FORCE ANALYSIS OF GEAR-LEVER DIFFERENTIAL TRANSMISSION MECHANISM OF A MACHINE WITH SHAFT Текст научной статьи по специальности «Механика и машиностроение»

UDC. 621

FORCE ANALYSIS OF GEAR-LEVER DIFFERENTIAL TRANSMISSION MECHANISM OF

A MACHINE WITH SHAFT

Abdullajonov Asrorbek Abduraxmon o'g'li Namangan Engineering-Construction Institute Email: [email protected]

Annotatsiya: Maqolada turli diametrga ega valli mashinaning ishchi vallari uchun mo'ljallangan tishli-richagli differensial uzatish mexanizmi nazariy tadqiq qilingan. Tadqiq qilinayotgan tishli-richagli differensial uzatish mexanizmi ishchi texnologik mashinaning turli diametrli, masalan yuqori vali kichik va ostki val katta bo'lgan holda ularning aylanishidagi chiziqli tezliklarini tengligini ta'minlaydi. Bunda ikki ishchi vallar orasida ishlov beriladigan mahsulotning bir tekisda harakatlanishini hamda ta'sir etishning yuqori aniqligini ta'minlab beradi. Ushbu tishli-richagli differensial mexanizmining asosiy, ya'ni ishchi vallariga, ular mahkamlanadigan richaglarga, tishli g'ildiraklariga ta'sir etuvchi kuchlar aniqlangan hamda tahlil qilingan. Natijada tishli-richagli differensial mexanizmiga ta'sir etuvchi kuchlarning kuch hisobi bajarilgan va tahlili keltirilgan.

Аннотация: В статье теоретически исследован зубчато-рычажный дифференциальный механизм, предназначенный для рабочих валов машины с валами разного диаметра. Исследуемый зубчато-рычажный дифференциальный механизм обеспечивает равенство линейной скорости вращения рабочей технологической машины с разными диаметрами, например, верхний вал малый, а нижний вал большой. Он обеспечивает равномерное перемещение обрабатываемого изделия между двумя рабочими валами и высокую точность удара. Определены и проанализированы основные силы данного зубчато-рычажного дифференциального механизма, т. е. силы, действующие на рабочие валы, рычаги, к которым они прикреплены, и зубчатые колеса. В результате выполнен силовой расчет сил, действующих на зубчато-рычажный дифференциальный механизм, и представлен анализ.

Abstract: In the article, the gear-lever differential mechanism designed for the working shafts of the machine with shafts of different diameters is theoretically researched. The researched gear-lever differential mechanism ensures the equality of the linear speed of rotation of the working technological machine with different diameters, for example, the upper shaft is small and the lower shaft is large. It ensures uniform movement of the processed product between two working shafts and high accuracy of impact. The main forces of this gear-lever differential mechanism, i.e., the forces acting on the working shafts, the levers to which they are attached, and the gear wheels have been determined and analyzed. As a result, the force calculation of the forces affecting the gear-lever differential mechanism was performed and the analysis was presented.

Kalit so'zlar: valli mashina, uzatish mexanizmi, richaglar, tishli g'ildiraklar, aylanish tezligi, aylanish tezlanishi, kuch hisobi.

Ключевые слова: валковая машина, передаточный механизм, рычаги, зубчатые передачи, частота вращения, ускорение вращения, силовой расчет.

Keywords: shaft machine, transmission mechanism, levers, gears, rotational speed, rotational acceleration, force calculation.

Introduction. Dynamics problems can be divided into two main types. These problems are as follows for a free material point:

In the first fundamental problem of dynamics, given the mass of a material point and its law of motion, it is asked to find the driving force.

The second main problem of dynamics is to determine the kinematic elements formed by the effect of this force when the mass of a material point and the force acting on it are known [1].

In the technique, it is necessary to solve many problems related to the verification of the motion of a material point involuntarily (in contact). In such cases, a bond placed on a point forces it to move on a fixed surface or line [2].

Taking into account the properties of various products, the work of scientists is dedicated to researching methods of processing them [3, 4].

In [5-11], the authors developed designs and studied the parameters of roller technological machines. The rational parameters of working bodies and actuators of machines were theoretically determined. The technological parameters of mechanical processing of fibrous materials, using raw hides as an example, were experimentally determined and scientifically substantiated.

We can see the scientific work carried out on the transmission mechanisms developed for machines with variable shaft interracial distance of the working shafts. 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 tooth lever 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 of 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 [12, 13, 20].

The transmission mechanisms of cotton raw material pretreatment machines are composed of gear and chain mechanisms and other devices. The following works are dedicated to their research [14-16].

Materials, methods, and objects of study. When solving problems related to the movement of a free material point, this point is freed from the bond and replaced by the imposed bond reaction force. As a result, the main equation of the dynamics of a material point is written as follows:

where N is the binding reaction force.

So, in the first basic problem of the dynamics of an involuntary material point, the reaction force is determined when the mass of the material point and its law of motion and the force acting on this point are known; and in the second problem, when the mass of the material point and the force acting on it are known, it is necessary to determine the reaction force of the material point by the law of motion [21-26].

Since the bonds are ideal, the reaction forces are zero. Equation (2) takes the following form

The general equation of dynamics can be written as follows through its projections on coordinate axes

ma = F + N

(1)

ma = F

(3)

mX = Fx my = Fy

here F x , F y are projections of the force on the OX , OY coordinate axes

(4)

(5)

Q'

\fJJ0— D

. h , P . 1 109 \

10 /

Figure 1. - Calculation scheme of the balance of the driven gear wheel with the driven working shaft

1 - we take the projections of the forces acting on the scheme on the OX , OY coordinate axes and put them in equations (4) and (5) [17-19]:

mX = Ril "P109 C0S (Pn - W) (6)

my = -Q + T - P109 sin( Pn -W) (7)

Integrating equations (6) and (7) once over time, we determine the projections of the speed on the Vx , Vy axes

l t

Vx = X = — I (R61 - P109 COs(Pn - W))dt

vn

Vx = X = — (R61 -P109sin(^„ -y))t m

1 t

Vy = y = - J (- Q + T - P109 sin(^„ - y))dt

s Wt J

1

Vy = y = — (-Q + T -P109 cospPn - W))t -P109 m

We determine the full speed using the defined expressions (8) and (9).

1

V10 = -\ /Vx2 + Vy2 =

(R61- P109sin(^« -Y))t I +

+ | -(-Q + T -P109COs(p„ -Y))t - P109

m

(8)

(9)

(10)

Using equations (6) and (7), we determine the projections of the acceleration on the coordinate axes ax, ay

ax = X = — (R61 -P109COs(fPn - Y)) m

ay = y = — (-Q + T - P109 sin(Pn - W)) m

We determine the total acceleration using expressions (11) and (12).

(11) (12)

R

61

0

0

m

2

a10 + al = ^/(^61 - P109cos(pn +(- Q + T - P109sin(pn

(13)

We take the projections of the forces acting on the scheme in Fig. 2 on the coordinate axes OX and OY and put them in equations (4) and (5).

Pn-y

Figure 2. - 9- calculation of the balance of the intermediate gear wheel

mx = Rg4cos(90-<n)-U9cos£-P910cos(<n -y)-P,8cosy (14)

my = R94 sin(90-<n )-U9sin£ + P,10sin(<n -y) - P98siny (15)

Integrating equations (14) and (15) once over time, we determine the projections of the speed on the Vx , Vy axes

1 1

Vx = x = — i (R94 cos(90 - <n) -U9 cos£ -P910cos(<pn - y) -P98 cosy)dt

tvi J

Vx = x = — (R94sin(90-pn)-U9sin£-P910sin(pn -y)-P98siny)t m

1 t

Vy = y = — {(R94 s in(90 -Pn) - U9 s in £ + P910 sin(Pn - y) - P98 s in y)dt

K = y =

1 (R94cos(90 -Pn)-U9cos£- J m P910 cos(Pn -y) -P98 cosy

t + R94 U9 + P910 P98

(16)

(17)

We determine the full speed using the defined expressions (16) and (17).

V9 = Jv2 + V2 =

^1 (R94sin(90 -Pn)- U9 sin£ -P910sin(Pn -y)-P98 siny)tj +

m

R94 cos(90 -pn) -U9cos£- ^

(18)

t + R94 U9 + P910 P98

P)10 cos(<n -y)" p98 cosyy

Using equations (16) and (17), we determine the projections of the acceleration on the coordinate axes OX , OY

ax = x = — (R94 cos (90 -<n) - U9 cos £ - P910 cos (<n - y) - P98 cos y) m

ay = y = — (R94 sin(90 - <n) - U9 sin £ - P910 sin(<n - y) - P98 sin y) m

We determine the total acceleration using expressions (19) and (20).

1 2 1 2 a = 1 la,. + ay =

^1 (R94 cos(90 - Pn) - U9 cos£ - P910 cos(Pn - y) - P98 cosy)jj + + [ m (R®4 sin(90 -Pn ) - U9 sin£ - P910 sin(Pn - y) - P98 siny)J

(19)

(20)

(21)

0

0

2

+

3 - we take the projections of the forces acting on the scheme on the OX , OY coordinate axes and put them in equations (4) and (5)

mx = R89 cosy - U8 cosy - R83 cos(90 - —) + R87 cos(pn + y)

v

my = R89 sin y - U8 sin y - R83 sin(90 - —) + R87 sin(pn + y)

(22) (23)

Figure 3. - 8- the calculation scheme of the balance of the intermediate gear wheel

(22) and (23) once over time, we determine the projections of the speed on the axes Vx and Vy

i t

if v

Vx = x = — i (R89cosy- U8 cosY~R83cos(90 —) + R87cos(p„ + \))dt

m J 1

1 V

Vx = x = — (R89 sin Y - U8 sin Y - R83 sm(9° - ^ + R87 Sin(Pn + Y))t m 2

i t

1 f v

Vy = y = — I (^89 Sin Y - U8 Sin Y - R-83 Sin(90 - -) + R87 sin(Pn + Y))dt 7 m^ 2

1 V

Vy = y = — (R89SinY -U8 SinY-R83Sin(90 - -) + R87 s mp +Y))t-m 2

-R89-U8 -R83 + R87

We determine the full speed using the defined expressions (24) and (25).

1 V

—(r89sinY-u8SinY-rS3sin(90 —) + r87sin(pn +y))11 + m 2 1

(24)

(25)

v =jvx2 + v2 =

(i v ^

—(r89sinY-U8 sinY-r83sin(90 -—) + r87sin(pn + \))t-m 2

v- R89 - U8 - R83 + R87

(26)

Using equations (22) and (23), we determine the projections of the acceleration on the coordinate axes OX , OY

11 " " " - V ~ ' '' (27)

(28)

ax =—I R89cos\-U8cos\-R83cos(90 —)+r87cos(p„ + \) m V 2

i ( v

av =—I R89sinY-U8sin y-R83sin(90 —) + R87sin(p„ + \)

m \ '' 2

We determine the total acceleration using expressions (27) and (28).

, 2 , 2 a8 = A K + ay =

i V

—(R89cosy-U8cos\-R83cos(90 —) + R87cos(p„ +\)) I + m 2

2

( i v

+ I — (R89sin\-U8sin\-R83 sin(90--) + R87sin(p„ +\))

V m 2

2

(29)

4 - we take the projections of the forces acting on the scheme on the OX , OY coordinate axes and put them in equations (4) and (5)

mx = R71cos(pn + y) - P78c°s(pn + Y) (30)

0

2

+

my = r71 sin(<n +y)- P78 sin(<n + y)- G (31)

Integrating equations (30) and (31) once over time, we determine the projections of the speed on the Vx and Vy axes

1

Vx = x = — i (R71 cos(Pn + y) - P78 cos(Pn + y)) dt

vu J

Vx = x = — (R71 sin(Pn +y)-P78 sin(Pn +y))t m

M,„

(32)

Figure 4. - Scheme of calculating the balance of the driving working shaft and the leading gear

wheel

1 t

Vy = y = — i (R71 s in(Pn + y) - P78 s in(Pn + y) - G)dt mi

1

Vy = y = — (( R71 cos (<n + y) + P78 cos (<n + y) - G)t + R71 - P78) m

Results . We determine the full speed using expressions (32) and (33).

V = , K2 + Vy2 =

m

(R71 sin(Pn + y) - P78 sin(Pn + y))t I +

+1 — ((-R7! cos(Pn + y) + P78 cos(Pn + y) - G)t + R7j - P78) m

(33)

(34)

Using equations (30) and (31), we determine the projections of the acceleration on the coordinate axes OX , OY

ax = — (R71 cos(Pn + y) -P78 cos(Pn + y)) m

= — (R71 sin(Pn +y) - P78sin(Pn +y) - G)

-y Vr71

m

We determine the total acceleration using expressions (35) and (36).

i 2 i 2 a7 = /ax + =

1 V

- (R71 cos(Pn +y) - P78cos(Pn +y))J +

rn J

+ [ m (R7rn sin(Pn + y) - P78 sin(Pn + y) - G)Jj

(35)

(36)

(37)

Discussion. Formulas (10), (18), (26), (34) determine the full speed of the four gear wheels of the mechanism. Formulas (13), (21), (29), (37) determine the full acceleration of the four gears of the mechanism. As a result, the gear wheels of the differential transmission mechanism with different diameters are considered as material points, and the main equation of their dynamics was developed. Based on the second problem of dynamics, the developed equations determined the

0

2

1

2

reaction force with the law of motion of the material point when the mass of the material point and the force acting on it are known.

Conclusion. The kinetostatic analysis of the gear-lever differential transmission mechanism with different diameters of the leading and driven gears intended for the working shafts of the shaft machine consists in determining the reaction forces falling on the joints, and is used in the synthesis and design of this mechanism. is used.

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