ON ONE FEATURE OF THE DESIGN OF BALL DRIVES

A. E. BELYAEV

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

In view of certain shortcomings of worm gears, the chief of which is a comparatively low efficiency, designs of worm gears with rolling friction have appeared recently. As one variety of such drives, the present article considers a worm gear with ball intermediate bodies.

However, not all proposed variants of ball meshing for transmitting motion between intersecting axes are acceptable for investigation; in particular, this applies to a drive of the following design (Fig. 1, a). The driving element of the drive is made in the form of a disk 1 with hemispherical grooves cut on its end face along an Archimedean spiral (in Fig. 1, a, a section of the disk is shown by the “meshing” plane passing through the axis \(O'O''\) and perpendicular to the end face of the disk). The driven element 3 is likewise a disk (with an axis of rotation perpendicular to the plane of the drawing), on the periphery of which balls 2 are mounted closely adjacent to one another, fastened in a definite manner from both end faces.

When considering any new variant of meshing (including ball meshing), it is necessary first of all to consider the kinematic and force parameters. In the proposed variant of the drive (Fig. 1, a), meshing takes place along two centroids: a straight line and a circle. In the mentioned section (“wheel—rack”), the normal to the profile of the ball (passing through point \(M\)—the point of contact of the intermediate body and the profile of the driving disk 1) is situated in a plane nonpermanently and, consequently, the pole of meshing will not be a fixed point. It is evident that in this case the contact ratio \(\varepsilon\) will prove to be less than unity, and the angular velocity \(\omega_2\) of the driven member will prove to be nonconstant, i.e., the fundamental law of meshing will not be satisfied. It is also difficult to speak of satisfactory strength of the pointed tooth of the driving element.

In order for the drive under consideration to prove workable, it is necessary to increase the distance between the intermediate elements to such a value that a tooth of height \(h\) can be profiled on the driving disk (Fig. 1, b). In this case the common normal \(NN\) at the point \(M\) of contact of the conjugate surfaces must pass through the pole \(P\) (the instantaneous center of rotation). The tooth height \(h\), evidently, also determines the value of the contact ratio, which, for the case shown—

Fig. 1

a) Ball drive with a flat driving disk;
b) ball drive with a flat driving disk, ensuring constancy of the transmission ratio with a contact ratio of not less than unity

...shown in Fig. 1,b, will be not less than unity (the engagement shown in Fig. 1,a may be regarded as a special case of the one just considered, with \(h=0\) and \(\varepsilon=0\)). Let us note that, in the relative motion of the two elements under consideration, point \(P\) will describe hypocycloids (\(PC\) and \(PC'\)); therefore the working profile of the tooth of the driving disk must be formed by equidistants of the hypocycloids (\(L'F\) and \(LF\)). With these modifications, however, the driven element is structurally rather complicated, since corresponding slots will have to be made in it (with depth greater than \(h\)), while the driving element will prove technologically difficult to manufacture because of the complexity of cutting the profile \(KLM\).

Further analysis shows that a drive with intersecting axes, satisfying all the requirements of kinematics and sufficiently simple in design and manufacturing technology, can be created if hemispherical grooves are cut on the end face of the driving disk along

Fig. 2. Ball drive with intersecting axes ($\omega_2 = \mathrm{const};\ \varepsilon > 1$)

of an Archimedean spiral on a toroidal surface (Fig. 2). In this case, the contact ratio can theoretically be infinitely large (in Fig. 2, $\varepsilon \approx 3$). In this variant, the angular velocity $\omega_2$ will be constant, and the strength of the tooth of the driving disk sufficiently high. Instead of point contact (Fig. 1, a), in the proposed drive (Fig. 2) the contact will be linear, and the number of contact lines will be in direct dependence on the contact ratio. Thus, the drive variant shown in Fig. 2 can be used (after appropriate experimental verification) both as a power drive and as a drive possessing high kinematic accuracy.