REFINEMENT OF THE DESIGN DEPENDENCE IN A BALL TRANSMISSION WITH PARALLEL SHAFTS

A. E. Belyaev, A. I. Kan, V. V. Gurin, L. D. Garanin

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

A transmission for parallel shafts with ball intermediate bodies consists of a driving race with balls located on its end surface (in spherical recesses) and a driven wheel with cut end teeth, whose working surface is made along a hypocycloid. A preliminary geometric calculation of this meshing was given by A. I. Nechaev [1]. However, engineering calculations by his method in some cases yielded unacceptable results. Thus, the author of [2] proposed the following dependences, which clearly are not mutually consistent:

\[ r_{в_1}=r_{\text{ш}}(0.4 \div 0.9) \quad \text{and} \quad r_{e_1}=r_{в_1}+r_{\text{ш}}, \]

where \(r_{в_1}\) is the radius of the circle of the internal protrusions of the ball;
\(r_{\text{ш}}\) is the radius of the ball;
\(r_{e_1}\) is the initial circle of the driving disk.

Since the radius of the ball \((r_{\text{ш}})\) is a function of several parameters, including \(r_{e_1}\), the radius \(r_{в_1}\) cannot be assigned arbitrarily. Otherwise, the first equality cited by us will not be satisfied. For clarity, two calculations were made for a transmission ratio \(i=5\) and number of balls \(z_{\text{ш}}=3\). With the arbitrary assignment \(d_{\text{ш}}=12.7\) mm and \(r_{в_1}=0.6r_{\text{ш}}\), according to [2] \(r_{e_1}=12\) mm and \(r_{e_2}=60\) mm were obtained.

If one specifies the minimum overlap coefficient (as is allowed in the calculation of gear transmissions in mechanical engineering) \(\varepsilon=1.1\), then by formula 6 of [2] the radius of the circle of the internal protrusions of the wheel teeth \(r_{n_2}\) proves to be equal to 49.25 mm (Fig. 1). The circle of radius \(r_n\) determines the beginning of conjugation of the end-tooth tip with the ball and is the initial one for determining the overlap coefficient. In this case the wheel tooth is formed by segments of two equidistants to the hypocycloids generated by the centers of the balls, and the intersection of these equidistants lies on a circle having radius \(r_в=51\) mm, i.e., much larger than \(r_{n_2}\) (Fig. 2 shows a transmission with the same initial parameters \(i\), \(z_{\text{ш}}\), \(r_{e_1}\) as the transmission in Fig. 1, but the diameters are selected so that \(r_в=r_{n_2}\)).

The inconsistency found indicates that the formulas proposed by the author of [1] and [2] are not universal, since, evidently, the quantities \(z_{\text{ш}}\), \(r_{\text{ш}}\), \(r_{e_1}\), and \(i\) must be interrelated. Let us note that the determination of such

Fig. 1. For calculation by method (2).

Fig. 2. Creation of a transmission with coefficient value \(\varepsilon = 1.1\), with parameters analogous to the parameters of the transmission in Fig. 1

The derivation of the design dependence is associated with certain difficulties, since on the inner radius of the driven wheels there must always be some platform, which will affect the magnitude of the overlap coefficient. Therefore, in deriving the design dependence, we assume that \(\varepsilon = 1\), \(z_{\mathrm{sh}} \geq 3\), and the tooth width of the wheel along the circumference of the tooth projections is equal to zero. Subsequently, all changes of quantitative order can be introduced into the theoretical formulas obtained.

The problem posed can be solved in two ways: graphically and analytically. In the first case, from a series of drawings constructed for different meshing parameters, one can create the so-called “blocking contours,” which limit the region of possible selection of combinations of the parameters \(i\), \(r_{\mathrm{sh}}\), \(z_{\mathrm{sh}}\), \(r_e\). However, this method requires an enormous amount of time and careful execution of the drawings. Therefore, we have chosen the second method.

Considering a mesh with ball intermediate bodies, one can be convinced that it is a special case of lantern gearing—

…engagement, and, consequently, it may be assumed that the practical line of engagement passes through pole \(P\) (Figs. 2 and 3). The ball enters engagement at the tooth tip and, at the moment of leaving engagement, its center is located at the pole. At the initial moment of engagement, the point of contact of the ball with the tooth profile lies at the intersection of two straight lines: the common normal \((AB—nn)\) to the two mating profiles at the contact point, passing through the pole and the center of the ball, and the axis of symmetry of the tooth. This point \(C\) is the tooth tip.

Fig. 3. Toward the derivation of the design dependence

Let us denote the angles:

\[ \beta=\frac{360^\circ}{2\cdot i\cdot z_{\mathrm{sh}}} =\frac{180^\circ}{z_{\mathrm{sh}}\cdot i},\qquad \alpha=\frac{360^\circ}{z_{\mathrm{sh}}},\qquad \delta=90^\circ\left(1-\frac{2}{z_{\mathrm{sh}}}\right). \tag{1} \]

From Fig. 3 it follows that \(BC=BA-CA\).

In this case

\[ BA=\frac{BL}{\sin\delta} =\frac{BO_1\cdot\sin(180^\circ-\alpha)}{\sin\delta} =\frac{r_{e_1}\cdot\sin\alpha}{\sin\delta}; \tag{2} \]

\[ CA=\frac{OA\cdot\sin\beta}{\sin\gamma} =\frac{r_{e_2}\cdot\sin\beta}{\sin\gamma} =\frac{r_{e_1}\cdot i\sin\beta}{\sin\gamma}. \tag{3} \]

Then, according to 1, 2, 3,

\[ BC=\frac{d_{\mathrm{sh\,max}}}{2} -\frac{r_{e_1}\cdot\sin\alpha}{\sin\delta} -\frac{r_{e_1}\cdot i\sin\beta}{\sin\gamma}, \tag{4} \]

where \(d_{\mathrm{sh\,max}}\) is the maximum ball diameter;
\(i\) is the transmission ratio;

\[ \gamma \text{ is angle } ACO,\qquad \gamma=180^\circ-(\delta+\beta) =90^\circ\left[1+\frac{2}{z_{\mathrm{sh}}}\left(\frac{i-1}{i}\right)\right]. \]

Solving equation 4 for \(d_{\mathrm{sh\,max}}\) and substituting the values of the angles \(\alpha,\beta,\gamma,\delta\), after simple transformations we obtain

\[ d_{\text{ш max}}=2\cdot r_{e_1}\left[ 2\sin\frac{180^\circ}{z_{\text{ш}}} - i\frac{\sin\frac{180^\circ}{i z_{\text{ш}}}} {\cos\left(\frac{180^\circ}{z_{\text{ш}}}\cdot\frac{i-1}{i}\right)} \right]. \tag{5} \]

Thus, to determine the meshing parameters \((d_{\text{ш max}}\) or \(r_{e_1})\), one may use equation (5) over practically any range of values of \(z_{\text{ш}}\) and \(i\) \((z_{\text{ш}}\ge 3,\ i\ge 2)\).

REFERENCES

1. A. I. Nechaev. Investigation of transmissions for parallel shafts with spherical intermediate bodies. Proceedings of STI, collection 41, Krasnoyarsk, 1967.

2. A. I. Nechaev. Investigation of transmissions for parallel shafts with spherical intermediate bodies (dissertation). Krasnoyarsk, 1967.