Series of gear speed reducers

[57] ABSTRACT

The invention relates to a series of n gear speed reducers with at least n unequal center distances in increasing order, the absolute values of the last center distance of each speed reducer in the series, i.e. of the specific center distances, form an improper geometric progression of which the successive values of the ratio form a geometric progression.

Claims, 14 Drawing Figures

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PATENTED JUL 4 1972

SHEET 1 OF 5

3,673,885

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INVENTOR.

David C Hansen

BY

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ATTORNEYS

PATENTEDM 41972

3,673,885

SHEET 2 OF 5

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INVENTOR.

David C. Hansen

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attorneys

PATENTED JUL 41972

3,673,885

SHEET 3 OF 5

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ATTORNEYS

PATENTED JUL 41372

3,673,885

SHEET U OF 5

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BY

INVENTOR.

David C Hansen attorneys

PATENTED JUL 41972

3,673,885

SHEET 5 OF 5

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INVENTOR.

David C. Hansen

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attorneys

3,673,885

SERIES OF GEAR SPEED REDUCERS

This invention has as its object a series of gear speed reducers fulfilling the optimal conditions for rational and profitable prefabrication, meeting at the same time the interests and convenience of the manufacturer and the interests and requirements of the user. The series in question is conceived in such a way that each speed reducer in a given series draws near to the specific requirements of the user in an ideal way. These requirements principally that dominate the problem.

The benefits of a rational prefabrication for the manufacturer are in fact well known. These benefits will be all the more saveguarded as will be the interests and requirements of the user. For this purpose, starting from options judiciously taken by the manufacturer, the user must be able to find in the given series, the appropriate reducer or reducers, which approximate most closely the ideal conditions, free from any under-dimensioning and with a least amount of overdimensioning of the components within a reducer with respect to one another.

These options lead the manufacturer to (1) choose beforehand a manufacturing program, determined by the well-considered choice of the smallest and the largest sizes; (2) find a judicious law of progression between the two limits of the program established at (1); (3) decide upon the number of sizes, considered to be sufficient to meet the demands of the market.

In presence of these options, the manufacturer finds himself facing the problem of the execution of all and every one of the gear speed reducers in the chosen series, an execution which must at all times be adequate and economical. Each reducer consists mainly of a housing and of gear sets. The problem of execution consists of imagining a solution, within the above mentioned options, which enables an optimal use of the same housings and the same gear sets in a maximum number of valid combinations, i.e. combinations which meet practical requirements, the reducers having in none of the many combinations components which are needlessly overdimensioned with respect to one another.

In other words, the series has to be conceived in such a way that, although it consists of prefabricated material, each reducer closely approximates that, which would have been adopted if each case had been calculated separately.

These requirements tie the prefabrication of both housing and gears to certain conditions. The series of gear speed reducers, according to the invention, is therefore characterized by very particular rules, on the one hand for the establishment of a series of center distances and — in the framework of options of a manufacturing nature — the spacing of the sizes of the gears suitable to the series considered; and, on the other hand for the judicious choice, in this spacing, of the center distances and thus of the gears for each size of reducer in the given series.

For more clarity, reference will always be made hereafter, and only as an example, to a series of reducers in which each reducer has three unequal center distances of increasing size. The last center distance in each reducer of the series, is called hereafter “specific center distance.”

The specific center distance of a speed reducer can be the first or second center distance of other reducers in the series, so that the number of center distances with different absolute values, may be very close to the number of reducers in the given series, without being less.

Therefore, according to the invention, the series of gear speed reducers is essentially characterized by the fact that the absolute values of the last center distance of each reducer in the series — or specific center distance — constitute an improper geometric progression, the successive values of whose ratio constitute a proper geometric progression.

On the other hand, again according to the invention, the series of gear speed reducers in question is characterized by the fact that each reducer of the series comprises gear sets such that the values of all primary and overall reduction ratios of said reducers are part of the same geometric progression.

For more clarity, use will be made hereunder of a model program of prefabrication of a series of gear speed reducers, according to the invention, on the understanding that it only has to be considered as an example, without restricting the ob5 ject and the scope of the invention.

In the enclosed drawings,

FIGS. 1 to 10 show, schematically and as a front view, the successive reducers of the series considered;

FIG. 11 represents a table giving the selection of the specific 10 center distances in the execution considered;

FIG. 12 represents schematically and in perspective, a gear speed reducer with parallel input and output shafts;

FIG. 13 represents schematically and in perspective, a gear speed reducer with perpendicular input and output shafts;

FIG. 14 represents schematically and in perspective, a gear speed reducer with a built-on motor, whose axis constitutes the primary axis of the reducer.

In the example chosen, the series comprises reducers only with helical gears and three unequal center distances of in20 creasing size, the third center distance being called the specific center distance. The number n of reducers A, B, C, D, E, F, G, Η, K, L (FIGS. 1 to 10) in the series equals 10, whereas the number of different center distances of the series in question is n + 2 = 12, designated by the series 0«,, a₀, a,, a₂, ²⁵ a₃ ... a₁₀. The number of specific center distances also equals 10 and each of the specific center distances i.e. a, to a₁₀, may be found as a first, a second or a third center distance in the reducers of the series. As shown in the table of FIG. 11, the __ -center distances a_M to u₁₀ are used respectively 1,3,3,4,5,3, 4, 2,2, 1, 1, and 1 times, which means that there are some 30 center distances obtained from 12 different center distances.

If this repetition in the use of the center distances is advantageous, still more is the way in which the range of as35 sociated torques, is established. This determines the range of characteristic torques of the speed reducers in the series in question, so as to meet the requirements of the user better, and also the way in which these center distances are selected in such a way as to harmonize with the center distances, used 4Q as specific center distances, in order to obtain as favorable as possible a power equilibrium.

To make this more understandable, the absolute values of the ten specific center distances a, up to a₁₀ in the series of the ten gear speed reducers may be considered to form a geomet45 ric progression in which the common ratio h is itself a variable, i.e. an improper geometric progression, and the successive values of h constitute a proper geometric progression, whose constant ratio is equal to p. The values and p are calculated in function of the characteristic torques of the different 50 sizes, which lies in the sphere of any qualified person familiar with the manufacture and the sale of gear speed reducers.

The series of center distances a_t to a_I0, is represented as follows, taking into consideration that a_t is any given center distance of the series in question:

α_Ι;β2=β₁·Αι;β₃=β₂·Α_!;...;

Oj = Oi_i’/l|_₁ . . . 4n~&amp;n-i'hn-l O)

As to the series of variable common ratios h, it is represented as follows, taking into consideration that h_t is any given variable common ratio in the series:

Ail /¾=hip; A3=hip;

hr=&gt;h-iP; hn-l^hn-iP (2) which can also be written:

Α,;Α_Ι=Α₁·ρ;λ3=Α_Ι·ρ^ί;...

A_i = A₁p^t-¹;...h_B.₁ = A_lff-² (3)

Considering series (1) the values of A in function of the center distances a may be deduced from it:

₇₀ (4) and

From (3) the following relationship is deduced:

3,673,885 n-2/ΚΪ ^P~ y ht (6)

Combining the two series (1) and (2), one deduces that the value of any center distance a, can be calculated from the relationship.

(i-l)(i-2) ai=ai-hi'~^l-p ² (7)

This shows that any qualified person can easily apply the essential characteristic of the invention, i.e. take care that, in the series, the absolute values of the specific center distances aj.. . a₂ . . . a„ form an improper geometric progression of which the variable common ratio itself h_lt h₂. .. follows the law of a proper geometric progression with constant common ratio p.

Between the face width b of the gear and the center distance a, the following relationship can be established:

b = K_Ia^r (8) which is valid for all the gears in the series in question and in which, for a given reduction ratio, K, and e are constants, which can be determined by experience in the manufacture of gears, in such a way that these gears can be produced with the necessary precision.

On the other hand and in order to approach the “optimal conditions” stated in this document, the characteristic torques of the successive sizes are chosen in such a way that they equal the characteristic torques of the last center distance of the reducer, i.e. of the specific center distance. Moreover, the utmost has been done to bring close together the torques, calculated in function of surface durability and of strength, establishing however the fact that the characteristic torque is determined by the calculation of the surface durability.

In these conditions, the characteristic torque T calculated according to a given process of calculation (in the example, the process put forward by A.G.M.A.: American Gear Manufacturers Association) may be expressed by the relationship;

T=K₂-^b° (9) in which K₂ is a constant exponent for all characteristic torques, being function of the raw material, the reduction ratio and the rotational speed of the secondary shaft; a is the center distance, b indicates the face width of the gears,/and g are constant numbers.

The characteristic torque T_t of any given specific center distance a_t may be written, according to the relationships (8) and (9):

7, = ¾^ (10) in which K₃ and m are constants withK₃ = K₂ K_Y° and m =/+€ g

In this way a series of characteristic torques may be placed alongside the series of center distances, so that the terms of both series correspond two by two. From the relationship (10) between T_t and a,, it is established that the series of characteristic torques is, as well as the series of specific center distances, a geometric progression having identical properties.

The series of characteristic torques T may be written as follows; taking in consideration that T_t is any given characteristic torque taken from the series 7, up to T_n:

7,;7₂=7,*,;7₃=T₂K₂;...;

7, = 7^,‘Λ^,;...; 7„ = (11)

By comparison with the series of specific center distances (1) and taking the relationship (10) into account, one can write:

/:,_,= λ,.,' (12)

The series of the variable ratios k is, as is also series (2), a geometric progression.

Indeed:

from (12) results ki=hi^mfrom (3) results =¹therefore Αι=^i—1from (12), is found Ai^m=fci or also hi=!yfki Taking q=p^m in which q is constant or, also p=ΐ/q (13) one can finally write:

k_t=ki^-' (14) which shows that the series of ratio k is a geometric progression.

The series of variable ratios k is expressed as follows, taking into consideration that A, represents any given ratio from the series £,; k2 = kiq;...; k^k^q;...

£, = ^-19,-.--(4-1 = 4-29(15) or also £1:^2 = ^19(... £₃=£i9²;...

£1 = ^9^5---:^1=^9^²(16)

Taking the relationship (16) into consideration, one can write _n-2/£_n_, V £i(17)

Combining the series of the torques T (11) and of the ratio k (16), one obtains:

(1-1)(1-2)

T_i=T₁fc₁ⁱ-&gt;9 ²(18)

When i=n, the relationship (18) becomes: (n—l)(n—2) ²(19) from which (n—l)(n—2) i-------^?== Λ Tlfci” -¹(20)

Replacing q by the relationship (17) n-2/K]_ y hr y TJcf-i (21)

One finds that, taking the above mentioned options as a starting point, the numerical values of the specific center distances as well as of the characteristic torques, may be determined from the above stated relationships.

It will be observed too, that the choice of the options is important.

It is a function of the manufacturing program. As an example it will be observed that the applicant has, among other things, drawn up a complete program having as its object the integral prefabrication of a series of ten sizes of reducers of medium capacity, of which the characteristic torque of the smallest size is equal to 90 mkg (7,800 Ibs.in) and the characteristic torque of the largest size is equal to 7,000 mkg (680,000 Ibs.in).

In the series in question, as already stated, one has, from the start, established that 7, = 90 mkg (7,800 Ibs.in) and 7₁₀ = 7,000 mkg (680,000 Ibs.in).

Based on the data of experience, the relationship

7₂/7, = £i = 2 between the smallest torque 7, and the second torque 7₂ has been established.

On the other hand, taking into account the best conditions of adequacy, one has admitted that the characteristic torque T_n of the size n is approximatively equal to 1,3 times the characteristic torque of size n— 1, which gives the following relationship:

3,673,885

1·3

The ratio k_t, which is the relationship between two successive characteristic torques, thus varies — by application of what precedes — in a continuous and known way from one ex- j treme value to the other one.

By choosing the values A, = 2 and k„._t «· 13, the series of Λ is a decreasing series.

From relationship (21), in which T„ T„, and kn._t are known from the options taken, the value of n can be deduced, jq

Since, as a general rule, one will find a fractional value for n, one will choose, for the number of sizes, the nearest full number. By then making this calculation in reverse, the exact value of A».) will be established, and this will be adopted in lieu of the approximate value originally fixed.

Indeed, one obtains for n = 10.11

From this, one deduces that 10 sizes must be provided and one calculates the exact value of £„_! =1.316.

If, according to the example, the following values are admitted:

e=0.9 /=1.96 g=0.9 the relationships (8)(9) and (10) become respectively:

b = K_la^0S

T=K₂a^1Mb^as

T' = K₃a?·”

On determining the values of the constants and K₂, the values of h_t, k_t, p and q may be calculated from the relationships (3), (16), (13) and (20) respectively, whereas the series 30 of center distances a and the series of characteristic torques T are easily determined by means of the relationships (7) and (18) respectively.

The values of the characteristic torques T of the successive sizes thus form a geometric progression, whose ratio k itself is 35 variable.

two center distances, selected from the series of specific center distances, shown above.

When considering a speed reducer, the permissible capacities of the incorporated gear sets can be compared to one another and the most economical speed reducer will naturally be obtained, when the various characteristic capacities are as close as possible to one another and aim towards equality. Indeed, in this case, one avoids as much as possible, that one gear set should be overdimensioned in regard of the other one.

In terms of torque values, it may be said, that the most economical speed reducer will be obtained, when the value of the torque of the penultimate center distance is as close as possible to — but not less than — the value of the characteristic torque of the gear set of the specific center distance in 15 the given speed reducer, divided by the reduction ratio used in that center distance.

One will proceed in the same way for the other center distances. The reduction ratio of the last center distance plays a part in establishing the criterion of equilibrium or of internal 20 economy of the speed reducer, various ratios being possible.

On the other hand, by repeated use of the gears, the gears used in center distances, other than the specific center distance, will be chosen from the series of specific center distances. Since, in order to create an adequate range of speed 25 reducers, this series must follow a special law (an improper geometric progression), it is understandable that the “technique” of this choice must take into account also the law mentioned with a certain amount of flexibility.

If one calls “reduced torque” the torque at the pinion of the specific center distance, of which the characteristic torque is of course the torque at the wheel, one can write:

T_r= T/R_r in which

T_r is the above described reduced torque

Τ'the characteristic torque of the last center distance

R_T any chosen reduction ratio of the last center distance.

_________TABLE I__________________ ki k₂ ka kj k₅ k₅ kr k₈ k₈

2.. 1.898 1.801 1.709 1.622 1.539 1.461 1.3861.316

TABLE II tJ ϊξ T₈ T₄ T₈ T₈ Tt T₈ T, T1|&gt;

90........ 180 324 615 1,050 1,700 2,620 3,830 5,3207,000

7,800””Ξ 15,600 30,000 43,000 91,000 147,000 228,000 332,000 462,000 610,000

TABLE III ai as &amp;3 a&lt; as as a? as asaio

111 143 180 222 269 321 376 430 484635

Said improper geometric progression has, as a variable ratio, nine terms of which the first and the last are established beforehand. As already stated in connection with the characteristics of the center distances, the different values of the ratio of said improper geometric progression vary themselves according to a proper geometric progression whose constant ratio is expressed by the relationship:

_{g =} 36/ Tip '

V TiX2°

After making the necessary numerical substitutions, one obtains:

the series of variable ratios A (see table I) the series of characteristic torques T (in mkg) &amp; T (in Ibs.in) (see table II) the series of specific center distances a (in mm) (see table III)

These are theoretical values, which will be rounded off in the establishment of the manufacturing drawings.

As it was said before, each speed reducer in the series considered, comprises, besides the specific center distance, one or

Since this ratio may vary from R_r „_tn to R_r the following is obtained:

Tr max T/R_{r m}&lt;_R and

Tr min ⁼ T/R_r max

The choice of a center distance from the series of specific center distances must be done in such a way that its torque T' is as close as possible to — but not less than — the reduced torque T_rmax.

Thus Τ' » T_{r m}ax

It must be noted that the absolute value of the reduction ratio R_{r m}in plays a part in establishing the criterion of selection and that the relationship R_{r ma}x/Rr mtn is a measure of the disequilibrium in the capacities introduced in the speed reducer.

The fact of keeping the relationship R_{r max}/R_{r ml}„ to a minimum, therefore aims directly at maintaining the interior harmony or the interior economy of the reducers by avoiding excessive deviations from the starting-situation in question,

3,673,885 that situation being the result of a judicious equilibration of the ratio R_{r mln} and the available specific center distances.

The table of FIG. 11 illustrates the application of what precedes.

Although not essential, it is very desirable and required by an evoluted practice anyhow, that the same range (or “the only range”) of reduction ratios are found in all sizes of gear reducers. On the basis of these very considerations, it is also good practice to adopt, as numbers, values that are generally used, in other words, normalized numbers or “standard numbers.”

In the present case the so-called series R 20 is applied (see Recommendation No. 497 from the International Standardization Organization).

It will be noted that it is sufficient for the primary reduction ratios (i.e. those realized in a given center distance) to be part of the stated series R 20, so that also the overall reduction ratios (obtained by combining the primary ratios) are part of the same series.

Combining primary ratios in order to obtain overall ratios is however conditioned — in the present case — by the repeated use of the same gears in the series of speed reducers (a rationalization which diminishes the number of different gears, whence a “total” economy). This repeated use is regulated by the “law of selectivity of the specific center distances,” which is itself the result of a search for an equilibrium of capacities in every reducer (which aims at obtaining an internal or intrinsic economy).

In addition to the above conditions, the hoped-for result, (i.e. a range of ratios which is complete and is the same in all sizes) must be obtained by a judicious choice, for each center distance, of primary reduction ratios from the series of standard numbers adopted. Thus, it appears that, faced with this problem and having a series of standard numbers at one’s disposal, the most economical solution must be looked for.

This solution is based upon the fact that the differences between the order numbers of the reduction ratios for the same center distance must be different from those found in the adjoining center distance.

It will be remembered that the R 20 standard numbers form a geometric progression in which the ratio s is /---₈=γ10=1.12...

The terms of this progression which are taken into account, are obtained by taking a full and non negative power of the ratio s.

If more particularly one writes that the respective terms are obtained by s (n being the exponent and called “order number”), to each order number corresponds naturally a term in the series R 20 and consequently a reduction ratio.

The table on page 19 gives the relation: order number / reduction ratio, for the ratios from 1 to 100.

The concern for economy, which is at the base of the present considerations, can also be expressed by the notion of efficiency, understanding from this, that, for a given number of primary reduction ratios, one must obtain, by combination, a number of different values, that is as close as possible to the ideal number. In a speed reducer with two center distances, that number is obtained by multiplying the numbers of reduction ratios of both center distances mutually.

<td> n</td><td> R = s</td><td> n</td><td> R = S'</td>

<td> 0</td><td> 1.0</td><td> 20</td><td> 10.0</td>

<td> 1</td><td> 1.12</td><td> 21</td><td> 11.2</td>

<td> 2</td><td> 1.25</td><td> 22</td><td> 12.5</td>

<td> 3</td><td> 1.4</td><td> 23</td><td> 14</td>

<td> 4</td><td> 1.6</td><td> 24</td><td> 16</td>

<td> 5</td><td> 1.8</td><td> 25</td><td> 18</td>

<td> 6</td><td> 2.0</td><td> 26</td><td> 20</td>

<td> 7</td><td> 2.24</td><td> 27</td><td> 22.4</td>

<td> 8</td><td> 2.5</td><td> 28</td><td> 25</td>

<td> 9</td><td> 2.8</td><td> 29</td><td> 28</td>

<td> 10</td><td> 3.15</td><td> 30</td><td> 31.5</td>

<td> 11</td><td> 3.55</td><td> 31</td><td> 35.5</td>

<td> 12</td><td> 4.0</td><td> 32</td><td> 40</td>

<td> 13</td><td> 4.5</td><td> 33</td><td> 45</td>

<td> 14</td><td> 5.0</td><td> 34</td><td> 50</td>

<td> 15</td><td> 5.6</td><td> 35</td><td> 56</td>

<td> 16</td><td> 6.3</td><td> 36</td><td> 63</td>

<td> 17</td><td> 7.1</td><td> 37</td><td> 71</td>

<td> 18</td><td> 8.0</td><td> 38</td><td> 80</td>

<td> 19</td><td> 9.0</td><td> 39</td><td> 90</td>

<td></td><td></td><td> 40</td><td> 100</td>

If one takes the concrete case of the specific and the penultimate center distances of the series of speed reducers, object of the present invention, it will be remembered that, on striving for an equilibrium of the capacities in each of those reducers, and although each center distance has several (generally more than two) predetermined reduction ratios, only two ratios are used if this center distance is used as the specific center distance. This is important for the establishment of the practical rule for the optimal choice of the reduction ratios.

Finally, it must not be forgotten that one of the aims in the field of the reduction ratios is to avoid gaps for shortcomings in the overall series.

All preceding considerations lead to the formulation of a practical rule, according to which the choice of the reduction ratios of each center distance in the series of center distances considered, will be made by taking into account the selection of these center distances in such a way that the differences between the order numbers of successive reduction ratios, classified in increasing order, shown as Series 1(. ..3, 1,3,1, 3 . . . ) and Series II ( . . . 2, 2, 2, 2, 2, . . . ) belong to the specific center distance and the penultimate center distance of a speed reducer respectively or conversely.

On respecting this rule, the number of combinations of different reduction ratios obtained, equals the ideal or maximum number, which means thus — for the whole range — a minimum of different gear sets.

In the case of gear speed reducers, the standard numbers stated must be realized naturally as reduction ratios, i.e. as ratios of full numbers of teeth. This means that the above mentioned values will be obtained with a certain tolerance, which does not affect at all the fundamental value of the established rule.

Therefore, thanks to this new conception in the establishment of the characteristics of all and of each gear speed reducer of a freely determined series in particular, one places at the disposal of industry, whatever kind of industry it may be, speed reducers, which, thanks to a relatively small number of different parts, enable not only a maximum number of combinations, but also such combinations as to meet almost all required applications with middle-sized speed reducers, with a harmoniously spread range of reduction ratios, being amply dimensioned, and showing no unnecessary overdimensioning of the components in a reducer with respect to one another.

It has to be noted too, that the very particular special characteristics, revealed by the present invention, do not introduce any difficulties in the possibilities of diversification of a series of speed reducers. Indeed, the new considerations that determine the absolute values of the center distances as well as the gears, do not influence the outward shape of the housing, i.e. of all speed reducers in the series.

Evidently, considering the experience already acquired in the field of the gear speed reducer, it is estimated that a maximum efficiency will be obtained if the rules stated are applied to speed reducers with a prism shaped housing, which facilitates the manufacture and lends itself admirably to manifold possibilities of execution and also the possibilities of mounting.

By belonging to a series of speed reducers, which is itself subject to the laws of mutual dependence, the object of the invention is essentially tied to the notion itself of a series of gear speed reducers. However, since each speed reducer considered individually benefits, in intrinsic value, from the ad3,673,885 vantages of the series in question, the invention concerns equally each speed reducer in such a series, considered individually.