US3730654A

US3730654A - Gear arrangement for providing an oscillating rotational motion

Info

Publication number: US3730654A
Application number: US00226127A
Authority: US
Inventors: W Mcmahon
Original assignee: Individual
Current assignee: Individual
Priority date: 1972-02-14
Filing date: 1972-02-14
Publication date: 1973-05-01
Anticipated expiration: 1990-05-01

Abstract

A pair of meshed gears each of which comprises a plurality of sectors having the configuration of a logarithmic spiral provide an oscillating motion with respect to each other as they rotate. The gears can be coupled to the rotors of rotary piston machines and the like to provide the desired oscillating rotational motion thereto.

Description

United States Patent 11 1 McMahon [451 May 1,1973

[ GEAR ARRANGEMENT FOR PROVIDING AN OSCILLATING ROTATIONAL MOTION [76] Inventor: William McMahon, 20 Hillside Ave., Summit, NJ. 07901 [22] Filed: Feb. 14, 1972 21 Appl. NO; 226,127

4 74/437 [51] Int. Cl. ..F0lcl/00,F16h35/()2,F()4c1/00 [58] Field ofSearch ..418/3336, 150; 74/393,437; 123/8,47

[56] References Cited UNITED STATES PATENTS 8/1959 Kitano ..74/437 3,061,180 10/1962 Durgin ..4l8/36 3,112,062 11/1963 Way 3,398,643 8/1968 Schudt 3,430,573 3/1969 Groeger ..41 8/36 Primary ExaminerCarlton R. Croyle Assistant Examiner-John J. Vrablik Attnrney-Alvin D. Hooper I 57] ABSTRACT A pair of meshed gears each of which comprises a plu rztlity of sectors having the configuration of a logarithmic spiral provide an oscillating motion with respect to each other as they rotate. The gears can be coupled to the rotors of rotary piston machines and the like to provide the desired oscillating rotational motion thereto.

16 Claims, 9 Drawing Figures Patented May 1, 1973 3 Sheets-Sheet 1 Patented May'l, 1973 3 Sheets-Sheet 2 Patented May 1, 1973 3,730,654

3 Sheets-Sheet 3 GEAR ARRANGEMENT FOR PROVIDING AN OSCILLATING ROTATIONAL MOTION BACKGROUND OF THE INVENTION adjacent pistons always rotate in the same direction but with alternating increasing and decreasing speeds with respect to each other.

Various gear arrangements have been disclosed in the prior art for providing an oscillating rotational motion. These arrangements have primarily utilized sector gears and elliptical gears. However, these types of gears are not ideally suited for such applications. The intermittent meshing of sector gears resultsiin high stresses and noise. These gears must be manufactured with very tight tolerances to prevent excessive wear which would increase the noise and stresses during meshing. When elliptical gears are used, the point of meshing does not remain on the line joining the axes or centers of two meshing elliptical gears. Thus the gears tend to unmesh or uncouple. Further, meshing away from the line joining the axes can produce high stresses in the gears.

Another disadvantage of the gear arrangements previously disclosed is the number of gears required and the resulting complexity of the arrangements.

SUMMARY OF THE INVENTION The foregoing disadvantages are overcome in accordance with this invention by a gear arrangement which utilizes meshed gears having sectors with configurations ofa logarithmic spiral for obtaining an oscillating rotational motion. Each gear has four identical quadrants each of which has a varying pitch diameter defined by a logarithmic spiral. The gears are meshed so that the mesh point is always on a line joining their axes and the rate of rotation of the gears alternately increases and decreases with respect to each other within every 90 of rotation, i.e., within every quadrant of the gears. The gears can be connected to the rotors of rotary piston machines and the like to provide an oscillating rotational motion for defining alternately expanding and contracting chambers therein.

BRIEF DESCRIPTION OF THE DRAWING The invention will be more fully comprehended from the following detailed description and accompanying drawing in which:

FIG. 1 is an exploded perspective view partly in section of a rotary piston machine utilizing a gear arrangement in accordance with this invention;

FIG. 2 is an elevation view of the logarithmic spiral gear utilized in FIG. 1;

FIG. 3A, 3B and 3C are schematic representations of the operation of the gears of FIG. 1;

FIG. 4A, 4B and 4C are schematic representations of the configurations of the machine of FIG. 1 with reference to the positions of the gears in FIG. 3A, 3B and 3C, respectively; and

FIG. 5 is a perspective view of a second gear arrangement utilizing intermediate gears to obtain different rotational speeds between the logarithmic gears and the associated rotors.

DETAILED DESCRIPTION The gear arrangements of this invention will be described with particular reference to rotary piston machines. However, it is to be clearly understood that the gear arrangements can be utilized anywhere an oscillating rotational motion is desirable.

In FIG. 1 is shown partly in section an exploded perspective view of a rotary piston machine 101 comprising two coaxial rotors or shafts 2 and 4 about each of which a plurality of vanes 6 and. 8, respectively, are mounted at substantially equal angular intervals. A plate or flange is mounted on one end of shaft2 to which vanes 6 are mounted so that the edges 12 of vanes 6 are spaced asmall distance 14 from shaft 2. Vanes 8 are mounted directly upon shaft or rotor 4. Flange 10 can have a series of ports or openings 11 therein which function as will hereinafter become apparent. Vanes 6 and 8 and rotors 2 and 4 are enclosed within a housing 16 which in the illustrative embodiment comprises a generally cylindrical container with one open end. Housing 16 is mounted to the outer peripheries of vanes 8 and rotates therewith.

Shaft 2 fits within shaft 4 and can be journaled therein by well known techniques. Plate 10 fits against and seals the open end of housing 16. Vanes 6 fit between vanes 8 in an alternate configuration around the circumference of shaft 4. The edges 5, 7, 9 and 12 of vanes 6 and 8 form seals where appropriate with corresponding portions of shaft 4, plate 10 and housing 16 so that eight independent chambers are defined.

Shaft 2 extends from the opposite end of shaft 4 and has fastened thereto a logarithmic spiral gear 18. Gear 18 can be coupled to a power source or power utilizing apparatus depending upon whether machine 101 is being used as a pump or an engine.

Coupled or meshed with gear 18 is another logarithmic spiral gear 20. Gear 20 is connected through shaft 23 and identical

intermediate gears

22 and 24 to shaft 4 so that shaft 4 rotates at the same rate but in the opposite direction as gear 20, i.e., shafts 2 and 4 rotate in the same direction.

Gears

22 and 24 can comprise circular spur gears well known in the prior art.

FIG. 2 is an elevation view of a logarithmic spiral gear like

gears

18 and 20 utilized in machine 101. Gear 30 has four identical sectors or

quadrants

32, 34, 36 and 38 which are joined as shown to form a roughly bow-like" configuration. By designating gear 30 as a logarithmic spiral, it is meant that any point 42 along the pitch diameter of eachquadrant is defined by the function: D=e" where D equals the radius 44 from the center 46 of gear 30 to point 42; e is the base of the natural logarithm; a is the angle 48 between the radius 50 to the starting point 52 of pitch diameter 40 and the radius 44 to point 42; and k is a constant determined by the dimensions of gear 30 such as starting point 52 and end point 54 of pitch diameter 40. The distance 56 along the pitch diameter 40 from point 52 to point 42 is given by: S=(D1) (l+l/k where S equals distance 56.

The teeth 60 and spaces 62 of gear 30 are formed about radii 44 by known techniques. With respect to radii 44 teeth 60 have the approximate shape of spur gear teeth with one side cut slightly deeper than the other side. The width 64 of teeth 60 and width 66 of spaces 62 are substantially equal along the pitch diameter 40.

In order to combine four logarithmic spiral quadrants into a gear 30 as shown, it is necessary that the combined number of teeth 60 and spaces 62 in each quadrant be an odd number. This is required in order that each quadrant have a half-tooth at point 52 and a half-space at point 54, or vice versa, which combines with the corresponding half-tooth or half-space from the adjacent quadrant to form a complete tooth 60 or space 62. Also the number of teeth 60 in any quadrant must equal the number of spaces 62 in that quadrant. The actual number of teeth 60 and spaces 62 utilized in each quadrant; and accordingly the tooth size, is determined by known gear design techniques which depend upon such factors as the forces being transmitted by the gears, etc.

The operation of two meshing logarithmic spiral gears such as gear 30, and accordingly the operation of the vanes or pistons on the rotors associated therewith, is illustrated in FIG. 3A, 3B and 3C and FIG. 4A, 4B and 4C, respectively. In FIG. 3A two logarithmic spiral gears 80 and 82 are oriented at 90 degrees with respect to each other and coupled or meshed. The position of vanes 90 A, B, C, D and 92 A, B, C, D associated with

gears

80 and 82, respectively, are shown in FIG. 4A. For example, gear 80 and vanes 90 A, B, C, D can correspond to gear 18 and vanes 6, respectively; gear 82 and vanes 92 A, B, C, D can correspond to gear 20 and vanes 8, respectively, and axis x--x corresponds to axis yy of FIG. 1. At this position which can be considered an initial or starting position the vanes 90 A, B, C, D and 92 A, B, C, D are equally angularly spaced around the circumference of

shafts

94 and 95 thereby defining eight equal chambers 96 A, B, C, I-I. When gears 80 and 82 rotate as shown, gear 82 initially rotates at a faster rate than gear 80 until some intermediate point illustrated in FIG. 3B is reached. At this point the angle of rotation 84 of gear 82 is greater than the corresponding angle of rotation 86 of gear 80. Thus, as shown in FIG. 4B vanes or pistons 90 A, B, C, D and 92 A, B, C, D associated with

gears

80 and 82, respectively, have also rotated different amounts, i.e., 92 A, B, C, D have rotated more, thereby causing some chambers 96A, C, E, G to expand and others 96 B, D, F, H to contract or compress.

As

gears

80 and 82 rotate beyond the intermediate point shown in FIG. 3B, gear 80 now rotates faster than gear 82 until the position illustrated in FIG. 3C is reached at which point both gears have rotated an equal amount of 90". As shown in FIG. 4C, during this segment of rotation pistons 90 A, B, C, D and 92 A, B, C, D have correspondingly reversed their relative speeds of rotation so that chambers 96 A, C, E, G have contracted and chambers 96 B, D, F, II have expanded so that all chambers have returned to their original sizes, i.e., the sizes shown in FIG. 4A but rotated 90 degrees therefrom. Upon continued rotation past the position shown in FIG. 3C, gear will continue to rotate faster than gear 82 until another intermediate point similar to FIG. 3B is reached at which point the relative rates of rotation again reverse. Accordingly during this continued rotation chambers 96 A, C, E, G continue to contract and chambers 96 B, D, F, H continue to expand until the intermediate point is reached at which the expansion and contraction again reverses.

From the foregoing, it is apparent that each chamber 96A, H contracts and then expands, or vice versa, the same amount with respect to its initial or nominal size indicated in FIG. 4A That is, with the indicated gear arrangement a chamber will contract and return to its beginning or initial configuration during one segment of rotation. During the subsequent 90 segment of rotation the chamber will expand and return to its initial configuration. Thus each chamber undergoes a complete expansion-contraction cycle during each of rotation.

The relative amount of contraction and expansion of chambers 96A, H with respect to their initial sizes depends upon such parameters as the dimensions of

gears

80 and 82, the angular thickness 97 of vanes or pistons 90A, D, and 92A, D, and the gear

arrangement connecting gears

80 and 82 with the vanes. When chambers 96A H define combustion chamber for an engine or pump chambers for a pump, it is desirable that the volume of a chamber be reduced essentially to zero during the contraction part of the cycle. Thus as shown in FIG. 4B chamber 968 has been reduced essentially to zero and the

vanes

92A and 90B defining this chamber are essentially touching at the end of the contraction phaseof the cycle. The angular thickness 97 of the vanes is determined by such factors as the strength required to transmit the forces generated or expended in the chambers. For purposes of illustration, vanes 90A, D, and 92A, D might have an angular thickness 97 of approximately 15. Thus in the configuration shown in FIG. 4A, the nominal or initial size or thickness 98 of a chamber would be 30, i.e., eight vanes having a thickness of 15 each and eight chambers having a nominal size of 30 each equals 360. Accordingly, in order to reduce a chamber to zero volume during its contraction part of the cycle, the gear arrangement must provide a relative oscillating motion of 30. This also means that a chamber will have a maximum angular thickness 98 of 60 during the expansion part of the cycle, i.e., 30 expansion above its nominal size of 30.

In order to provide approximately 30 relative motion between

gears

80 and 82 during each quadrant or 90 of rotation, it can be established mathematically that the major axis of the gear quadrant in logarithmic spiral gears 80 and 82 must be approximately four times the minor axis thereof. Referring to FIG. 2, this means that radius 58 is four times greater than radius 50. These dimensions also establish the value of the constant k, previously discussed, as approximately 0.8825 as determined from the boundary conditions: D=l when the angle a=0 and D=4 when the angle a=90With the value of k thus established, the coordinates of any point along the pitch diameter 40 can be readily determined by the equations previously given.

As previously discussed, with the gear arrangement shown in FIG. 1, each chamber undergoes a complete expansion-compression cycle during each two quadrants or 180 of rotation. Thus if machine 101 were used as a two-cycle engine, two two-stroke cycles could be obtained from each cylinder or chamber during each 360 of rotation. The expansion and contraction of the chambers could be synchronized, as is known in the art, with the passage of openings or port therein by appropriate fuel inlet ports, intake and exhaust ports to provide a fuel mixture to the chambers, ignite such fuel mixture, and exhaust the gases, respectively. For example, one of the openings 11 communicates with each chamber throughout the entire expansion-contraction cycle and could be used as such opening or port for the chamber.

The gear arrangement shown in FIG. 1 would provide only one fourstroke cycle for each 360 of rotation. However, it is often desirable to have a symmetrical or balanced machine which provides a complete four-stroke cycle every 180. This can be accomplished as shown in FIG. 5 by adding intermediate gearing between the logarithmic spiral gears 18 and and the shafts 2 and 4, respectively. Shaft 2 has a circular gear 106 on the end thereof which is connected through intermediate circular gears 104, 103 and 102 and shaft 107 to logarithmic spiral gear 18 in such a way as to give a two-to-one reduction between gear 18 and shaft 2, i.e., shaft 2 rotates at half the rate of gear 18. Likewise, intermediate circular gears 108 and 110 give a two-to-one reduction between gear 20 and shaft 4. Thus for each complete revolution of

gears

18 and 20, the chambers would undergo two complete expansioncontraction cycles, i.e., one four-stroke cycle, while rotating only 180. Chambers utilizing the gear arrangement of FIG. 5 will expand or contract only onehalf as much as those in FIG. 1. Accordingly, additional chambers could be added, the thickness of the vanes could be increased, or the design of the

gears

18 and 20 could be changed as previously discussed to insure that the volume of a chamber is essentially reduced to zero during the contraction or compression part of the cycle.

From the foregoing, it is clear that the number of vanes and the gearing between the

logarithmic gears

18 and 20 and rotors 2 and 4 can be varied to obtain varia tions in the operation of machine 101. However the logarithmic spiral gears 18 and 20 provide the basic oscillating motion required in all such variations. The design of these logarithmic spiral gears 18 and 20 can be varied as previously described to provide a wide range of magnitudes in such oscillating motion. The logarithmic spiral gears remain smoothly meshing at all times thereby avoiding the high stresses and noise caused by the intermediate meshing of sector gears. As shown in FIG. 3A, B, C, the mesh point 87 of the logarithmic gears always remains along the line 88 joining the centers of the gears thereby decreasing the likelihood that such gears will become unmeshed.

Although the operation of the logarithmic spiral gear arrangements of this invention have been described with particular reference to rotary piston machines, it is to be clearly understood that the applications of the invention are not to be limited to such machines. The gear arrangement is intended to have application wherever an oscillating rotational motion is needed.

What is claimed is:

1. A pair of meshed non-circular gears each of which has a plurality of gear sectors, each of said sectors having a pitch diameter defined by a logarithmic spiral, said gears being meshed so that said gears contact each other at different relative points along said respective pitch diameters as said gears rotate whereby one of said gears alternately rotates at a faster and then a slower rate than the other of said gears.

2. A pair of meshed non-circular gears for causing a first rotor connected to one of said gears alternately to rotate at a faster and then a slower rate than a second rotor connected to the other of said gears thereby to produce an oscillating rotary motion between said rotors, each of said gears having a plurality of identical sectors each of which has a pitch diameter defined by a logarithmic spiral of the form:

where: D is the radius from the center of said gear to any point along said pitch diameter;

e is the base of the natural logarithm;

a is the angle between the radius to the starting point of said pitch diameter and said radius D; and

k is a constant determined by the ratio of the radii to the ending point and said starting point of said pitch diameter, said gears being rotated with respect to each other before meshing so thatsaid gears contact each other at different relative points along said pitch diameters as said gears rotate, whereby said oscillating rotary motion is obtained.

3. Apparatus in accordance with claim 2 wherein said ratio of said radii is four whereby said oscillating motion of said rotors with respect to each other is approximately 30.

4. Apparatus in accordance with claim 2 wherein the number of teeth and tooth spaces in each of said sectors is the same and each is an integer plus one half so that adjacent ones of said sectors can be joined to form a continuous set of teeth and spaces.

5. Apparatus in accordance with claim 2 including intermediate gears interconnecting said non-circular gears with respective ones of said rotors so that said ro tors can rotate at speeds different than said non-circular gears.

6. Apparatus in accordance with claim 2 wherein said gear has four sectors forming four gear quadrants, said quadrants being symmetrical about each of two orthogonal axes through the center of said gear, said gears being meshed so that one complete oscillating cycle of said rotors is obtained as said gears rotate through two of said quadrants corresponding to of rotation.

7. Apparatus in accordance with claim 6 wherein 'said gears are rotated with respect to each other so that said starting point of one of said gears meshes with said ending point of the other of said gears.

8. In a rotary piston machine including a housing, a pair of coaxial rotors within said housing, and an equal number of pistons connected to each of said rotors for movement therewith within said housing, said pistons on said rotors alternating about the circumference of said housing and defining a plurality of chambers therein; means for interconnecting said rotors and adapted to cause adjacent ones of said pistons to alternately approach and recede from each other as said rotors rotate within said housing thereby to cause said chambers to alternately contract and expand, including:

a pair of meshed non-circular gears one of which is connected to each of said rotors, said gears having a plurality of identical sectors, each of said gear sectors having a pitch diameter in the form of a logarithmic spiral, said gears being meshed so that said gears contact at different points along said respective pitch diameters of said sectors whereby said gears and said rotors rotate at varying speeds with respect to each other to cause said pistons to alternately approach and recede.

9. Apparatus in accordance with claim 8 wherein each of said rotors has four pistons connected thereto so as to define eight of said chambers.

10. Apparatus in accordance with claim 8 including intermediate gears for interconnecting said non-circu lar gears with respective ones of said rotors so that said rotors can rotate at speeds different from said non-circular gears.

11. Apparatus in accordance with claim 8 wherein each of said gears has four sectors forming four identical quadrants for said gears, said gears being meshed so that said pistons approach and recede from each of the adjacent pistons one time thereby undergoing -a complete contraction-expansion cycle during each 180 degrees of rotation of said gears. I

12. Apparatus in accordance with claim 8 wherein said logarithmic spiral defining said pitch diameter of said sectors is given by the equation:

2 is the base of the natural logarithm;

k is a constant determined by the ratio of the radii to the ending point and the starting point of said pitch diameter.

13. Apparatus in accordance with claim 8 wherein said housing is connected to one of said rotors and rotates therewith, said housing including a first intermediate gear on the exterior thereof;

a second intermediate gear identical to said first intermediate gear and meshing therewith, said second intermediate gear being connected to one of said meshed non-circular gears to rotate therewith so that said rotors rotate in the same direction.

14. Apparatus in accordance with claim 13 including means for sealing said housing so that said chambers are isolated from each other, said machine including inlet and outlet ports therein for permitting movement of substance into and out of said chambers as said chambers expand and contract.

15. Apparatus in accordance with claim 8 wherein said gear has four sectors comprising four identical gear quadrants, said quadrants being symmetrical with respect to two orthogonal axes through the center of said gear, said gears being meshed such that they are rotated with respect to each other when the mesh point thereof is on either of said axes so that said gears undergo one complete oscillating cycle with respect to each other during each of rotation.

16. Apparatus in accordance with claim 15 wherein said logarithmic spiral is given by the equation:

where: D is the radius from the center of said gear to any point along said pitch diameter of said quadrant;

e is the base of the natural logarithm;

k is a constant determined by the ratio of the radii to the ending point and said starting point of said pitch diameter, said ratio having a value of four so that said chambers expand and contract approximately 30 during said oscillating cycle.

Claims

2. A pair of meshed non-circular gears for causing a first rotor connected to one of said gears alternately to rotate at a faster and then a slower rate than a second rotor connected to the other of said gears thereby to produce an oscillating rotary motion between said rotors, each of said gears having a plurality of identical sectors each of which has a pitch diameter defined by a logarithmic spiral of the form:D eka, where: D is the radius from the center of said gear to any point along said pitch diameter; e is the base of the natural logarithm; a is the angle between the radius to the starting point of said pitch diameter and said radius D; and k is a constant determined by the ratio of the radii to the ending point and said starting point of said pitch diameter, said gears being rotated with respect to each other before meshing so that said gears contact each other at different relative points along said pitch diameters as said gears rotate, whereby said oscillating rotary motion is obtained.

3. Apparatus in accordance with claim 2 wherein said ratio of said radii is four whereby said oscillating motion of said rotors with respect to each other is approximately 30*.

5. Apparatus in accordance with claim 2 including intermediate gears interconnecting said noN-circular gears with respective ones of said rotors so that said rotors can rotate at speeds different than said non-circular gears.

6. Apparatus in accordance with claim 2 wherein said gear has four sectors forming four gear quadrants, said quadrants being symmetrical about each of two orthogonal axes through the center of said gear, said gears being meshed so that one complete oscillating cycle of said rotors is obtained as said gears rotate through two of said quadrants corresponding to 180* of rotation.

7. Apparatus in accordance with claim 6 wherein said gears are rotated with respect to each other so that said starting point of one of said gears meshes with said ending point of the other of said gears.

8. In a rotary piston machine including a housing, a pair of coaxial rotors within said housing, and an equal number of pistons connected to each of said rotors for movement therewith within said housing, said pistons on said rotors alternating about the circumference of said housing and defining a plurality of chambers therein; means for interconnecting said rotors and adapted to cause adjacent ones of said pistons to alternately approach and recede from each other as said rotors rotate within said housing thereby to cause said chambers to alternately contract and expand, including: a pair of meshed non-circular gears one of which is connected to each of said rotors, said gears having a plurality of identical sectors, each of said gear sectors having a pitch diameter in the form of a logarithmic spiral, said gears being meshed so that said gears contact at different points along said respective pitch diameters of said sectors whereby said gears and said rotors rotate at varying speeds with respect to each other to cause said pistons to alternately approach and recede.

10. Apparatus in accordance with claim 8 including intermediate gears for interconnecting said non-circular gears with respective ones of said rotors so that said rotors can rotate at speeds different from said non-circular gears.

11. Apparatus in accordance with claim 8 wherein each of said gears has four sectors forming four identical quadrants for said gears, said gears being meshed so that said pistons approach and recede from each of the adjacent pistons one time thereby undergoing a complete contraction-expansion cycle during each 180 degrees of rotation of said gears.

12. Apparatus in accordance with claim 8 wherein said logarithmic spiral defining said pitch diameter of said sectors is given by the equation: D eka, where: D is the radius from the center of said gear to any point along said pitch diameter; e is the base of the natural logarithm; a is the angle between the radius to the starting point of said pitch diameter and said radius D; and k is a constant determined by the ratio of the radii to the ending point and the starting point of said pitch diameter.

13. Apparatus in accordance with claim 8 wherein said housing is connected to one of said rotors and rotates therewith, said housing including a first intermediate gear on the exterior thereof; a second intermediate gear identical to said first intermediate gear and meshing therewith, said second intermediate gear being connected to one of said meshed non-circular gears to rotate therewith so that said rotors rotate in the same direction.

15. Apparatus in accordance with claim 8 wherein said gear has four sectors comprising four identical gear quadrants, said quadrants being symmetrical wiTh respect to two orthogonal axes through the center of said gear, said gears being meshed such that they are rotated 90* with respect to each other when the mesh point thereof is on either of said axes so that said gears undergo one complete oscillating cycle with respect to each other during each 180* of rotation.

16. Apparatus in accordance with claim 15 wherein said logarithmic spiral is given by the equation: D eka, where: D is the radius from the center of said gear to any point along said pitch diameter of said quadrant; e is the base of the natural logarithm; a is the angle between the radius to the starting point of said pitch diameter and said radius D; and k is a constant determined by the ratio of the radii to the ending point and said starting point of said pitch diameter, said ratio having a value of four so that said chambers expand and contract approximately 30* during said oscillating cycle.