=50\ \text{cm}\).

On the basis of (14), we obtain the following cubic equation:

\[ h_1^3 - 150 h_1^2 + 5000 h_1 - 10400 = 0. \]

Assigning various values to \(h_1\) within the limits from \(h_{1n}=2,10\ \text{cm}\) to \(h_{10}=3,45\ \text{cm}\), we construct the graph (Fig. 3), from which we find \(h_1=2,23\ \text{cm}\). By condition (12), we find \(a_1=4,55\ \text{cm}\), and accordingly from (9) and (11): \(A_1=51,73\ \text{cm}\), \(B_1=15,66\ \text{cm}\).

The specified acceleration \(j_1\), with the obtained values of the parameters of the profiling ellipse, in accordance with condition (1), at \(x=a_1\), retains its value.

We find the parameters of the profiling curves on the portion of retarded motion of the follower from equation (15):

\[ a_{2n}=8,34\ \text{cm}, \qquad h_{2n}=4,17\ \text{cm}, \]

\[ A_2=103,46\ \text{cm}, \qquad B_2=31,32\ \text{cm}, \]

\[ a_2=9,1\ \text{cm}, \qquad h_2=4,45\ \text{cm}, \]

\[ R_2=23,6\ \text{cm}, \qquad a_{20}=16,7\ \text{cm}, \]

\[ h_{20}=6,91\ \text{cm}. \]

Construct the cam profiles using arcs of parabolas, ellipses, and circles, and also the acceleration curves (Fig. 4).

Fig. 3. Graphical determination of the quantity \(h_1\).

Fig. 4. Profiling of a cam by arcs: 1) parabolas, 2) ellipses, and 3) circles, and the corresponding acceleration curves.

Comparing the obtained cam profiles and follower accelerations with one another, the following should be noted:

1. The dimensions of the cam when it is profiled by arcs of ellipses are considerably smaller than the dimensions of cams when it is profiled by arcs of circles, and differ hardly at all from the dimensions of a parabolic cam.

2. The acceleration of the follower when the cam is profiled by arcs of ellipses varies within considerably smaller limits than is the case when it is profiled by arcs of circles, and in its average value approaches the constant acceleration of a parabolic cam.

REFERENCES

1. Toropygin E. I. Profiling of cylindrical cams by arcs of ellipses. Izvestiya vuzov, Mashinostroenie, No. 1, 1960.
2. Reshetov L. N. Cam Mechanisms. Mashgiz, Moscow, 1953.