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References:
1. Mishchenko E. F. Differential Equations with Small Parameter and Relaxation Oscillations. - Moscow: Nauka, 1975. - 248 p. (in Russian).
2. Barbashin E. A. Introduction into the Theory of Stability. - Moscow: Nauka, 1967. - 224 p. (in Russian).
3. Poincare A. Selected Works. - Moscow: Nauka, 1972. -V. 2. - 543 p. (in Russian).
4. Annakulova G. K., Igamberdiev K. A., Sattarov B. B., Abdullaeva M. Study of Lyapunov's Functions of Strongly Non-linear Dynamic System//Proc. of the XI All-Russian Congress in Fundamental Problems of Theoretical and Applied Mechanics. - Russia, Kazan, 20-24 August, 2015. - P. 172-175. (in Russian).
Romashin Valerij Nikolaevich, Technological Institute of the National Research Nuclear University MEPhI Candidate of Technical Sciences, Associate Professor E-mail: valeryromashin@yandex.ru; Romashin Roman Valerevich, Technological Institute of the National Research Nuclear University MEPhI
Senior Lecturer E-mail: rvromashin@mephi3.ru
Forgotten alternative to crank mechanism
Abstract: In the history of the industry of Europe, there was the mechanism with a screw surface for reception of rotation movement from linear movement. Having found new outlines in XIX century, but using the same principle, it took replaced the crank in the steam engine. Because of the big dimensions, it has ceased to be used and has been forgotten. The new design has appeared now, allowing to return to an old principle.
Keywords: crank mechanism, engine, drive, torque moment, screw surface, cylindrical wedge mechanism.
Transformation of translational motion into rotational motion is traditionally associated with a crank mechanism. They surround us everywhere and have become usual and indispensable. But why are screw surface mechanisms not used for this purpose? What obstructed their wide spread? Let's take a look at history.
The authors associate the first use of a screw surface in mechanics with Archytas, 428-365 BC. Archimedes went down in history as the author of a screw pump. The Mechanica by Hero of Alexandria considers simple mechanisms, including a screw. But all these examples describe the use of a screw surface as a transformer of rotational motion into translational motion.
The use of a water wheel in Europe in X-XV centuries was a foundation for future mechanization. Some processes required not rotational motion, but reciprocal motion of an output element, for instance, in blacksmithing, metal industry, construction and weaving. Hence, crank mechanism gained widespread.
Although, according to literature sources, another mechanism with a screw surface was used apart from the water wheel. Thus, the book of an Italian architect and engineer Giovanni Branca «Le Machine», 1629 depicts such mechanism in figures XVI, XIX, XX in the first part [1]. It transforms linear motion into rotation. The mechanism is based on a drum capable of revolving about its axis — it is installed vertically in the support blocks. Two spiral pipes installed at the angle of elevation over 45° are fixed on the cylindrical surface of the drum. Water is used as a vertically moving body. Supplied from top, under gravitation force, it runs down spiral pipes; herewith, affecting the walls of the pipes, radial component of gravitation force makes the light drum revolve. A toothed wheel is attached to it at the bottom and the revolving is transferred to burr stones as shown in figure XVI (Fig. 1).
Figure XIX depicts a cascade of three such drums, one below another (Fig. 2).
Figure XX shows the use of such machine in household — the drum is located in the cellar of the house and it puts the spinning wheel upstairs in motion (Fig. 3).
Thus, one can conclude that such mechanisms were widely applied in Europe in XVI-XVII centuries.
100 years before this Middle, or as it is also called «High» Renaissance, in one of his codices, Codex Madrid I, 1493, kept at the National Library of Spain under registration number 8937, Leonardo Da Vinci (1452-1519) described the structure of an endless screw in page 70. The mechanism is known under the code Madrid Ms. I (BNM), MSS/8937, fol. 70r [2]. The entire mechanism transforms the rotation from one plane into the rotation in the other plane. It is possible due to screw surfaces with a high angle of elevation. Let us suppose that if one makes such surfaces on a cylinder, installs a screw in such a way that it cannot move along axis, and makes the cylinder move linearly, the screw will revolve sliding along the turns. In the mechanism of Leonardo, such cylinder is convolved and forms a torus (Fig. 4). With the help of an input element — gear wheel m-n, that has projections, which correspond to pockets in the torus supported by worm rollers, starts rotating in the horizontal plane, and the output element — screw, sliding on the screw surfaces, also starts rotating but in the vertical plane. The figure does not show that the screw is fixed against the movement on the axis of the torus, but, according to the comment by Leonardo to the figure, «if one makes the endless screw rotate with the help of gear wheel m-n, and firmly keeps the screw f-s in place to make it possible for it to rotate, the rotation of the screw will undoubtedly take place with a strong force» [3]. Thus, one can assume that it is the first, from archive data, mechanism using screw surfaces with the angle of elevation over 45° to create rotation of the output element. Although, due to the difficult production, it was most probably never manufactured as well as many constructions of Leonardo that were ahead of time.
Fig. 1.
Fig. 2.
Fig. 4.
In the Modern age, when the energy of falling water was replaced by the steam, and Watt steam engines were widely applied, the mechanism with a screw surface to obtain rotational motion from linear motion gained a new shape. Thus, there is an
element under number 167 in the collection of illustrations and descriptions contained in the book of Henry T. Brown «Five hundred and seven mechanical movements» published in New York in 1871 [4] (Fig. 5).
Fig. 5.
It is described as follows: «A drum or a cylinder with an endless helical groove spread around it; one half of the groove has an inclination in one direction and the other half has an inclination in the opposite direction. The pin on the rod performing reciprocal straight line motion moves in the groove thus transforming reciprocal motion into rotational motion. It was used as a replacement of crank in the steam engine».
Now, mechanisms transforming reciprocal motion of the input element into reverse rotational motion of the output element with the help of a screw surface with an angle of elevation over 45° are rarely applied. Series-produced ball screw gears (BSG) allow transforming translational motion into rotational one. There are BSGs whose lead is twice as big as the nominal diameter of the screw, i. e. the elevation angle is 32° 28'. There is no BSG with an angle of elevation of 45° because, then, the screw will be too long as, constructively, the number of turns in the screw is usually from 1 to 6 [5]. But when it is required to create a high pulling torque, when it is required to obtain only rotational motion — it is not effective to use BSG, and even impossible. It makes sense to use a screw pair for this purpose, whose angle of elevation of helical curve is over 45°, i. e. when the screw pair becomes a mechanical enhancer. All mechanisms using such transformation have a significant advantage — developed torque moment is even in the time function under constantly acting force and there is no one-sided effect on the output element. However,
there is a significant disadvantage as well — longer helical curve, i. e. large size and a small angle of turn of the output element. The effort to reduce height, i. e. reduce the motion of the input element, leads to the limitation ofturning angle and, consequently, to inevitable reversibility — during reverse motion there is a turn in the reverse direction. These disadvantages overshadow the advantages of screw pair, where the elevation angle of helical curve is over 45° in mechanisms for transformation of translational motion into rotational motion. Thus, the opportunity to obtain even torque moment on the output element without one-sided effect on it is not used. This explains a rare use of similar mechanisms in the past and today.
However, one can limit oneself to a small part of the screw surface with an angle of elevation over 45°, and, herewith, obtain rotational motion in one direction from reciprocal motion. Such drive constructions are described in a patent for the invention RU 2389925 C2 [6], the authors of which are the authors of the present article.
Torque moment in all variants of drives is created with the help of wedge mechanism folded around rotation axis. For the sake of convenience, instead of the phrase «power mechanism with a screw kinematic pair with a screw surfaces made under the required law, with the elevation angle significantly bigger than the friction angle, with intermediate bodies of rolling or sliding between the screw and nut or with their direct contact», the authors of the invention introduce a new notion — cylindrical wedge mechanism (CWM). Unlike
the known wedge mechanism, the members ofwhich form only rectilinear pair, CWM contains a screw pair that allows transforming translational motion into rotational one. Considering the forces in a regular screw pair, it is convenient to turn the screw to medium diameter in the inclined plane, and replace the nut with a sliding member. The screw pair allows, just like the wedge, gaining advantage in force. The notion of CWM was introduced to emphasize this fact, as well as due to a significant similarity to wedge mechanism, the input element of which the translational motion is reported to, and also to make it clear which screw pair is meant. As an example of cylindrical wedge
mechanism — CWM, let's consider the drive construction based on item 2 of the formula similar to figure 1 of the patent [6].
The drive in figure 6 contains a fixed member 1 of the wedge mechanism, which is simultaneously a drive case, in this example — the case of hydro-cylinder, firmly tied with the second fixed member 9 of the wedge mechanism due to functional and constructive reasons, as well as moving element 2 of the drive, input element 3, intermediate element 7, mechanism of one-way action 14 and output element 11. Control system is not shown in figure 6 — it may be of any type.
Fig. 6. Construction with cylindrical wedge mechanism
A piston, the moving element 2 ofthe drive, can be placed inside the fixed member 1. Any linear drive can be used in the description of the invention [6]. Due to functional and constructive reasons, the moving element 2 is firmly tied with the input element 3 of the wedge mechanism that has several helical grooves 6. The intermediate element 7 has response helical grooves 8. Screw working surfaces of the grooves 6 and 8 is made according to the required law, have the elevation angle significantly bigger than the friction angle and have a needed shape in the section. To reduce losses during operation, elements 3 and 7 are divided by the rolling elements 29 located in the helical grooves 6 and 8. The moving element 2 has additional helical grooves 4 — directional grooves setting the law of its motion, and the motion of the input element 3 of the wedge mechanism firmly tied to it. The surfaces of these directional grooves 4 can slide along several directional elements 5 fastened in the fixed member 1 of the wedge mechanism. Since the elevation angle of the helical grooves 6 and 8 is significant, the change of the position of the moving element 2 with the input element 3 allows gaining turning motion of the intermediate element 7 of the wedge mechanism in accordance with the law set by the helical grooves 6 and 8 as well as the law set by the additional helical grooves 4. To simplify this, let's consider that the grooves 4 are vertical. Elements 3 and 7 with screw working surfaces, contacting with each other through rolling element 29, are the basic elements of the cylindrical folded around the axis turn of the output element 11 of the wedge mechanism. It acts as a power mechanism. Plates 30 and 31 are fixed on the end surface of elements 3 and 7 to prevent the release of rolling
elements 29 beyond the sphere of their mutual contact with helical grooves 6 and 8. It is possible to apply separator for rolling elements or use a by-pass channel to make close the chain of rolling elements. Intermediate element 7 relies upon the ring-shaped row of rolling elements 10 installed between the intermediate element 7 and fixed element 9 of the wedge mechanism.
Between the intermediate element 7 and output element 11 of the wedge mechanism, mechanism of unilateral action 14 intended for rendering of torque moment from the intermediate element 7 to the output element 11 only in one direction is installed with the help of fixed connections 12 and 13. For instance, unilateral free-wheel clutch or ratchet gear can serve as such mechanism.
The drive operates according to fig. 6 as follows.
Operating environment is supplied to the upper pocket 15 of the drive under pressure. At the same time, moving element 2, firmly tied with the input element 3 of the wedge mechanism, starts moving downwards sliding along the directional elements 5 that render the appearing reaction moment to the fixed member 1 of the mechanism. Axial force developed by the moving element 2 and rendered to the input element 3 of the wedge mechanism and further through contact surfaces of the helical grooves 6 and 8 with rolling elements 29 to the intermediate element 7, is taken by the ring-shaped row of rolling elements 10. Thus, input element 3 of the wedge mechanism going downwards turns the intermediate element 7, which renders torque moment through the fixed connection 12 to the mechanism ofunilateral action 14. This mechanism renders rotation to the output element 11 through fixed connection 13.
In the bottom position of the moving element 2, there is a double line sensor installed in the control system (control system is not depicted in fig. 6) that sends a signal for reversing of movement of the moving element 2. During this free movement of the moving element 2, the output element 11 does not turn. Now, the operating environment under pressure is supplied to the bottom pocket 16 of the drive. The moving element firmly tied with the input element 3 of the wedge mechanism starts moving upwards; herewith, the intermediate element 7 turns backwards because the directions of axial and peripheral forces, as well as contact surfaces of helical grooves 6 and 8 of the elements 3 and 7 with rolling elements 29 changed. But now, the intermediate element 7 cannot turn the output element 11 because the mechanism of unilateral action 14 in this direction does not render torque moment and the output element 11 is still. When the moving element 2 reaches the upper position, one double line is completed and the intermediate element 7 turns to the same angle as when moving downwards.
If the required angle position is achieved through one double line, the sensor of angle position of the output element 11 installed in the control system sends a signal for the completion of the drive's work. Otherwise, there is another double line of the moving element 2. Thus, the turn can be performed to any required angle.
In case of repeated double lines, interrupted rotation will take place. One should remember that patent [6] has variants of constructions that have both lines as operating.
The considered construction has more than one double line, which, on the one side, leads to reduction of height of the drive, and on the other side, increase of the time of turning to a required angle. If the second factor is important for the drive, it can be kept within required range at the expense of increasing the speed of movement of the moving element and input element 3 firmly tied with it. The increase of the speed of the input element allows construction with CWM serve as alternative to the constructions with crank mechanism.
As a result of experimental studies of cylindrical wedge mechanism, with same parameters for all samples:
• ball diameter Dw = 7.94 mm.;
• diameter of circumference of ball location Dpw = 50 mm.;
• number of operating grooves on a sample i = 3;
• number of balls in one groove z = 2,
dependences on axial force under different profiles and elevation angles were obtained (fig. 7). Samples № 21, 22, 23 have a trapezoid profile and elevation angle of the groove 62.25°, 70°, 80° respectively. Samples № 25, 27, 29 have an arc profile and elevation angle of the groove 62.25°, 70°, 80° respectively.
Fig. 7. Dependence of the moment on axial force under different profiles and elevation angles
Fig. 7 depicts the influence of the profile and elevation angle on angle of the helical groove, the moment increases and the required the change of the moment and axial force in case of same properties axial force reduces. The increase of the permitted contact stress al-of the material for all samples. With the increase of the elevation lows increasing the moment significantly.
References:
1. Giovanni Branca. Le Machine/Kinematic Models for design - digital library; Cornell University Library, Ithaca NY 14853//[Electronic resource]. - Available from: http://ebooks.library.cornell.edu/k/kmoddl/pdf/049_Section_I.pdf
2. Leonardo da Vinci. Codex Madrid I/Kinematic Models for design - digital library; Cornell University Library, Ithaca NY 14853// [Electronic resource]. - Available from: http://ebooks.library.cornell.edu/k/kmoddl/pdf/020_009.pdf
3. Leonardo da Vinci. Circular worm screw/Museo Galileo - Institute and Museum of the History of Science; Piazza dei Giudici 1-50122 Florence - Italy//[Electronic resource]. - Available from: http://brunelleschi.imss.fi.it/genscheda.asp?appl=LIR&xsl=pag inamanoscritto&lingua=ENG&chiave=100806
4. Henry T. Brown. Five Hundred and Seven Mechanical Movements/Kinematic Models for design - digital library; Cornell University Library, Ithaca NY 14853//[Electronic resource]. - Available from: http://ebooks.library.cornell.edu/k/kmoddl/pdf/005_003.pdf
5. Precision ball screw assemblies [Text]: R310EN 3301 (2009.08)/Bosch Rexroth AG. - Printed in Germany, 2009. - 164 p.
6. Turn drive (versions) [Text]: pat. RU 2389925 C2/Inventors and proprietors: Romashin V. N. (RU); Romashin R.V. (RU). - 2007.
