05/412469

01/14/1975

11/02/1973

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74/82

74/25, 74/89.210

F16H35/02; F16H35/00; F16H27/04

74/141.5,242.1,42,112,25,89.2,89.22,96,82

Ratliff Jr., Wesley S.

Barnes, Kisselle, Raisch & Choate

I claim

1. A multiple step reversible intermittent motion mechanism comprising:

2. A mechanism as defined in claim 1 in which said idler means are positioned on said oscillating member such that the line of said flexible drive means from either side of that said drive member within the said topological loop of said flexible drive means to the tangency point with the first engaged of said multiple idler means, and the line from said tangency point to the axis of pivot of said oscillating member, form an angle whose mean value during a given oscillation cycle of said oscillating member lies within the range of 70° to 110°.

3. A multiple step reversible intermittent motion mechanism comprising:

4. A mechanism as defined in claim 3 in which said idler means are positioned on said oscillating member such that the line of said flexible drive means from either side of said input member to the tangency point of contact with the first of said idler means, and the line from said tangency point to the axis of pivot of said oscillating member, form an angle whose mean value during a given oscillation cycle of said oscillating member lies within the range of 70° to 110°.

5. A mechanism as defined in claim 3 in which said idler means comprise two idler members, with one of said idler members rotatably mounted on one side of said oscillating member with respect to said first axis and the other of said idler members rotatably mounted on the other side of said oscillating member with respect to said first axis.

6. A mechanism as defined in claim 3 in which said idler means comprise multiple idler members, with one or more of said idler members rotatably mounted on one side of said oscillating member with respect to said first axis and one or more of said idler members rotatably mounted on the other side of said oscillating member with respect to said first axis.

7. A mechanism as defined in claim 3 in which said idler means comprise multiple idler members, with multiple said idler members rotatably mounted on one side of said oscillating member with respect to said first axis and multiple said idler members rotatably mounted on the other side of said oscillating member with respect to said first axis.

8. A mechanism as defined in claim 3 in which said means connecting said eccentric drive shaft with said oscillating member comprises a link member rotatably connected at one end to said eccentric shaft and rotatably connected at its other end to said oscillating member at a point spaced from said first axis.

9. A mechanism as defined in claim 3 in which said means connecting said eccentric drive shaft with said oscillating member comprises:

10. A mechanism as defined in claim 3 in which said means connecting said eccentric drive shaft with said oscillating member comprises:

11. A mechanism as defined in claim 3 in which said means connecting said eccentric drive shaft with said oscillating member comprises:

12. A multiple step reversible intermittent motion mechanism comprising:

13. A mechanism as defined in claim 12 in which said idler means are positioned on said oscillating member such that the line of said flexible drive means from either side of said output member to the tangency point of contact with the first of said idler means and the line from said tangency point to the axis of pivot of said oscillating member form an angle whose mean value during a given oscillation cycle of said oscillating member lies within a range of 70° to 110°.

14. A mechanism as defined in claim 12 in which said idler means comprise two idler members, with one of said idler members rotatably mounted on one side of said oscillating member with respect to said second axis and the other of said idler members rotatably mounted on the other side of said oscillating member with respect to said second axis.

15. A mechanism as defined in claim 12 in which said idler means comprise multiple idler members, with one or more of said idler members rotatably mounted on one side of said oscillating member with respect to said second axis and one or more of said idler members rotatably mounted on the other side of said oscillating member with respect to said second axis.

16. A mechanism as defined in claim 12 in which said means connecting said eccentric shaft with said oscillating member comprises:

17. A mechanism as defined in claim 12 in which said eccentric drive shaft with said oscillating member comprises:

18. A multiple step reversible intermittent motion mechanism comprising:

19. A mechanism as defined in claim 18 in which said means connecting said eccentric drive shaft with said oscillating member comprises:

20. A mechanism as defined in claim 18 in which said means connecting said eccentric drive shaft with said oscillating member comprise:

This invention relates to a cyclicly repetitve motion generating system.

Various motion devices which utilize complicated cam grooves and followers can accomplish predetermined acceleration, deceleration and dwell and even reversing characteristics; but the cam configurations are very expensive to build and they admit of no flexibility or adjustment short of changing or substituting any particular cam section.

It is an object of the present invention to provide a motion generating system that can be constructed of relatively simple parts in the form of gears, levers, and chains which can generate repetitive cycles which can rather easily be altered in any portion without expensive machining.

Another object is to provide such a mechanism in which second and third harmonics may be readily added to increase the versatility of the mechanism.

It is a further object of this invention to provide a mechanism for superimposing a reversing motion on a substantially constant motion.

It is another object of this invention to provide a mechanism capable of generating repetitive index cycles consisting of a short dwell, smooth acceleration to a maximum velocity and smooth deceleration to a following short dwell.

A still further object of this invention is to provide a mechanism capable of generating a wide variety of repetitive motion patterns or profiles.

Other objects and features of this invention relating to the details of construction and operation will be apparent in the following description and claims in which the principles of operation are set forth together with the best modes presently contemplated for the practice of the invention.

Drawings accompany this disclosure and the various views thereof are briefly described as:

FIG. 1, a side view of a mechanism for practicing the invention.

FIG. 2, an end view of the mechanism illustrated in FIG. 1 taken on line 1--1.

FIG. 3, a view taken on line 3--3 of FIG. 1.

FIG. 4, a view of the power input taken on line 4--4 of FIG. 1.

FIGS. 5 to 7, diagrammatic views of the mechanism of FIG. 1 illustrating the motion.

FIG. 8, a modified mechanism with the addition of a second harmonic.

FIGS. 9, 10, 11 illustrate in graph form, respectively, the relative displacement, velocity and acceleration which can be obtained by the previously illustrated mechanisms.

FIG. 12, a schematic drawing of the mechanism illustrated in FIG. 8.

FIGS. 13, 14 and 15 illustrate, respectively, in graph form the displacement, velocity, and acceleration utilizing a third harmonic.

FIG. 16, a modified mechanism illustrating the addition of two distinct and higher harmonics.

FIGS. 17, 18, 19 illustrate, respectively, in graph form, the displacement, velocity and acceleration of a mechanism utilizing multiple harmonics.

FIG. 20 illustrates an embodiment in which the idler arm pivots about the input axis.

FIG. 21 illustrates a mechanism similar to that shown in FIG. 20 with a single higher harmonic added.

FIG. 22, a mechanism for increasing the angle of chain wrap for high torque applications.

FIG. 23, a modified arrangement for introducing a second or third harmonic into the output motion.

Referring to the drawings FIGS. 1, 2, 3 and 4, a frame 2 supports an output shaft 4 through bearings 6 and 8. It will be understood that this output shaft in turn drives any one of a variety of mechanisms, such as an indexing elevator, an indexing conveyor or other type of mechanism requiring an intermittent indexing rotary drive.

The frame 2 also supports a gear reducer 10 (FIG. 4) driven by a suitable motor 12 through belt 14 and pulleys 16 and 18. The output shaft 20 of the gear reducer 10 acts as the input shaft of this mechanism. It will be understood that the shaft 20 may be driven by any one of many other types of input systems, such as a direct drive from high torque prime movers such as air or hydraulic motors, or from some related and interconnecting mechanism. The shaft 20 has mounted on it an input sprocket 22, which is the input member of the mechanism.

An idler arm 24 is mounted to the output shaft 4 through bearings 26 and 28. An idler sprocket 30 is mounted to one end of the idler arm 24 through shaft 32, bearings 34 and 36 and retainer nuts 38 and 40. A second idler sprocket 42 is mounted to the other end of the idler arm 24 through shaft 44, bearings 46 and 48 and retainer 50.

An oscillating link 52 is connected at one end to the idler arm 24 through shaft 44, bearing 54 and retainer 56. At its other end the link 52 is driven by an eccentric shaft 58, bearing 60 and retainer 62 (FIG. 3) mounted on the input sprocket 22. The shaft 58 revolves on an axis which is eccentric to the axis of the shaft 20.

A flexible drive chain 64 interconnects the input sprocket 22 and an output sprocket 66 mounted on the output shaft 4. The chain passes over the idler sprockets 30 and 42 in completing its driving loop.

If it is hypothetically assumed that the idler arm 24 is maintained in a stationary position (as by disconnecting the link 52 from the shaft 54), it can be seen that the input sprocket 22 will drive the output sprocket 66 through a ratio determined by the numbers of teeth on the respective sprockets, and, further, that the speed ratio will be constant.

If it is assumed that the input sprocket 22 is stationary, while the idler arm 24 is moved through some small angle, the output sprocket 66 will rotate through some proportional angle.

Therefore, it can qualitatively be seen that if the idler arm 24 is connected to the eccentric shaft 58 on the input sprocket 22 through the link 52, that the motion of the output sprocket 66 is the superposition or summation of motions caused by the rotation of the input sprocket and the oscillation of the idler arm. Indeed, if the proportions of the system are properly chosen, the output sprocket can be made to stop momentarily or to "dwell" once during each complete rotation of the input sprocket.

To construct a viable system based on this concept, it is necessary that the chain length remain substantially unaltered over the useful range of angular positions of the idler arm. How this is accomplished is represented schematically in FIG. 5. The output sprocket 66 rotates about an axis A ₁ and the input sprocket 22 rotates about an axis A ₂ ; the idler arm 24 also rotates about axis A ₁ ; and the idler sprockets 30 and 42 rotate about axes A ₃ and A ₄ on the idler arm 24. The line of centers from A ₁ to A ₃ is defined as C ₁ and the line of centers of A ₁ to A ₄ is defined as C ₂ . If the angle α, between C ₁ and C ₂ is chosen such that, at the approximate center of the oscillation of idler arm 24, the angle γ between C ₁ and the chain centerline C ₃ is 90°, and the angle β between C ₂ and the chain centerline C ₄ is also 90°, then the total developed chain length will remain substantially constant even through the idler arm 24 is rocked about an angle of ± 20° about axis A ₁ . The formula for the total developed chain length may be developed using straightforward trigonometric relationships but becomes extremely long and cumbersome. It has been evaluated with the aid of a computer and the total developed length of the chain found to vary less than 1 part in 10,000 for a total oscillation angle of ± 20°. This is negligibly small for all practical purposes. It should be noted that the pitch diameters of the idler sprockets 30 and 42 need not be identical, nor is it necessary that the distance from A ₁ to A ₃ be the same as the distance from A ₁ to A ₄ .

The amplitude and character of oscillation of the idler arm 24 determines the dynamic characteristics of the output sprocket 66. The quantitative analysis refers to FIGS. 6 and 7 which are the schematic kinematic drawings of the system. The variables are defined as follows:

θ = rotation of idler arm 24 about axis A ₁

Ψ ₁ = rotation of output sprocket 66 about axis A ₁ caused by rotation of idler arm 24

Ψ ₂ = rotation of output sprocket 66 about axis A ₁ caused by rotation of input sprocket 22

Ψ = total rotation of output sprocket 66

φ = rotation of input sprocket 22

R ₁ = pitch radius of output sprocket

R ₂ = pitch radius of input sprocket

R ₃ = pitch radius of the arc which is the envelope of the pitch circle of the idler sprocket during the oscillation of the idler arm 24

L ₁ = distance from axis A ₁ to pivot connection of link 52 on arm 24

L ₂ = eccentric distance from axis A ₂ to pivot connection of link 52 on input sprocket 22

Referring to FIG. 6 which is a schematic drawing showing the rotation of the output sprocket 66 due to a rotation of the idler arm 24, assuming the input sprocket 22 is stationary, it can be seen that the total rotation of the output sprocket 66 is caused by two effects. If the idler arm 24 rotated through an angle θ, the output sprocket rotates through that angle θ also, assuming there is no relative movement of the chain between idler sprocket 42 and output sprocket 66. However, there is a movement of the chain shown as S ₁ which in radian measure is

S ₁ = R ₃ θ

The chain length S ₁ subtends an angle on output sprocket 66 equal to S ₁ /R ₁

Therefore, the total angle of rotation of the output sprocket 66 is

Ψ ₁ = θ + (S ₁ /R ₁ )

Ψ ₁ = θ + (R ₃ /R ₁ ) θ (1)

where θ is the rotation of the idler arm 24.

If we now consider the idler arm 24 stationary, it is easily seen that the rotation of the output sprocket 66 due to a rotation of the input sprocket 22 is

Ψ ₂ = (R ₂ /R ₁ ) θ (2)

therefore, the total rotation of the output sprocket 66 is

Ψ = Ψ ₁ + Ψ ₂

Ψ = [(R ₃ + R ₁ )/R ₁ ] θ + (R ₂ /R ₁ ) θ (3)

fig. 7 is a schematic drawing relating the rotation, θ, of the idler arm 24 due to a rotation θ of the input sprocket 22. The chain and sprocket pitch diameters are omitted for clarity.

It can be seen that if the input sprocket 22 is rotated through an angle φ about axis A ₂ , the link 52 is moved through a distance S ₂ where

S ₂ ≅ L ₂ Sin φ

In radian measure

- θ = S ₂ /L ₁

Therefore,

θ ≅ - (L ₂ sin φ/L ₁ ) (4)

substituting equation (4) in the general relationship established by equation (3), we obtain

Ψ ≅ (R ₂ /R ₁ ) φ - (L ₂ /L ₁ ) (R ₃ + R ₁ /R ₁ ) sin φ

Ψ ≅ (R ₂ /R ₁ ) φ (L ₂ /L ₁ ) (R ₂ /R ₁ ) (R ₃ + R ₁ /R ₂ ) sin φ

Ψ ≅ (R ₂ /R ₁ ) [φ - (L ₂ /L ₁ ) (R ₃ + R ₁ /R ₂ ) sin φ]

If it is desired to obtain a cycloidal output motion characteristic in which the output sprocket 66 has a momentary dwell once for each revolution of the input sprocket 22, the coefficient of the sin φ term must be made equal to 1, i.e.,

(L ₂ /L ₁ ) (R ₃ + R ₁ /R ₂ ) = 1

Or

l ₂ = (l ₁ r ₂ )/(r ₃ + r ₁ ) (5)

then equation (4) becomes

Ψ = (R ₂ /R ₁ ) (φ - sin φ) (6)

which is the classical equation for cycloidal motion.

The relationship between the idler arm angle θ and the input angle φ need not be the simple harmonic established by the crank L ₂ as represented by equation (4). It is possible to add a second harmonic to the output motion using the technique shown schematically in FIG. 8.

Referring to FIG. 8, the input sprocket 22, output sprocket 66, idler arm 24, idler sprockets 30 and 42, chain 64 and all their associated bearings, shafts and retainers remain the same as shown in FIGS. 1, 2, 3 and 4. However, the technique for oscillating the idler arm 24 is modified through the addition of a second harmonic or Fourier component.

A secondary drive sprocket 70 is mounted parallel to and concentric with the input sprocket 22 on the gear reducer output shaft 20 and both sprockets rotate with the shaft. An intermediate sprocket 72 is mounted on a suitable shaft and bearings from the frame 2; this intermediate sprocket 72 is driven by sprocket 70 through chain 74. If there are twice as many teeth on sprocket 70 as on sprocket 72, then sprocket 72 will rotate at twice the angular velocity of sprocket 70. An idler link 76 is pivotally connected to the sprocket 72 through an eccentric pivot 78. The other end of this link 76 is pivotally connected to a primary link 80 through a shaft 82; and the other end of the link 80 is driven through an eccentric shaft 84 on the input sprocket 22.

It can be seen, therefore, that the point 82 on the link 76 makes one vertical oscillation for each revolution of the sprocket 22, and that the point 78 on the link 76 makes two vertical oscillations for each revolution of the sprocket 22.

A drive link 86 is connected at one end to the idler arm 24 through the shaft 44 and at its other end to the link 76 through a pivot shaft 88. It can be seen, therefore, that the idler arm 24 will make one complete oscillation for each rotation of the input sprocket 22, to which is added a component of second harmonic whose magnitude is dependent upon the ratio of the distances along link 76 between points 78, 88 and 82, and the relative eccentricities of shafts 78 and 84 on their respective sprockets.

The effect of the addition of this second harmonic component may best be illustrated through several examples. In the general case, through an extrapolation of formula 6, it may be hypothesized that the equation of motion for the output movement would become:

Ψ (R ₁ /R ₂ ) = φ + F ₁ sin φ + F ₂ sin 2 φ

Note: If F ₁ = -1 and F ₂ = 0, we obtain equation (6).

For simplicity, Ψ (R ₁ /R ₂ ) is defined as relative output motion, U, and therefore:

U = φ + F ₁ Sin φ + F ₂ sin 2 φ (7)

By successive differentiation

dU/dφ = 1 + F ₁ Cos φ + 2 F ₂ Cos 2 φ (Velocity) (8)

d ² U/dφ ² = -F ₁ Sin φ -4 F ₂ sin 2 φ (Acceleration) (9)

d ³ U/dφ ³ = -F ₁ Cos φ -8 F ₂ cos 2 φ (10)

d ⁴ U/dφ ⁴ = F Sin φ + 16 F ₂ Sin 2 φ (11)

These relationships may now be utilized to establish the values of F ₁ and F ₂ that best meet the design requirements, and, subsequently, these values of F ₁ and F ₂ will be converted into usable geometric parameters.

EXAMPLE I

It is desired to design the mechanism to achieve a long dwell. Mathematical "dwell" consists of a single or, at best, a group of points; practical dwell, considering lost motions, bearing clearances, and the demands of the application may be a band whose width will vary from application to application. In a typical case, the dwell may be considered that portion of the cycle in which the output motion does not exceed one one-thousandth of the total output movement of an index cycle.

With this in mind, we examine the behavior of equation (7), (8), (9), (10), and (11). It will be noted that when φ = 0, then

U = d ² U/dφ ² = d ⁴ U/dφ ⁴ = 0

for all values of F ₁ and F ₂ .

To achieve the maximum dwell, it is clear that the velocity, dU/dφ, should be set equal to 0; therefore, from equation (8)

1 + F ₁ + 2 F ₂ = 0 φ = 0 (12)

noting that d ³ U/dφ ³ is the slope of the acceleration, we knowledgeably, but arbitrarily, may assign a value to equation (10) equal to -M which indicates the slope of the acceleration is made slightly negative at φ = 0. Therefore, the acceleration "starts out" from φ = 0 and becomes slightly negative before starting its positive rise.

From equation (10), we obtain

-F ₁ -8 F ₂ = -M (13)

solving equations (12) and (13) for the values of F ₁ and F ₂ , the following are obtained:

F ₁ = -1 - 1/3 (M + 1)

F ₂ = (M + 1)/6

Then, for several values of M, the final values of F ₁ and F ₂ are as follows:

Curve in M F ₁ F ₂ FIG. 9, 10, 11 ______________________________________ 0 -1.3333 .16667 A ₁ .1 -1.3667 .18333 A ₂ .2 -1.400 .20000 A ₃ ______________________________________

When these values of F ₁ and F ₂ are utilized in equation (7), and the values of U calculated for various values of φ, the curves A ₁ , A ₂ and A ₃ in FIG. 9 are obtained. It will be noted that the displacement axis is generally magnified and further that the output displacement is relative to a total cyclic displacement of 2π. Considering curve A ₂ , it will be noted that the output movement is negative for the first 31° of input movement, reaching a value of -0.001 at that point, and then not reaching +0.001 until 42° of input rotation.

Since these curves are symmetrical about the origin, it may be seen that the output "dwells" for 84° (2 × 42°) of input rotation if the dwell is defined as a band whose width is allowed to be ± 0.001 × 1/2π × output angle per index.

It will be further noted that if the definition of dwell width is widened and M made appropriately larger, the dwell is lengthened in terms of input angle. From curve A ₃ where M = 0.2, the total negative movement is -0.0048 at 42° and crosses +0.0048 at 58°, and the total dwell becomes 116° of input movement if the dwell is a band of output movement equal to ± 0.0048 × 1/2π × output angle per index.

The corresponding velocity and acceleration for an entire half cycle are shown by curve A ₂ in FIGS. 10 and 11.

It will be understood that the velocity curves are symmetrical about the line where φ = 180° and that the acceleration curves are symmetrical about the point U = 0, φ = 180°, i.e., the velocity curves remain positive between φ = 180° and 360°, while the acceleration becomes negative or becomes a deceleration.

The velocity and accceleration scales are relative to an index of 2π output units per unit of time.

It will be noted that only curve A ₂ is presented in FIGS. 10 and 11 since curves A ₁ and A ₃ would be extremely close to it on the scale used.

EXAMPLE II

If it should be desired to make the output velocity as flat as possible near the center of the index stroke, while still retaining a dwell of some reasonable value, a different set of conditions must be placed on equations (7), (8), (9), (10) and (11), as follows:

In order to achieve a dwell at φ = 0, dU/dφ = 0 or to "widen" that dwell dU/dφ = - V. Therefore,

1 + F ₁ + F ₂ = -V

Defining

V + 1 = K

F ₁ + 2 F ₂ = -K (14)

in order to achieve a flat velocity at φ = 180°, d ² U/dφ ² = 0, however d ² U/dφ ² = 0 at 180° for all values of F ₁ and F ₂ . Therefore, the next condition may be employed, which is:

d ³ U/dφφ ³ = 0 φ = 180°

Substituting in equation (10), the following is obtained.

F ₁ -8 F ₂ = 0 (15)

solving equations (14) and (15):

F ₁ = -0.8K

F ₂ = -0.1K

Then for several values of K the final values of F ₁ and F ₂ are as follows:

Curve in K F ₁ F ₂ FIGS. 9, 10, 11 ______________________________________ 1.00 -.8 -.1 B ₁ 1.01 -.808 -.101 B ₂ 1.02 -.816 -.102 B ₃ ______________________________________

When these values of F ₁ and F ₂ are utilized in equation (7), and the values of U calculated for various values of φ, the curves B ₁ , B ₂ , and B ₃ in FIG. 9 are obtained. Here again, it is obvious that as the permissable dwell band width is increased, the total length of dwell in terms of input angles is increased.

Curve B ₁ in FIG. 10 represents the relative velocity of the output for a standard output index of 2π, obtained by substituting the value of F ₁ = -0.8 and F ₂ = -0.1 in equation (8). The curves B ₂ and B ₃ are omitted in both FIGS. 9 and 10 because of their near coincidence with curve B ₁ . It should be noted that the velocity reaches a maximum of 1.6 and is nearly constant over a 70° span of input movement from φ = 145° to 100 = 215°.

Curve B ₁ in FIG. 11 represents the relative acceleration for these values of F ₁ and F ₂ when inserted in equation (10).

EXAMPLE III

If it is desired that the maximum relative velocity, wherever it should be reached during the index cycle, be as low as possible, a more subtle technique is required. It will also still be desired to achieve the maximum practical dwell. Therefore, as before from equation (8):

(14) F ₁ + 2F ₂ = - K

Then set

F ₁ = N F ₂ where N is an unknown parameter, whereupon:

F ₁ = -NK/(N+2) (16)

f ₂ = -k/(n+2) (17)

the point of maximum velocity, dU/dφ wherever it is reached in the cycle occurs at a point where d ² U/dφ ² = 0. Substituting equations (16) and and (17) into equation (N ) and setting it equal to 0.

NK/(N+2) Sin φ + 4K/(N+2) Sin 2 φ = 0 (18)

which may be solved to yield

Cos φ = - N/8 (19)

this establishes a relationship between N and φ at any point where the velocity is at an intermediate peak. Equation (18) may then be substituted back into equation (8) to yield a value for maximum velocity at such a point in terms of N only, which is:

dU/dφ = 1 -[NK/(N+2)] (-N/8) - [2K/(N+2)] (2 (N ² /64) -1)

which reduces to

dU/dφ = 1 + K [(N ² + 32)/16 (N+2)] (20)

to find that specific point whwere dU/dφ is a minimum with respect to N, equation (20) is differentiated with respect to N and the result set equal to 0, which when solved, yields

N = 4

whereupon

F ₁ = -0.66667K

F ₂ = -0.16667K

Then for several values of K, the final values of F ₁ and F ₂ are as follows:

Curves in K F ₁ F ₂ FIGS. 9, 10, 11 ______________________________________ 1.00 -.66667 -.166667 C ₁ 1.01 -.67333 -.16833 C ₂ 1.02 -.68 -.17 C ₃ ______________________________________

When these values are utilized in equation (7), and the values of U calculated for various values of φ, the curves C ₁ , C ₂ , and C ₃ are obtained. It can be seen that for identical values of K, the dwell characteristics are slightly inferior to the B curves for flatest midstroke velocity.

Curve C ₁ (curves C ₂ and C ₃ are again omitted because of near coincidence) in FIGS. 10 and 11 show the behavior of the relative velocity and acceleration throughout the cycle.

It will be noted that the peak relative velocity reaches a maximum value of only 1.5 at 120° then drops to an intermediate minimum of 1.33 at 180°, as compared to a constant 1.6 for curve B ₁ .

EXAMPLE IV

In this case, the objective is to find those values of F ₁ and F ₂ which achieve the lowest peak acceleration wherever it may occur during a cycle while still maintaining a maximum practical dwell. Therefore as before from equation (8)

(14) F ₁ + 2 F ₂ = -K

And again setting

F ₁ = N F ₂ (N is now a new unknown)

Then

(16) F ₁ = -NK/(N+2)

(17) F ₂ = -K/(N+2)

The point of maximum acceleration, d ² U/dφ ² , wherever it occurs in the cycle must occur where d ³ U/dφ ³ = 0. Therefore, equations (16 ) and (17) are substituted into equation (10) which is then set equal to 0.

NK/(N+2) Cos φ + 8K/N+2 Cos 2 φ = 0 (21)

which reduces to:

N = (8 - 16 Cos ² φ)/Cos φ (22)

This establishes a relationship between N and φ at any point where the acceleration is at an intermediate peak. Equations (22), (16) and (17) may then be substituted into equation (9) to yield a value for the maximum velocity at such a point in terms of φ only which reduces to:

d ² V/dφ ² = K [Sin ³ φ/(1 - 2 Cos ² φ + Cosφ/4)] (23)

This may now be differentiated with respect to φ and the result set equal to 0, which in turn reduces to:

-2 Cos ³ φ + 1/2 Cos ² φ - Cos φ + 1/2 = 0

The only real root of this equation is:

Cos φ = 1/4 (24)

When this solution is substituted back into equation (22)

N = 28

And, therefore, from (16) and (17)

F ₁ = - 14/15 K

F ₂ = - K/30

Then for several values of K the final values of F ₁ and F ₂ are as follows:

Curves in K F ₁ F ₂ FIGS. 9, 10, 11 ______________________________________ 1.00 -.93333 -.0333 D ₁ 1.01 -.94267 -.0336 D ₂ 1.02 -.952 -.034 D ₃ ______________________________________

As before, these values of F ₁ and F ₂ are utilized in equations (7), (8) and (9) to calculate the curves D ₁ , D ₂ , and D ₃ in FIGS. 9, 10 and 11. As expected, the value of the peak relative acceleration is lower than that found in the other curves satisfying other conditions.

The foregoing examples are illustrative only to exemplify several pure mathematical techniques for evaluating the coefficients F ₁ and F ₂ to obtain the desired output conditions. Other conditions may be imposed on the output characteristics and other techniques may be employed with equal validity.

It will now be shown how these pure coefficients F ₁ and F ₂ may be converted into useful dimensions for the additive system shown in FIG. 8. FIG. 12 represents the schematic drawing of the linkage and eccentricities which create the harmonic superposition and establish the relationship between the rotation, θ, of the idler arm 24 due to the rotation φ of the input sprocket 22 and the rotation 2φ of the sprocket 94. The chain and sprocket pitch diameters are omitted for clarity.

It can be seen that if the input sprocket 22 is rotated through an angle φ about axis A ₂ the link 80 is moved through a distance L ₂ Sin φ. If it is temporarily assumed that pivot 78 is stationary, the resultant movement of link 86 is

[L ₄ /(L ₄ + L ₅ )] L ₂ Sin φ.

Similarly, it is temporarily assumed that the pivot 82 is momentarily stationary while the sprocket 72 rotates through an angle 2 φ about axis A ₃ the resultant movement of link 86 is

[L ₅ /(L ₄ + L ₅ )] L ₃ Sin 2 φ.

By superposition, therefore, the total movement of link 86, as represented by the distance S ₂ is

S ₂ ≅ [L ₄ /(L ₄ + L ₅ )] L ₂ Sin φ + [L ₅ /(L ₄ + L ₅ )] L ₃ Sin 2 φ

This is an approximation which is nearly perfect if the links 80 and 86 are substantially perpendicular to the link 76.

As Before:

(4) - θ = S ₂ /L ₁

Therefore

θ = - [L ₄ /(L ₄ + L ₅ ) (L ₂ /L ₁ )] Sin φ - [L ₅ /(L ₄ + L ₅ ) (L ₃ /L ₁ )] Sin 2 φ (25)

Substituting equation (25) in the general relationship established by equation (3) and simplifying, the following is obtained ##SPC1##

Therefore

F ₁ = - [(R ₃ + R ₁ )/R ₂ ] (L ₂ /L ₁ ) [L ₄ /(L ₄ + L ₅ )] (26)

f ₂ = - [(r ₃ + r ₁ )/r ₂ ] (l ₃ /l ₁ ) [l ₅ /(l ₄ + l ₅ )] (27)

in a practical application, R ₁ , R ₂ , R ₃ , L ₁ , L ₄ , and L ₅ will generally be chosen for mechanical convenience; the eccentricities L ₂ , and L ₃ may then be calculated from these restatements of (26) and (27).

L ₂ = - L ₁ [R ₂ /(R ₃ + R ₁ )] [(L ₄ + L ₅ )/L ₄ ] F ₁ (28)

l ₃ = - l ₁ [r ₂ /(r ₃ + r ₁ )] [(l ₄ + l ₅ )/l ₅ ] f ₂ (29)

in all four mathematical examples, the value for F ₁ was always minus, making L ₂ positive, so that the direction of L ₂ from axis A ₂ is as shown in FIG. 12. In the case of L ₃ , if it is positive, its direction from A ₃ is as shown in FIG. 12; if it is negative, its direction is 180° opposite from its indicated position in FIG. 12.

While not shown in the examples, a phase shift may be added to the second harmonic component to meet those situations where non symmetry of output movement is desired.

Referring again to FIG. 8, if the number of teeth on the sprocket 70 is made three times as great as the numer of teeth on the sprocket 72, the sprocket 72 will rotate at three times the angular velocity of sprocket 70. The motion of the idler arm 24 is now a sum of the fundamental and some component of third harmonic whose magnitude is dependent upon the ratio of the distance along link 76 between points 78, 88 and 82 and the relative eccentricities of shafts 78 and 84 on their respective sprockets, as was the case with the second harmonic addition.

The effect of the addition of this third harmonic component may again best be illustrated through several examples. The equation for the output motion of the system may now be hypothesized as follows:

U = φ + F ₁ Sin φ + F ₃ Sin 3 φ (Displacement) (30)

By successive differentiation:

dU/dφ= 1 + F ₁ Cos φ + 3 F ₃ Cos 3 φ (Velocity) (31)

d ² U/dφ ² = - F ₁ Sin φ - 9 F ₃ Sin 3 φ (Acceleration) (32)

d ³ U/dφ ³ = - F ₁ Cos φ - 27 F ₃ Cos 3 φ (33)

d ⁴ U/dφ ⁴ = F ₁ Sin φ + 81 F ₃ Sin 3 φ (34)

These relationships may be utilized to establish values of F ₁ and F ₃ that best suit specific design requirements. Several illustrative example cases will be outlined.

EXAMPLE V

It is desired to design the mechanism to achieve a long dwell using the third harmonic (rather than the second as in Example I).

By a process completely identical with that used in Example I, it is found that

F ₁ = - 1 - 1/8 (M+1)

F ₂ = M+1/24

Then for several values of M, the final values of F ₁ and F ₃ are as follows:

Curves in M F ₁ F ₂ FIGS. 13, 14, 15 ______________________________________ 0 -1.125 .04167 E ₂ .2 -1.150 .05 E ₂ .4 -1.175 .05833 E ₃ ______________________________________

When these values are utilized in equation (30), and the value of U calculated for various values of φ, the curves E ₁ , E ₂ , and E ₃ in FIG. 13 are obtained. It can be seen that the dwell characteristics so achieved are slightly inferior to those achieved with the second harmonic addition as shown by curves A ₁ , A ₂ and A ₃ in FIG. 9.

Only the curve E ₂ is plotted in FIG. 14 from equation (31) representing the relative velocity using these values of F ₁ and F ₃ , because of near identity of curves E ₁ and E ₃ ; similarly, curve E ₂ is plotted alone in FIG. 15, from equation (32) representing the relative acceleration using these values of F ₁ and F ₂ . A significant reduction in the values of peak velocity and peak acceleration will be noted in comparison to the curves A ₂ in FIGS. 10 and 11 which represent the velocity and acceleration for maximum dwell conditions using the second harmonic addition. Therefore, using the third harmonic instead of the second to increase the dwell is a compromise that can be chosen only in the light of a specific application's requirements.

EXAMPLE VI

In this case, it is desired that acceleration be as flat as possible at an input angle of 90° and that the practical dwell again be as large as possible. It is obvious from the form of equation (32) that the relative acceleration must be symmetrical about the line φ = 90°; it is further obvious that the slope of the acceleration, as expressed by equation (33), is always 0 at φ = 90° regardless of the values of F ₁ and F ₃ . The curvature of the acceleration is proportional to d ⁴ U/dφ ⁴ ; therefore, by setting d ⁴ U/dφ ⁴ = 0 at φ = 90° the optimum flatness of the acceleration is achieved; therefore, from equation (32)

F ₁ -81 F ₃ = 0

By again setting the slope slightly negative at φ = 0, from equation (31)

F ₁ +3 F ₃ = -K

Therefore

F ₁ = - 27/28 K

F ₃ = - 1/84 K

Then for several values of K, the final values of F ₁ and F ₃ are as follows:

Curves in K F ₁ F ₃ FIGS. 13, 14, 15 ______________________________________ 1.00 -.96429 -.011905 G ₁ 1.01 -.97393 -.012024 G ₂ 1.02 -.98357 -.012143 G ₃ ______________________________________

When these values are utilized in equations (30), (31) and (32) and the output displacement, velocity, and acceleration calculated the curves G ₁ , G ₂ and G ₃ in FIGS. 13, 14 and 15 are obtained. It will be noted that in FIG. 15, the acceleration curve G ₁ is substantially flat from φ = 60° to φ = 120° at a value of 0.85, meeting the set-up boundary condition.

EXAMPLE VII

In this case, it is desired that the peak acceleration be as low as possible wherever it occurs in a cycle and that the practical dwell again be as large as possible. This situation occurs when an intermediate peak is reached at some input angle less than 90°, and at some symmetrical angle beyond 90° (due to the aforementioned acceleration symmetry about 90°). It is, therefore, hypothesized that:

F ₁ = N F ₂ (where N is again a new unknown) (35)

It is also known that wherever the acceleration is a maximum that d ³ U/dφ ³ = 0. Therefore, by substituting (35) into equation (33), setting equation (33) equal to 0 and clearing, the following is obtained:

Sin ² θ = 27=N/108 (36)

this now establishes a relationship between N and θ at any point of maximum acceleration. As before, to achieve a slight negative velocity at φ = 0

F ₁ + 3 F ₃ = -K

By combining with equation (35), the following are obtained:

F ₁ = -NK/N+3 (37)

f ₃ = -k/n+3 (38)

equations (36), (37) and (38) are substituted into equation (32), to establish a relationship between N and K and d ² U/dφ ² at all points where d ² U/dφ ² is a maximum. In other words, this establishes (d ² U/dφ ² ) _max = f (N,K). After clearing, this is found to be

(d ² U/dφ ² ) _max = 2/3 [K/(108) ¹ /2 ] [(N+27) ³ /2/ N+3] (39)

in order to find at what specific value of N, (dU/dφ ² ) _max has is least value, equation (39) is differentiated with respect to N, and the result set equal to 0.

After clearing

N = 45

When this is substituted back into equations (37) and (38)

F ₁ = - 45/48 K

F ₃ = - 1/48 K

Then for several values of K, the final values of F ₁ and F ₃ are as follows:

Curves in K F ₁ F ₂ FIGS. 13, 14, 15 ______________________________________ 1.00 -.9375 -.020833 H ₁ 1.01 -.9469 -.02104 H ₂ 1.02 -.95625 -.02125 H ₃ ______________________________________

When these values are utilized in equations (30), (31) and (32), and the output displacement, velocity, and acceleration calculated, the curves H ₁ , H ₂ and H ₃ are obtained. In FIG. 15 it will be noted that the acceleration curve H ₁ reaches a peak of 0.81 at φ = 50° and φ = 130°, which is some 5 percent less than the more flattened top of G ₁ .

Examples V, VI and VII are again intended as illustrative only. Whether the value of the factors F ₁ and F ₃ are obtained by these methods or by other techniques to meet other requirements, they may again be converted to usable gometric parameters for the linkage using the methods outlined above in connection with the first and second harmonic system.

While the mechanism illustrated in FIG. 8 is capable of superimposing either a second, third or any other single higher harmonic to the output, additional adder links may be added so that multiple higher harmonics may be incorporated into the output characteristics. The schematic mechanism for adding 2 distinct and separate higher harmonic is shown in FIG. 16.

Referring to FIG. 16, the input sprocket 22, output sprocket 66, idler arm 24, idler sprockets 30 and 42, chain 64 and all associated bearings, shafts and retainers again remain the same as shown in FIGS. 1, 2, 3 and 4, except that the center distance from the output sprocket 66 to the input sprocket 22 may be increased slightly to provide space for the addition of multiple additional sprockets, and as will be shown, the technique for oscillating the idler arm 24 is again modified.

A secondary drive sprocket 90 and a tertiary drive sprocket 92 are mounted parallel to and concentric with the input sprocket 22 on and to rotate with the gear reducer output shaft 20. A first intermediate sprocket 94 and a second intermediate sprocket 96 are each mounted to the frame 2 through suitable shafts and bearings. The sprocket 94 is driven by sprocket 90 through chain 98, and sprocket 96 is driven by sprocket 92 through chain 100. As shown in FIG. 16, sprocket 96 rotates at three times the angular velocity of the input sprocket 22, and sprocket 94 rotates at twice the angular velocity of the input sprocket 22.

A first adder link 102 is pivotally connected to the sprocket 94 through an eccentric pivot 104. The other end of the link 102 is pivotally connected to a primary link 106 through a shaft 108; the other end of the link 106 is driven through an eccentric shaft 110 on the input sprocket 22.

A second adder link 112 is pivotally connected to the sprocket 96 through an eccentric pivot 114. The other end of the link 112 is pivotally connected to a secondary link 116 through a shaft 118; the other end of the link 116 is driven through a pivot connection 120 on the link 102.

A drive link 122 is connected at one end to the idler arm 24 through a shaft 44 and at its other end to the link 112 through a pivot 124.

It can be seen, therefore, that for each revolution of the input sprocket 22, the idler arm 24 will make one complete oscillation, which is the summation of a fundamental component dependent on the eccentricity of the pivot 110 on the input sprocket 22, a second harmonic component dependent on the eccentricity of the pivot 104 on the sprocket 94 and the ratio of the distances between points 104, 120 and 108 on the link 102, and a third harmonic component dependent on the eccentricity of the pivot 114 on the sprocket 96 and the ratio of the distances between points 114, 124 and 118 on the link 112.

The effect of the addition of second and third harmonic may again best be illustrated through several examples. The equation of motion of the output movement may now be hypothesized as follows:

U = φ + F ₁ Sin φ + F ₂ Sin 2 φ+ F ₃ Sin 3 φ (Displacement) (40)

By successive differentiation:

dU/dφ = 1 + F ₁ Cos φ + 2F ₂ Cos 2 φ+ 3 Cos 3 φ (Velocity) (41)

d ² U/dφ ² = -F ₁ Sin φ - 4F ₂ Sin 2 φ- 9 Sin 3 φ (Acceleration) (42)

d ³ U/dφ ³ = -F ₁ Cos φ - 8F ₂ Cos 2 φ -27 Cos 3 φ (43)

d ⁴ U/dφ ⁴ = F ₁ Sin φ + 16F ₂ Sin 2 φ + 81 Sin 3 φ (44)

d ⁵ U/dφ ⁵ = F ₁ Cos φ + 32F ₂ Cos 2 φ + 243 Cos 3 φ (45)

These relationships may be utilized to meet an extremely wide variety of requirements; two examples will be shown.

EXAMPLE VIII

It is again desired to design the mechanism to achieve a long dwell using both the second and third harmonics (rather than either the second or third harmonic as in the earlier cases).

It will be noted that when φ = 0, equations (40), (42) and (44), representing U, d ² U/dφ ² , and d ⁴ U/dφ ⁴ are equal to 0 for all values of F ₁ , F ₂ and F ₃ . It will be assumed that the maximum dwell can be achieved by requiring that the velocity, dU/dφ = 0 at φ= 0. Therefore, from equation (41),

1 + F ₁ + 2F ₂ + 3F ₃ = 0 (46)

it will be further assumed that the maximum dwell will be achieved by requiring that the slope of the acceleration, d ³ U/dφ ³ = 0 at φ = 0. Therefore, from equation (43)

-F ₁ - 8F ₂ -27F ₃ = 0 (47)

since three unknowns, F ₁ , F ₂ and F ₃ are to be derived, a third independent relationship between these unknowns must be established. To do this, equation (45) is utilized with the stipulation that d ⁵ U/dφ ⁵ = -H, where -H in the physical sense represents the curvature of the slope of the acceleration, and H becomes the controlling parameter of the dwell characteristics. Therefore, from equation (45),

H + F ₁ + 32 F ₂ + 243 F ₃ = 0 (48)

when equations (46), (47) and (48) are solved, the following values are found

F ₁ = - (H+36)/24

F ₂ = (H+9)/30

F ₃ = - (H+4)/120

Then, for several values of H, the final values of F ₁ , F ₂ and F ₃ are as follows:

______________________________________ Curves in H F ₁ F ₂ F ₃ FIGS. 17, 18, 19 ______________________________________ 0 -1.5 .3 -.0333 J ₁ 1 -1.5417 .3333 -.0417 J ₂ 1.5 -1.5625 .35 -.0458 J ₃ ______________________________________

When these values are utilized in equation (40), and the values of U calculated for various values of φ, the curves J ₁ , J ₂ and J ₃ in FIG. 17 are obtained. It will be noted that the dwell can be made appreciably greater than was possible using only a single harmonic addition.

Curve J ₂ in FIGS. 18 and 19 represents the relative velocity and relative acceleration characteristics under these conditions. Curves J ₁ and J ₃ are omitted in these two figures because of their near identity with curve J ₃ .

EXAMPLE IX

In this example, it is desired that the velocity be as uniform as possible over the center portion of a cycle and that the dwell be as long as possible consistent with this first requirement. To meet the latter condition, the dwell effect parameter K is again introduced into equation (41) at φ = 0 whereupon:

K + F ₁ + F ₂ + 3 F ₃ = 0 (49)

in order to achieve the most uniform possible velocity over the center portion of the cycle, equations (42), (43), (44) and (45) must be examined at φ = 180°; at φ = 180°, it will be seen that equations (42) and (44), representing d ² U/dφ ² and d ⁴ U/dφ ⁴ = 0 for all values of F ₁ , F ₂ and F ₃ .

Therefore, the second relationship established to find F ₁ , F ₂ and F ₃ is to set d ³ U/dφ ³ = 0 at φ = 180° in equation (43), whereupon

F ₁ - 8 F ₂ = 27 F ₃ = 0 (50)

the final relationship is established through equation (45). Here it is hypothesized that the curvature of the slope of the dwell, d ⁵ U/dφ ⁵ , as represented by factor -C should become the controlling parameter; therefore, at φ = 180° from equation (45):

C - F ₁ + 32 F ₂ - 243 F ₃ = 0 (51)

solving equations (49), (50) and (51), the following values are found:

F ₁ = - (180 K + 13 C)/264

F ₂ = (C - 9 K)/66

F ₃ = (5C - 12 K)/792

Through numerical representative calculations for several values of C, an excellent solution of near perfect constant velocity over the center of the cycle is found for C = -1. Therefore, the factors become:

F ₁ = (- 180 K + 13)/264

F ₂ = (- 1 + 9 K)/66

F ₃ = (- 5 + 12 K)/792

Then, for several values of K, the final values of F ₁ , F ₂ and F ₃ are as follows:

Curves in K F ₁ F ₂ F ₃ FIGS. 17, 18, 19 ______________________________________ 1.00 -.63258 -.15152 -.02146 K ₁ 1.01 -.63939 -.15288 -.02162 K ₂ 1.02 -.64621 -.15424 -.02177 K ₃ ______________________________________

When these values are utilized in equation (40), the curves K ₁ , K ₂ and K ₃ in FIG. 17 are obtained, indicating the variation in dwell characteristics for the various values of K.

Only the curves K ₁ are shown in FIGS. 18 and 19 because of the near identity of curves K ₂ and K ₃ to K ₁ , representing the relative velocity and acceleration under these conditions. It will be noted that the relative velocity is substantially constant at a value of 1.4 from φ = 115° to φ = 245°, thereby meeting the original design requirements.

The sprocket ratios shown in FIG. 16 are two and three times the fundamental. Any other usable combination could also be used such as 2 and 4 or 3 and 5. Furthermore, one or more additional sprockets and adder links may be added if specific requirements demand. Using this technique it becomes possible to construct an almost limitless variety of kinematic output characteristics using Fourier analysis as a mathematical basis to determine the Fourier coefficients required to approximate the desired kinematic profile.

A single example will be presented to characterize a usage of third and fifth harmonic addition. In this case the equation of motion would become

U = φ + F ₁ Sin φ + F ₃ Sin 3 φ + F ₅ Sin 5 φ (52)

By successive differentiation

dU/dφ = 1 + F ₁ Cos φ + 3 F ₃ Cos 3 φ + 5 F ₅ Cos 5 φ (53)

d ² U/dφ ² = -F ₁ Sin φ - 9 F ₃ Sin 3 φ - 25 F ₅ Sin 5 φ (54)

d ³ U/dφ ³ = -F ₁ Cos φ - 27 F ₃ Cos 3 φ -125 F ₅ Cos 5 φ (55)

d ⁴ U/dφ ⁴ = F ₁ Sin φ + 81 F ₃ Sin 3 φ + 625 F ₅ Sin 5 φ (56)

d ⁵ U/dφ ⁵ = F ₁ Cos φ + 243 F ₃ Cos 3 φ + 3125 F ₅ Cos 5 φ (57)

These relationships may again be utilized to satisfy many requirements in addition to their usage in the following example.

EXAMPLE X

The design requirements are the same as those for Example VIII, i.e., achieve as long a dwell as possible, but in this instance using the third and fifth harmonic in place of the second and third.

By a process completely analogous to that of Example VIII, the following values are found for F ₁ , F ₃ and F ₅ .

F ₁ = - (H+225)/192

F ₃ = (H+25)/384

F ₅ = - (H+9)/1920

Then, for several values of H, the final values for F ₁ , F ₃ and F ₅ are as follows:

Curves in H F ₁ F ₃ F ₅ FIGS. 17, 18, 19 ______________________________________ 0 -1.1719 .0651 -.0047 M ₁ 3 -1.1875 .0729 -.0063 M ₂ 6 -1.2031 .0807 -.0078 M ₃ ______________________________________

When these values are utilized in equations (52), (53), and (54), curves M ₁ , M ₂ , and M ₃ in FIG. 17 are obtained indicating the dwell characteristics, curve M ₂ in FIG. 18 representing the relative velocity characteristics and curve M ₂ in FIG. 19 representing the relative acceleration characteristics. It is especially noteworthy that when using the third and fifth harmonic, those factors which give the best dwell also provide a very flat velocity characteristic through the center portion of the stroke.

In all the foregoing mechanisms, the idler arm was arranged to pivot about the axis of the output shaft. Under some conditions, it becomes more convenient to have the idler arm pivot about the input axis as is shown in FIG. 20.

Referring to FIG. 20, an output sprocket 130 rotates about an axis 132 on a suitable shaft mounted in suitable bearings in the frame 2. An input sprocket 134 is driven by external means about an axis 136 on a suitable shaft. An idler arm 138 is independently mounted to pivot on axis 136. At one end the idler arm 138 has connected to it, through a suitable shaft and bearing, the idler sprocket 140 rotating on an axis 142. At its other end the idler arm 138 has connected to it through a suitable shaft and bearing another idler sprocket 144 rotating on an axis 146. A drive chain 148 is in driving engagement with sprocket 134 and sprocket 130 and in engagement with the idler sprockets 140 and 144. It can be seen that in the absence of motion of the idler arm 138, the angular velocity ratio between the input sprocket 134 and the output sprocket 130 is the ratio of their pitch diameters.

A link 150 is connected to the input sprocket 134 through a suitable eccentric shaft and bearings on an eccentric axis 152; the other end of the link 150 is connected to a beam link 154 through a pivot joint 156 at some suitable point along its length. One end of the beam link 154 is connected to the frame 2 through a suitable shaft and bearings on an axis 158; the other end of the beam link 154 is pivot connected to a drive link 160 with a pivot joint 162. The other end of the drive link 160 is connected to the idler arm 138 at the pivot joint 146.

It can be seen that as the input sprocket 134 rotates about its axis 136, the beam link 154 is caused to oscillate about pivot 158 as driven by link 150; this in turn causes the idler arm 138 to oscillate about axis 136 as driven by link 160 from beam link 154.

The magnitude of oscillation of the idler arm 138 is determined by the eccentricity of pivot 152 from axis 136, and the ratio of distances between points 158, 156 and 162 on beam link 154. By an analysis comparable to that employed in the earlier description, it is possible to show that the motion of the outut sprocket may be made to be substantially cycloidal through the proper selection of parameters. Therefore, the choice of positioning the idler arm to oscillate on the output axis or on the input axis is dependent only on the design factors of a particular application.

As was the case with the designs with the idler arms oscillating about the output axis, it is equally possible to employ superposition to construct Fourier additions when the idler arm is pivoted about the input axis.

A typical mechanism to add a single higher harmonic is shown in FIG. 21. The basic mechanism is the same as in FIG. 20 with the following modifications and additions. A secondary drive sprocket 164 is mounted parallel to and concentric with the input sprocket 134 on the input shaft. An intermediate sprocket 168 is mounted on a suitable shaft and bearings from the frame 2; this intermediate sprocket 168 is driven by sprocket 164 through chain 170. As shown in FIG. 21, the sprocket 168 rotates at twice the angular velocity of sprocket 164 but any other multiple ratio may be employed.

The beam link 154 is connected to the sprocket 168 through a pivot joint 172 which is eccentric to the axis of rotation of the sprocket 168. It can be seen, therefore, that a second harmonic is added to the motion of link 160 and idler arm 138 whose magnitude is determined by the eccentricity of pivot 172 on the sprocket 168 and the ratio of the lengths between points 172, 156 and 162 on the beam link 154.

A second intermediate sprocket and associated linkage may be added just as in the mechanisms in which the idler arm pivoted on the output sprocket.

In all the cases where the intermediate sprockets are used, it is possible to apply the input power from some external power source to that intermediate sprocket, rather than to the input sprocket directly. For example, in FIG. 21, the power may be applied to sprocket 168, in which case sprocket 164 is driven through chain 170 and in turn drives sprocket 134 which is the input sprocket of the variable loop.

The system is also available for situations in which the intermediate sprockets are larger than their associated primary sprocket, in which case subharmonics are generated. Therefore, a given cycle will not repeat until the slowest turning of the intermediate sprockets has made one complete revolution.

The drive link which causes the oscillation of the idler link was, in all cases, shown as being attached to the idler arm on an axis which was coincident with the axis of one of the idler sprockets. This was done for mechanical convenience and not kinematic necessity. The link may be attached to the idler arm at any point where an effective torque on the idler arm can be generated about the axis of rotation of the idler arm.

For light duty applications, any or all chain loops in all mechanisms may be replaced by cogged belts and suitably matching pulleys.

For high torque applications, it may become desirable to increase the angle of chain wrap about the output sprocket. A system for accomplishing this is shown schematically in FIG. 22. The input sprocket 22 again rotates on the axis A ₂ and the output sprocket 66 rotates on the axis A ₁ as in FIGS. 1 and 5. The idler arm 180 is enlarged to support two additional idler sprockets 184 and 186, through suitable shafts and bearings, in addition to the original idler sprockets 30 and 42. The idler arm 180 is free to pivot on axis A ₁ and is oscillated by the link 52 from input sprocket 22 as before. The chain 64 passes from input sprocket 22 to the idler sprocket 30, then to the added idler sprocket 186, then to the output sprocket 66, then to the other added idler sprocket 184, then to the idler sprocket 42, and finally back to the input sprocket 22. It can be seen that the addition of the additional idler sprockets 184 and 186 significantly increases the angle of chain wrap about the output sprocket 66, and that this addition will have no effect on the performance of the system if sprockets 30 and 42 are positioned as earlier described.

It is further evident that a marked increase in wrap may be accomplished through the addition of only one additional idler sprocket, either 184 or 186.

This increase in chain wrap around the output sprocket, through the addition of one or two idler sprocket, is also applicable to all the embodiments in which one or more higher harmonics is added. Furthermore, it may be seen that in those embodiments in which the idler arm 138 pivots about the axis A ₂ of the input sprocket 134, the angle of chain wrap about that input sprocket 134 may be similarly increased through the addition of one or more idler sprockets to the idler arm 138.

Another arrangement for introducing a second or third harmonic into the output motion is shown in FIG. 23 (see also FIGS. 1 and 5). As shown, the idler sprocket 42 has one-half the pitch diameter of the input sprocket 22; therefore, the idler sprocket 42 will make two complete revolutions during the interval that the input sprocket 22 makes one revolution. It will be noted that the link 52 is no longer attached to the arm 24 through the shaft 44, but is attached to an eccentric pivot shaft 190 mounted on sprocket 42. This then superimposes a modified second harmonic component on the motion of the arm 24 and, consequently, to the output motion of the output sprocket 66. The modification of the second harmonic arises out of the fact that the sprocket 42 does not rotate at a constant angular velocity about shaft 44 but undergoes cyclical velocity changes as arm 24 oscillates about axis A ₁ .

A modified third harmonic may be introduced into the output motion by designing the system such that the sprocket 42 is one-third the pitch diameter of the sprocket 22.