Кinematic analysis of a new gear-lever differential transmission mechanism with symmetrical displacement of the centers of rotation of the driven and driving gears

ABSTRACT

One of the main directions in the development of domestic machinery is the improvement and creation of high-performance, resource-saving devices and mechanisms of rolled technological machines. This is especially common in machines with a symmetrical moving roller, the center of rotation of the free-running shaft, and is therefore a very important issue in this field of mechanical engineering.

АННОТАЦИЯ

Одним из основных направлений развития отечественной техники является совершенствование и создание высокопроизводительных, ресурсосберегающих устройств и механизмов прокатных технологических машин. Это особенно часто встречается в машинах с симметричным движущимся роликом, центром вращения свободно вращающегося вала, и поэтому является очень важной проблемой в этой области машиностроения.

Keywords: link, plane, convulsive, power, linear, rings, lever arm.

Ключевые слова: звено, плоскость, судорожный, степень, линейным, колец, рычаг.

Below is considered one of the options developed by us toothed-lever differential transmission mechanisms, which can be used in roll machines with a variable center distance of the working shafts, where the axes of both working shafts have the ability to rotate around their own axes and move symmetrically along a straight line passing along the axes rotation of these working shafts.

Figure 2.1. Structural - kinematic diagram of a roller machine with a gear - lever differential transmission mechanism

1,2 - working shafts; 3.4 - tires of the working shafts; 5.6 - driven and driving gears; 7,8,13 - levers; 9.10 - axles of intermediate gear wheels; 11.12 - intermediate gear wheels; 14-18 – sliders

Figure 2.1. the structural and kinematic diagram of a roller machine with a developed transmission mechanism is shown

A mechanism installed on a two-roll technological machine, for example, on a machine for squeezing wet leather, consisting of two working shafts 1 and 2 (Figure 2.1) on the surface of which tires 3 and 4 are put on, where both working shafts have the ability to rotate around their own axes O1 and О4 and move using levers 15, 16, 17 and 18, in a straight line passing through the axes of rotation (О1 and О4) of the working shafts (1, 2) consists of: driven and driving gears 5 and 6 rigidly mounted at the output ends of the working shafts 1 and 2 and pivotally mounted levers 7 and 8, which are supports for axes 9 and 10. On axes 9 and 10, intermediate gears 11 and 12 are installed. The axes of rotation of the intermediate gears are kinematically interconnected by a lever 13, which is located parallel to the line passing along the axis of rotation of the working shafts 1, 2. The lever 13 in its middle is rigidly fixed with the slider 14 at a right angle.

The mechanism works as follows: the torque from the working shaft 1 to the working shaft 2 is transmitted by means of the gear wheels of the drive 6, intermediate 11, 12 and driven 5. When the processed material with variable thickness passes between the working shafts 1 and 2, the center distance О1О4 changes (Fig. .one). In this case, the slider 14 provides the lever 13 (O2O3) with a movement perpendicular to the line passing through the axes of rotation (O1 and O4) of the working shafts 1 and 2. The system of levers and gear wheels provide synchronous movement and rotation of the working shafts 1 and 2, since the axes of rotation of the intermediate gears (O2 and O3 i.e. 9 and 10) will receive a relative speed equal in magnitude and parallel in direction, and intermediate gears will receive angles of rotation equal in magnitude and opposite in direction, thereby compensating for the angles of rotation of each other.

Consequently, with a change in the center distance O1O4, there is no difference in the angles of rotation of the gear wheels of the driven 5 and the driving 6, and the synchronization of rotation of these gears is ensured. Since the gear wheels, driven 5 and driven 6, are rigidly fixed at the output ends of the working shafts 1 and 2, these working shafts will also rotate synchronously.

The kinematic analysis of the above-described mechanism is carried out in order to determine the angular and linear velocities and accelerations of the links of the mechanism depending on the angular speed and acceleration of the driving link (gear 5 or 6) and the linear speed and acceleration of the centers of rotation of the driven links (gear 5 or 6), and also to prove the equality of the linear velocities of the surfaces of the working shafts at the points of contact of these shafts with the material being processed.

Figure 2.2 shows the design diagram of the kinematics of the mechanism under consideration, figure 2.3 shows the design diagram of the kinematics of the lever contour of this mechanism.

In the considered mechanism, the diameters of the driving and driven gears and parasitic gears are equal in pairs or the diameters of all gears are equal to each other. Thus, the transmission mechanism we are considering are special cases of some generalized transmission mechanism. Synchronous rotation of the driven and driving gears (working shafts 1 and 2) at the time of changing the center distance of the working shafts is performed under the condition

,                                             (2.1)

or

                                      (2.2)

and with a plane-parallel movement of the lever relative to the line passing through the points and (Figure 2.1).

1, 4 - driving and driven gear wheels; 2, 3 - intermediate gear wheels; 5, 6, 8, 9 - levers; 7 - slider.

Figure 2.2. Design diagram of the kinematics of a gear-lever differential transmission mechanism with a symmetrical movement of gear wheels

The laws of motion of the levers 5 and 9 are determined by the vibration equation, which depends on the change in the thickness of the material being processed and, which we will consider given. Also known are angular velocities and angular accelerations in the relative motion of the driving and driven gears. The lever moves together with the slider 7 along the guide 00 perpendicular to the line of the gear wheels, which are rigidly fixed to the output ends of the working shafts (Figure 2.2).

Figure 2.3. Design scheme for determining the speeds and accelerations of the characteristic points of the lever contour

The lever moves together with the slider 7 along the guide 00 perpendicular to the line passing along the axes of rotation of the driving and driven gears. The lever contour of the gear-lever differential transmission mechanism (Figure 2.3) is designed so that the centers of rotation of the master with the driven (,) and the centers of rotation of the intermediate gears (,) move mirror-symmetrically relative to the cross slide 7, therefore, it can be written

,                                                    (2.3)

,                                                     (2.4)

,                                                    (2.5)

.                                                    (2.6)

by virtue of formulas (2.1) - (2.6), we write

         , ,                                      

, ,                                         (2.7)

Where ,,,‒ angular velocities and angular accelerations arising from a change in the center distance of the working shafts ,  hence

Determine the speed and acceleration of points  and . Lever8 ()  moves in a plane-parallel manner. The directions of the velocities (and )  of the two points ( and )  and the speed value is known ()  points    lever  8 (),  therefore, one can find the speed ()  second point (),  having previously determined the instantaneous center of rotation ()  this link (8)  and angular velocity ()  this lever.

.                                                 (2.8)

  is determined from the solution of the triangle  

.                                  (2.9)

Taking into account formula (2.9), formula (2.8) takes the form

.                                    (2.10)

Taking into account formula (2.2), formula (2.10) takes the form

.                                             (2.11)

Angular velocity ()  points   is determined by the formula

.                                           (2.12)

Taking into account formula (11), formula (12) takes the form

.                                       (2.13)

Out of the triangle                              

.                                  (2.14)

Taking into account formulas (2) and (14), formula (13) takes the form

.                                            (2.15)

Point speed    in lever 8 is determined by the formula

,                                             (2.16)

where   is determined by the solution of the triangle ,

.                      (2.17)

Taking into account formula (2.2), formula (2.17) takes the form

.                                                     (2.18)

Hence,

.                                                (2.19)

Point speed  ()  in the gear wheel 4 is determined by the formula

,                                             (2.20)

  is determined by the solution of the triangle

.                     (2.21)

Taking into account formulas (2.21) and (2.10), formula (2.20) takes the form

.                   (2.22)

Taking into account formula (2.2), formula (2.22) will be written

.                                  (2.23)

Point speeds  и  в  master and slave ( и )  gear wheels corresponding to the contact points of the working shafts with the processed material is determined by the formula

.    (2.24)

For a mechanism with the same cogwheels

                (2.25)

Thus, the final angular speeds of the working shafts are determined by the formula,                                                (2.26)

where

.                                                 (2.27)

The (+) or (-) sign is determined depending on the direction..  For roller machines with working shaft diameters D, the speed of the point of contact of the working shafts with the processed material is determined by the formula

       (2.28)

and by the formula

                      (2.29)

respectively, for mechanisms with pairwise identical (2.28) and identical (2.27) diameters of gears.

The acceleration of the characteristic points of the transmission mechanism and the working shafts of roll machines arising from the speed and acceleration of the centers of rotation of the working shafts is determined by the formulas

,                                                   (2.30)

,                                       (2.31)

,                     (2.32)

.                     (2.33)

For mechanism with pitch diameters ,

,        (2.34)

,   (2.35)

,         (2.36)

.    (2.37)

For mechanism with pitch diameters

,                         (2.38)

,                       (2.39)

,               (2.40)

.                (2.41)

The results of the kinematic analysis show that the rotational and symmetric displacement of the driving and driven working shafts was determined at the same angular velocities. as a result of changing the axial distances of the differential transmission mechanism and the drive gears, their gear ratios also change, which creates a geometric displacement between the working shafts and the workpiece. Thus, taking into account the results of the analysis performed, it is possible to reduce the geometric displacement between the working shafts and the workpiece by using the proposed transmission mechanism.

List of literature:

1. Abdukarimov A. Analysis and synthesis of transmission mechanisms of roll machines with variable center distance of the working shafts: Dis. ...Cand. tech. sciences. -Tashkent: IMSS, 1995.-158 p.
2. Usmonxo'jaev H.H. Mechanism and machine theory. - Tashkent: Teacher, 1970. –576
3. Jo‘rаyеv А.D. Mechanism and machine theory. Tashkent .: G. Gulom Publishing House, 2004. - 592 p