Synthesis of a ten-link tooth-lever differential roller transmission mechanism

This article deals with the synthesis of a ten-link tooth-lever differential transmission mechanism. The article contains an analytical review of modern scientific research on the synthesis of tooth-lever differential transmission mechanisms of roller machines with a variable center distance of the working shafts; a method for the synthesis of toothlever differential transmission mechanisms of roller machines with a variable center distance of the working shafts described on the example of a ten-link tooth-lever differential transmission mechanism; the conditions for the synthesis of the mechanism given and substantiated when this mechanism is used in a roller machine; one of its working shafts has the ability to rotate about its own axis, and the second working shaft, in addition to rotation about its own axis, has the ability to move relative to the first working shaft along a line passing through the center the axes of rotation of both working shafts; the geometric synthesis of the tooth and lever contours of the mechanism, the dynamic synthesis of the mechanism, taking into account the angles of pressure between the lever link of the lever contour of the mechanism, which allows us to determine the optimal working position of the mechanism where the angles of pressure are within acceptable limits; the graphs of changes in the angles of pressure between the links of the lever contour of the mechanism, plotted depending on its position.


Introduction
Despite the widespread use of roller machines and the numerous studies of these machines conducted in recent years [1][2][3][4][5][6][7][8][9], roller transmission mechanisms of roller machines with variable center-to-center distances of the working shafts are studied insufficiently [10]. A roller transmission mechanism of roller machines with a variable center distance of the working shafts, transmitting torque from one working shaft to another, must provide, among other conditions, one basic condition -to ensure the rotation of the working shafts identical in value and in direction of the linear velocities of the contact point of these shafts with the processed material, both with a constant and changing center distance of the working shafts. However, many roller machines do not meet these conditions; this ultimately leads to deterioration in the processed material quality, sometimes to their damage, and to a decrease in productivity and durability of the machines [11,12]. This problem can be solved using tooth-lever differential transmission mechanisms (TLDTM). An analysis of the design and research on tooth-lever mechanisms shows that the authors mainly considered the synthesis of tooth-lever mechanisms, in which the periodic rotational motion of the input link was converted into a periodic rotational motion of the output link with a delay, or the periodic rotational motion of the input link was converted into a rotational motion of the output link at variable speed. The weak sides of the methods for the analysis and synthesis of tooth-lever mechanisms, especially when these mechanisms are used as roller transmission mechanisms for roll modules, were pointed out by numerous authors such as N.I. Levitsky, K.V. Frolov, V.S. Karelin and others [13][14][15]. The authors of this study developed [16][17][18][19][20] and investigated [21,22] a transmission mechanism that prevents the above disadvantages in roller machines. The article discusses the synthesis of a tooth-lever differential transmission mechanism with a parallelogram lever contour, in which the rotational motion of the input link is converted into a synchronous rotational motion of the driven link in the opposite direction, but without change in value when the center distance between the driving and driven gearwheels changes.

Methods and Results
In Figure 1 shows the design diagram of a roller pair with a developed TLDTM. From the kinematic and dynamic analysis of this mechanism, conducted earlier, it can be seen that in order to meet the main condition, when designing a two-roll module with the same diameters of work rollers, it is necessary [22][23][24]: a) The tooth contour of the toothed-lever mechanism should consist of four gearwheels if the gears have external mesh. b) The number of teeth of the gearwheels must be identical or pairwise identical in all wheels, two idle gears and a driven with a driving one. c) One of the lever-type contours of the tooth-lever mechanism must be a parallelogram. d) The second lever-type contour must be axial one. e) The mechanism must ensure a change in the center distance of the working shafts during the technological process by the value of 1 W , and during the maintenance and repair work of the roller machine by the value of 2 W .
f) During the technological process, the angles of pressure between the lever links should be within the permissible range. g) The diameters of the tip circles of the driven and driving gears must be less than the minimum diameter of the working shafts to ensure the guaranteed pitch play between the teeth tips of the gearwheels. Let us conduct a synthesis of such a mechanism. We assume that it is necessary to design a roller machine with the diameters of the working shafts W . Based on the above conditions, we can write: driving gearwheels; Δ is the guaranteed pitch play between the teeth tips of the driven and driving gearwheels. The reference point to the teeth tips of the driving and driven gearwheels at the beginning of the design was dictated by the fact that when the center distance of the working shafts changes, the levers supporting the intermediate wheels, experience a load from the inertial force, which depends on the mass and acceleration of the centers of rotation of these gearwheels. Therefore, at a large acceleration of the change in the center distance, it is advisable to take the geometric parameters of the driving and driven gearwheels as maximum as possible, and the diameters of the intermediate gearwheels as minimum as possible in the scheme allowing to provide the value of м ах n АD . .
With the maximum moment transmitted by the gearwheels, we determine the preliminary basic parameters of the gear transmission -the center distance of the gearwheels ) ( p a , the preliminary width of the gearwheels ) ( p в , the preliminary modulus of the tooth ) ( p m , the gear ratio ( u ), the coefficient of the width ( а  ), and the angle of inclination of the tooth line (  ) assuming, that the driven and driving gears are in mesh [25]. The gear ratio u is determined by dividing the total gear ratio by stages. In our case 1  u , the width factor ( а  ) is chosen according to the instructions given in [26].
where К = 315 for spur gear pairs;   N  is the permissible contact stress; ' a T is the estimated moment; T is the greatest moment of a normally operating technological process, the opening of cylindrical gears is used where the speed of the gearwheel is less than 1 m/s. The opening of cylindrical gears is narrow with a coefficient of width  (9) in this case, the load distribution factor  N K and the dynamic factor ND K included in the formula (8) are considered equal to one. One of the features of open gears calculation is their intense wear. Therefore, they are made from normalized or heat-hardened steel. The permissible wear of open gears is up to 25% of the original thickness, in terms of the pitch circle. In this case, the bending strength is double less [26]. Therefore, the permissible stress is halved, therefore, formula (9) Since the diameters of the teeth tips of the driven and driving gears are given, we can determine the preliminary pitch diameter of these gearwheels we can also determine the preliminary number of teeth The number of teeth p Z is rounded down to a whole number Z and the preliminary module of the tooth is determined as Consider the lever contours of the TLDTM. Figure 2 shows the design scheme for the synthesis of the lever contours of the developed TLDTM. When designing a mechanism, it is necessary to take into account a very important parameter -an angle of pressure that characterizes the condition for the transfer of forces and the operability of the mechanism. The maximum value of the angle of pressure must not exceed the permissible value, that is In the considered transmission mechanism (Figure 1), the lever part consists of two contours: 1) Rocker-slider contour; 2) Lever four-link unit (a parallelogram). In the first contour, the leading link is slider 7, and the driven link is rocker 5. When slider 7 moves from the driving gear, the angle of pressure in the kinematic pair D is в The design scheme shows that we can determine the angle of position of the mechanism In the second contour, the leading link is lever 5, therefore, the angles of pressure in the kinematic pair C between links 5 and 4 are   ( Equating the right-hand sides of formulas (22) and (27)  Subtracting formula (32) from formula (33) and taking into account formulas (35)   Next, we determine the length of the arms and the maximum value of the roller distance