ASSEMBLY EQUATIONS FOR COAXIAL GEAR MECHANISMS

Yu. Ya. Kovyilin

(Presented by the scientific seminar of the Departments of Theoretical and Applied Mechanics)

To obtain minimum dimensions and weight, coaxial mechanisms (Fig. 1) are usually made with several planet gears. In doing so it is necessary to ensure the so-called “assembly condition.” This condition, as is known, requires that the teeth of the planet gears freely enter the corresponding tooth spaces of the central gear wheels, even in the case of ideal (tight, backlash-free) meshings.

Fig. 1. Diagrams of the most commonly encountered coaxial multi-planet mechanisms.

The assembly condition is expressed in the literature by various equations. Thus, for example, in work [1] an assembly equation was obtained that can be reduced to the form

\[ \frac{z_1 z_{2'} \pm z_2 z_3}{k z_2}=\gamma, \tag{1} \]

where \(z_1, z_2, z_{2'}\), and \(z_3\) are the numbers of teeth of the gear wheels in accordance with Fig. 1;

\(k\) is the number of uniformly arranged mutually meshing planet gears, i.e., such planet gears for which the relative position of the gear wheels 2 and \(2'\) is the same;

\(\gamma\) is a certain integer.

The plus sign corresponds to the \(\overline{AI}\) scheme, and the minus sign to the \(AA\) and \(II\) schemes.

In [2], for the scheme \(\overline{AI}\) an assembly equation of the form is recommended

\[ \frac{z_1 z_{2'} + z_2 z_3}{k z_{2'}} = r, \tag{2} \]

which differs from equation (1) (as applied to the scheme \(\overline{AI}\)) only in that, in the denominator of the left-hand side, the number \(z_2\) has been replaced by the number \(z_{2'}\).

In [3], for two-row coaxial mechanisms, an assembly equation is given (without derivation) having the form

\[ \frac{z_1 z_{2'} + z_2 z_3}{k M} = N. \tag{3} \]

Here, in addition to already known quantities, there appear:

\(M\)—the greatest common divisor of the numbers of teeth of gear wheels 2 and \(2'\);

\(N\)—some integer.

As applied to the scheme \(\overline{AI}\), which may be regarded as a special case of the scheme \(\overline{AI}\) \((z_2 = z_{2'} = M)\), all three equations give one and the same result

\[ \frac{z_1 + z_3}{k} = N. \tag{4} \]

When two-row schemes are calculated, the results obtained are substantially different. Thus, for example, for the mechanism \(\overline{AI}\), with \(z_1 = 24\), \(z_2 = 36\), \(z_{2'} = 27\), and \(z_3 = 87\), the possible values of \(k\) are represented by the following series. From equation (1) we obtain \(k = 1, 3, (5, 7, \ldots)^{1)}\); from equation (2) we find \(k = 1, 2, 4, (5, 7, \ldots)\); from equation (3) it follows that \(k = 1, 2, 3, 4, (5, 6, 7, \ldots)\). If the numbers of teeth are taken equal to \(z_1 = 18\), \(z_2 = 39\), \(z_{2'} = 21\), and \(z_3 = 78\), then according to equations (1) and (2) the assembly of the mechanism will be impossible for any \(k\) (even for \(k = 1?!\)), whereas in accordance with (3) assembly proves possible for \(k = 1, 2, 3, (4, 5, \ldots)\). The results of checking the assemblability of the mechanism \(AA\) by equation (1) depend on the order in which the gear wheels are numbered, i.e., on which of the central gear wheels is taken as the first. When equation (3) is used, the order of numbering is immaterial.

Equations (1), (2), and (3) are insufficient for a complete calculation of assembly, since none of them can answer a number of questions of interest to the designer, for example:

1) what should be the mutual arrangement of the gear wheels 2 and \(2'\) of the planet gears if, for instance, in order to achieve the greatest compactness of the mechanism it is expedient to make them noninterchangeable;

2) how to check the possibility of assembling the mechanism if the planet gears are arranged nonuniformly;

3) how, during installation, the planet gear should be positioned relative to the central gear wheels.

A complete calculation of the assembly of multi-planet coaxial mechanisms can be carried out with the aid of the so-called generalized assembly equations [4]. In practice, however, these equations are not entirely convenient, since in order to obtain an answer to the question of whether assembly of the mechanism is possible or not, it is sometimes necessary to investigate a large number of

\(^{1)}\) Here and below, the values of \(k\) indicated in parentheses do not satisfy the “adjacency” condition.

solutions of one of the generalized equations. Meanwhile, in designing a mechanism, the designer is interested first of all in the answer precisely to this question.

Below is set forth a derivation of assembly equations that make it possible, in a convenient form, to calculate the assemblability of multi-satellite mechanisms¹.

The teeth of each of the satellite gear wheels are numbered (Fig. 2, a and b) clockwise by the numbers \(0, 1, 2, \ldots, x, \ldots\) (on gear wheel 2) and \(0, 1, 2, \ldots, x', \ldots\) (on gear wheel \(2'\)). Teeth designated by the digit 0 are called reference teeth.

Fig. 2. Schemes for coordinating the teeth and gear wheels of the satellites.

The reference teeth for \(\psi = 0\) are located on different sides of the center of the satellite (Fig. 2, a) in the mechanism \(\overline{AI}\), and on one side (Fig. 2, b) in the mechanisms \(AA\) and \(II\). The mutual position of gear wheels 2 and \(2'\) is coordinated by the angle \(\psi\) between the axes of symmetry of the selected reference teeth (Fig. 2, a and b). The mutual arrangement of the teeth with numbers \(x\) and \(x'\) is determined by the angle \(\Theta\) (Fig. 2, a and b) between their axes of symmetry, which is obviously equal to

\[ \Theta = \psi + \frac{360}{z_2}x - \frac{360}{z_{2'}}x'. \tag{5} \]

Figure 3 shows a diagram of the mechanism \(\overline{AI}\) with one satellite \(A\). The angle \(\psi\) for satellite \(A\) is taken equal to zero (with one satellite, assembly is possible for any value of \(\psi\)).

Next, suppose it is required to install a second satellite \(B\) at an angle \(\varphi\) degrees relative to the first.

Let us denote the engagement poles by the letters \(a, b, a'\), and \(b'\). Suppose that along the arc \(ab\) there fits the greatest integer number of pitches \(e_1\), with a remainder \(s_1\) still remaining, and along the arc \(a'b'\), \(e_3\) whole pitches and a remainder \(s_3\). The remainders of the arcs are equal to:

\[ s_1 = \left(\frac{z_1}{360}\varphi - e_1\right)t_{12}, \tag{6} \]

\[ s_3 = \left(\frac{z_3}{360}\varphi - e_3\right)t_{2'3}. \tag{7} \]

Here \(t_{12}\) and \(t_{2'3}\) are the pitches of engagement of the corresponding pairs of gear wheels, measured along the pitch circles.

Next, on the pitch circles of gears \(2\) and \(2'\) of planet gear \(B\), from the poles \(b\) and \(b'\) we lay off the arc \(bd=s_1\) and the arc \(b'd'=s_3\).

In order for planet gear \(B\) to mesh with the central gears, it must have a pair of teeth with numbers \(x\) and \(x'\), the acute angle \(\Theta\) between whose axes of symmetry is equal to the sum of the angles \(dBb\) and \(d'Bb'\), i.e., the equality is required

Fig. 3.

\[ \Theta=\angle dBb+\angle d'Bb'. \]

Having determined the angles \(dBb\) and \(d'Bb'\) from the ratios of the corresponding arc lengths to their radii, expressed according to the general formula

\[ R=\frac{zt}{2\pi}, \]

and taking into account the dependences (6) and (7), we obtain

\[ \Theta=360\left(\frac{\dfrac{z_1}{360}\varphi-e_1}{z_2} + \frac{\dfrac{z_3}{360}\varphi-e_3}{z_{2'}}\right). \tag{8} \]

Combining (5) and (8), we find:

\[ \frac{\dfrac{z_1}{360}\varphi-e_1}{z_2} + \frac{\dfrac{z_3}{360}\varphi-e_3}{z_{2'}} = \frac{\psi}{360} + \frac{x}{z_2} - \frac{x'}{z_{2'}}. \tag{9} \]

With the reverse order of numbering of the gear teeth, the right-hand side of (9) should be multiplied by minus one.

It is easy to show that equation (9) will also be valid for mechanisms \(AA\) and \(II\), if in its left-hand side the plus sign is replaced by a minus sign and, in addition, when calculating mechanism \(II\), the right-hand side is multiplied by minus one.

Thus, the general assembly equation, suitable for all the schemes shown in Fig. 1 with any order of numbering of the gears, may be represented in the form

\[ \frac{\dfrac{z_1}{360}\,\varphi-e_1}{z_2} \ \pm\ \frac{\dfrac{z_3}{360}\,\varphi-e_3}{z_{2'}} = \sigma\left(\frac{\psi}{360}+\frac{x}{z_2}-\frac{x'}{z_{2'}}\right). \tag{10} \]

Here and below, \(\sigma=1\) if the central gear designated by the number \(1\) has external teeth, and \(\sigma=-1\) if it has internal teeth.

The upper sign (plus) is taken in calculating the schemes \(\overline{AI}\) and \(\overline{AI}\); the lower sign (minus), in calculating the schemes \(AA\) and \(II\).

From equation (10), incidentally, it is clear that, for given numbers of teeth of all gears and the angle \(\varphi\), installation of planet \(B\) can always be effected. For this it is sufficient to choose some tooth numbers \(x\) and \(x'\) on planet \(B\) (for example, \(x=0\) and \(x'=0\)), and then, from equation (10), to calculate the required angle \(\psi\). In this case, planets \(A\) and \(B\) will in general not be interchangeable.

For the case of interchangeable planets, in equation (10) one should take \(\psi=0\).

Then we obtain:

\[ \frac{z_1 z_{2'} \pm z_2 z_3}{\dfrac{360}{\varphi}} = \sigma\left(z_{2'}x-z_2x'\right)+z_{2'}e_1\pm z_2e_3. \]

The right-hand side of this equation is an integer. This integer has the property that it is divisible without remainder by the greatest common divisor \(M\) of the numbers of teeth of gears \(2\) and \(2'\). Consequently, the left-hand side of the equation must also be divisible without remainder by the number \(M\). Thus, instead of one assembly equation, it is now possible to write the following two:

\[ \frac{z_1 z_{2'} \pm z_2 z_3}{\dfrac{360}{\varphi}M}=N, \tag{11} \]

\[ \sigma\left(z_{2'}x-z_2x'\right)+z_{2'}e_1\pm z_2e_3=MN. \tag{12} \]

Here \(N\) is some integer.

Equations (11) and (12) are valid for mechanisms both with uncorrected and with corrected gears, since in deriving them no restrictions of any kind were introduced in this respect.

Equation (11) makes it possible, by the shortest route, to check the possibility of installing an interchangeable planet \(B\) at a given nominal angle \(\varphi\) to planet \(A\), and, if necessary, to determine the required value of the angle \(\varphi\) that deviates least from the nominal one. This deviation, as calculations show, usually lies within a few minutes and only in the worst case reaches several degrees.

Equation (12) serves to determine the numbers \(x\) and \(x'\) of the teeth of the double planet \(B\), which, during assembly of the mechanism, must be dis-

are to be placed opposite the corresponding spaces \(e_1\) and \(e_3\) of the central gear wheels. The number of nonrepeating variants for installing the planet gear is obviously equal to the number \(M\). For the scheme \(\overline{AI}\), solving equation (12) is pointless, since the number of possible variants for installing a single planet gear is equal to the number of teeth \(z_2\) [provided, of course, that equation (11) is satisfied].

We shall show the procedure for calculation by equations (11) and (12) on a concrete example.

Example. In the mechanism \(\overline{AI}\), for \(z_1=16\), \(z_2=39\), \(z_{2'}=21\), and \(z_3=76\), check the possibility of installing two mutually replacing planet gears \(A\) and \(B\) at the nominal value of the angle \(\varphi=\varphi_{\text{nom}}=120^\circ\). If such installation is impossible, determine by how much the angle \(\varphi\) must be changed so that the mechanism can be assembled. Determine the numbers of the “meshing” teeth of planet gear \(B\).

1) We check the possibility of assembling the mechanism at \(\varphi=\varphi_{\text{nom}}\) by equation (11).

\[ \frac{z_1 z_{2'}+z_2 z_3}{\dfrac{360}{\varphi}M} = \frac{16\cdot21+39\cdot76}{\dfrac{360}{120}\,3} = 366\frac{2}{3}\ne N. \]

Since a fractional number has been obtained, assembly of the mechanism is impossible at \(\varphi=120^\circ\).

2) We determine the required angle of installation of the planet gears, taking the number \(N=367\).

\[ \varphi=\frac{360MN}{z_1 z_{2'}+z_2 z_3} = \frac{360\cdot3\cdot367}{16\cdot21+39\cdot76} = 120.109^\circ\;(120^\circ6'32''). \]

As we see, the required angle \(\varphi\) differs from the nominal one by only \(+6'32''\).

3) We determine the numbers of the “meshing” teeth of planet gear \(B\).

\[ \frac{z_1}{360}\varphi=\frac{16}{360}\,120.109\simeq5.3, \qquad e_1=5; \]

\[ \frac{z_3}{360}\varphi=\frac{76}{360}\,120.109\simeq25.4, \qquad e_3=25. \]

Substituting all quantities into equation (12), we obtain

\[ x=1+\frac{13}{7}x'. \]

It follows from this that the integer values \(x=1, 14, 27\) correspond to the values \(x'=0, 7, 14\). Three possible variants \((M=3)\) are obtained for installing planet gear \(B\): opposite the space \(e_3\) there must be either tooth 0, or tooth 7, or tooth 14 of gear \(2'\) (Fig. 3); then opposite the space \(e_1\) there will be, respectively, either tooth 1, or tooth 14, or tooth 27 of gear 2.

When designing mechanisms with several planet gears \(A, B, C, D,\ldots\), it is generally necessary to calculate the installation of each planet gear separately (with the exception of planet gear \(A\)). In this case the angle \(\varphi\), coordinating the positions of the planet gears \(B, C, D,\ldots\), must always be measured from planet gear \(A\).

The calculation of assemblability with noninterchangeable planet gears should be carried out by equation (10). With interchangeable planet gears, the calculation must be carried out by equations (11) and (12).

From equation (11) it is easy to obtain particular assembly equations known from the literature. Thus, for example, for the case of a uniform arrangement of \(k\) interchangeable planet gears, \(\varphi = 360 : k\), and equation (11) passes into equation (3).

If in equation (11) we substitute \(\varphi = 360 : k\), \(N = \gamma \dfrac{z_{2'}}{M}\), and omit the minus sign, then we obtain equation (2). Finally, upon substituting into equation (11) \(\varphi = 360 : k\) and \(N = \gamma \dfrac{z_2}{M}\), we arrive at equation (1).

Taking the value of \(N\) from equation (12), we find that in equation (2)

\[ \gamma = x - \frac{z_2}{z_{2'}} x' + e_1 + \frac{z_2}{z_{2'}} e_3, \tag{13} \]

and in equation (1)

\[ \gamma = \sigma \left( \frac{z_{2'}}{z_2} x - x' \right) + \frac{z_{2'}}{z_2} e_1 + e_3. \tag{14} \]

From expressions (13) and (14) it is clear that the number \(\gamma\) in equations (1) and (2) can be an integer only in certain particular cases, when, for example, the ratios of the corresponding numbers of teeth of the planet gears are integers. The requirement that \(\gamma\) be only an integer [1], [2], etc., very strongly restricts the possibilities for synthesizing mechanisms and is compensated in many cases either by a considerable deviation from the required kinematic ratios, or by an unjustified increase in dimensions, and often by both together. In this respect equation (3) undoubtedly has a great advantage, since it considerably expands the designer’s possibilities. However, equation (3) is insufficient for a complete calculation of assembly even with a uniform arrangement of interchangeable planet gears.

The equations derived above, (10)–(12), make it possible easily and quickly to carry out a complete calculation of the assembly of any of the mechanisms shown in Fig. 1, for any arrangement of interchangeable and noninterchangeable planet gears.

They may be used for calculating the assembly not only of two-row but also of multi-row plane mechanisms, as well as spatial mechanisms composed of bevel gears.

LITERATURE

1. Dobrovolsky V. V. Selection of gears for reducers with several planet gears, “Bulletin of Engineers and Technicians,” 1937, No. 3.
2. Mechanical Engineer’s Handbook, vol. 1, Mashgiz, 1954.
3. Merritt H. E. Gears, Mashgiz, 1947.
4. Kronin D. V. Assembly conditions for planetary reducers, Proceedings of MAI, issue 18, Oborongiz, 1953.

1. Reported at the scientific seminar of the Department of Theoretical and Applied Mechanics, TPI, December 4, 1958. ↩