DYNAMIC SYNTHESIS OF PLANAR CAM MECHANISMS WITH TRANSLATING CAMS AND ROLLER FOLLOWERS

Yu. Ya. Kovylin

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

1. The dynamic synthesis of a mechanism is the problem of determining such parameters of its scheme as satisfy the necessary dynamic conditions ensuring the prescribed motion of the designed mechanism without jamming and with a sufficiently high efficiency. These conditions, as is known, reduce to requiring that the magnitude of the pressure angle in the cam pair, for any position of the mechanism, not exceed the admissible value of this angle.

In the present work, the magnitude of the admissible pressure angle is regarded as given.

1. Let us consider some \(i\)-th instantaneous position of the mechanism within the stroke \(H\) of the cam (Fig. 1, \(a\)). Suppose that, in the chosen position of the mechanism, the velocity of motion of the cam is \(\overline{u_i}\), the velocity of the center \(C\) of the follower roller is \(\overline{v_i}\), and the sliding velocity of the center \(C\) of the follower roller along the pitch profile of the cam is \(\overline{w_i}\).

These three velocities, as is known, are related to one another by the vector equation

\[ \overline{v_i}=\overline{u_i}+\overline{w_i}, \tag{1} \]

where the vector \(\overline{w_i}\) must be perpendicular to the normal \(nn\) to the pitch profile of the cam at the point \(C_i\).

The acute angle \(\alpha_i\) between the normal to the pitch profile of the cam and the direction of the velocity \(\overline{v_i}\) of the follower is the pressure angle in the \(i\)-th instantaneous position of the mechanism.

If the magnitudes and directions of the vectors \(\overline{v_i}\) and \(\overline{u_i}\) are known, then the pressure angle \(\alpha_i\) can be determined without constructing the normal to the pitch profile of the cam and even without constructing the cam profile itself. To do this it is sufficient, at an arbitrary place on the drawing (Fig. 1, \(b\)), to construct the velocity diagram according to equation (1) and determine the acute angle \(\gamma_i\) between the lines of action of the velocity vectors \(\overline{v_i}\) and \(\overline{w_i}\), and then also the pressure angle \(\alpha_i\), using the obvious relation

\[ \alpha_i = 90^\circ - \gamma_i. \tag{2} \]

Fig. 1

1. In studying the pressure angles of a cam mechanism, it is expedient to divide equation (1) by a scalar numerically equal to \(u_i\), and then transform it to the form

\[ \left|\,l \frac{d\psi}{dS}\,\right|_i \overline{v_i}^{\,\circ} = \overline{u}^{\,\circ} + \frac{\overline{w_i}}{u_i}. \tag{3} \]

The following enter into equation (3):

\(l = l_{OC}\) — the length of the follower;

\[ \frac{d\psi}{dS} \]
— the derivative of the angular displacement of the follower with respect to the displacement of the cam;

\(\overline{v_i}^{\,\circ}\) and \(\overline{u}^{\,\circ}\) — vectors of the same direction as the vectors \(\overline{v_i}\) and \(\overline{u_i}\), with length equal to unity, i.e. \(v_i^{\circ}=1,\ u^{\circ}=1\).

In the dimensionless equation (3), all vectors have the same direction as in equation (1). Therefore it may be called the dimensionless vector equation of velocities of the mechanism.

Using the data of the kinematic analysis of the mechanism (Fig. 1, c), with the aid of equation (3) it is easy to determine the pressure angles in any position of the mechanism. For this it is sufficient to construct, with the side \(\vec{u}^{\,0}\) coincident, the plans of the dimensionless velocities of the mechanism (Fig. 1, d). The construction is carried out in the following order.

a) From an arbitrary point \(p\), in the direction of motion of the cam, lay off an arbitrary segment \(pa\), representing the vector \(\vec{u}^{\,0}\) with a modulus equal to unity. b) Along the directions of the instantaneous velocities of point \(C\) of the follower, from point \(p\) lay off the segments (arms)

\[ pb_i=\left|\, l\frac{d\varphi}{dS}\,\right|_i(pa), \tag{4} \]

which represent, on the scale of the vector \(\vec{u}^{\,0}\), the dimensionless velocities of point \(C\) of the follower; the locus of the points \(b_i\) is the hodograph of the vector of the dimensionless velocity of point \(C\) of the follower—the arm diagram.

c) Connecting point \(a\) with the points \(b_i\) of the arm diagram by straight-line segments, determine the magnitudes of the angles \(\gamma_i\), and then also of the pressure angles \(\alpha_i\) by formula (2).

1. We shall now consider the problem of the dynamic synthesis of a cam mechanism with a translating cam and a rotating roller follower.

Given:

a) the form of the graph \(\psi=\psi(S)\) of the follower motion,
b) the swing \(X\) of the follower,
c) the length of the follower \(l\),
d) the motion of the cam—to the center \(O\) of rotation of the follower,
e) the allowable pressure angle \([\alpha]\).

It is required to determine the basic dimensions of the mechanism scheme: the initial angle \(\delta\) of the follower and the stroke \(H\) of the cam.

Procedure of solution

a) According to the prescribed form of the graph \(\psi=\psi(S)\) of the follower motion, plot it in the rectangular coordinate system \(S-\psi\) (Fig. 1, c). The scale along the ordinate axis is

\[ \mu_{\psi}=\frac{X}{mn}, \tag{5} \]

where \(mn\) is the ordinate of the graph \(\varphi=\varphi(S)\), representing the swing \(X\) of the follower.

The scale along the abscissa axis is undetermined.

b) Construct the graph

\[ \frac{d\psi}{dS}=\frac{d\psi}{dS}(S) \]

by graphical differentiation of the curve \(\psi=\psi(S)\) (Fig. 1, c).

Both scales of this graph are undetermined.

c) In accordance with the prescribed law \(\psi=\psi(S)\), draw the successive instantaneous positions of the follower (Fig. 2, b). The initial position is arbitrary.

The construction scale \(\mu_l\) is arbitrary.

d) From an arbitrary point \(p\) (Fig. 2, \(a\)) we draw rays perpendicular to the plotted positions of the follower (parallel to the instantaneous velocities of the follower).

Fig. 2

e) On these rays, to an arbitrary scale, we lay off the arms \(\overline{p b_i}\), proportional to

\[ \left|\,l\frac{d\psi}{dS}\,\right|_i \overline{a_i c_i}. \]

Since the scale of construction is arbitrary, on the rays, instead of the indicated arms, one may simply lay off the segments \(ii'\), taken directly from the graph of the first derivative.

e) Through the ends \(b_i\) of the arms we draw a curve—the arm diagram (Fig. 2, a).

f) From the points \(b_i\) of the arm diagram we draw straight lines making with the corresponding arms angles equal to

\[ [\gamma]=90^\circ-[\alpha]. \tag{6} \]

The straight lines drawn in this way bound a region (shaded in Fig. 2, a) in which any point may be taken as the end of the vector \(\dot u^\circ\); in this case the pressure angle \(\alpha\) will not exceed the permissible value in any position of the mechanism.

Let, for example, point \(a'\) be chosen as such a point. Connecting point \(a'\) with point \(p\) by a straight-line segment, we find the vector \((\dot u^\circ)'\), whose modulus is equal to unity. The direction of the vector \((\dot u^\circ)'\) indicates the direction of motion of the cam relative to the previously constructed positions of the follower (angle \(\delta'\)).

g) From (4), replacing \(pa\) by \(pa'\), we find the value \(\left|\dfrac{d\psi}{dS}\right|_i\) for the \(i\)-th position of the mechanism, and then the scale along the ordinate axis of the graph

\[ \frac{d\psi}{dS}=\frac{d\psi}{dS}(S), \]

as

\[ \mu_{\psi'}= \frac{\left|\dfrac{d\psi}{dS}\right|_i}{ii'} = \frac{p b_i}{l(pa')(ii')}. \tag{7} \]

Here \(ii'\) is a segment taken from the graph of the first derivative (Fig. 1, c).

i) The scale along the abscissa axis of both graphs (Fig. 1, c) is determined from the known relations in graphical differentiation:

\[ \mu_s=\frac{\mu_\psi}{k\mu_{\psi_1}} = \frac{\mu_\psi\, l(pa)(ii')}{k(p b_i)}, \tag{8} \]

where \(k\) is the pole distance chosen arbitrarily in graphical differentiation.

k) Then we determine the cam stroke

\[ H=\mu_s(On)= \frac{\mu_\psi\, l(pa')(ii')(On)}{k(p b_i)}, \tag{9} \]

where \(On\) is the length of the base of the diagram \(\psi=\psi(S)\).

It is evident from expression (9) that, all other conditions being equal, the magnitude of the cam stroke \(H\) is the smaller, the smaller the segment \(pa'\) representing the dimensionless velocity of the cam \((\dot u^\circ)'\). The smallest cam stroke \(H\) will obviously occur when point \(a'\) coincides with point \(a\), the intersection of the straight lines drawn at angles \([\lambda]\) to the rays of the arm diagram, i.e., when

\[ pa'=pa. \]

In this case the initial angle of the follower will be equal to \(\delta\) (Fig. 2, a and b).

l) After determining \(H\) and \(\delta\), the construction of the pitch curve and the working profile of the cam is carried out in the usual way, i.e., effectu—

kinematic synthesis of the mechanism is carried out (Fig. 2, б). The roller size is determined from known relations [2], [3].

m) The smallest length \(L\) of the cam is determined directly from the drawing (if \(\psi_n=0\), then \(L=H\)).

1. The dynamic-synthesis method considered is evidently also applicable in the case when the follower moves in translation.

In this case, in equation (3) the quantity \(l\dfrac{d\psi}{dS}\) need only be replaced by the quantity \(\dfrac{d\sigma}{dS}\), where \(\sigma\) is the linear displacement of the follower.

1. The proposed method can also be used for the dynamic synthesis of cylindrical cams of spatial mechanisms, since a cylindrical cam can be reduced to a translating cam if it is developed onto a plane along the cylinder of mean diameter [2], [3].

2. On the basis of the proposed graphical construction, in certain special cases it is possible to obtain a final expression in analytical form.

Conclusions

1. The proposed method requires simple auxiliary constructions.

2. The method makes it possible to determine directly the smallest length \(L\) of the cam and the direction of its motion (angle \(\delta\)).

3. For a given direction of motion of the cam, it is possible to determine the smallest length of the cam, and conversely.

4. The method can be used for the synthesis not only of planar mechanisms, but also of certain spatial mechanisms.

References

1. Basic Problems of the Theory of Machines and Mechanisms, Academy of Sciences of the USSR, Moscow, 1956.

2. Artobolevsky, I. I. Theory of Mechanisms and Machines, GITTL, 1940—1953.

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