Kinematic analysis of a new harrow type wool transport mechanism Текст научной статьи по специальности «Математика»

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Kinematic analysis of a new harrow type wool transport mechanism Kayumov J.1, Castelli V.2, Isaxanov H.3, Kozokboyev D.4, Norinova R.5 (Republic of Uzbekistan, Italian Republic) Кинематический анализ нового типа боронного механизма в шерстомоечной машине Каюмов Ж. А.1, Кастелли В. П.2, Исаханов Х.3, Козокбоев Д. Х.4, Норинова Р. О.5 (Республика Узбекистан, Итальянская Республика)

1Каюмов Журамирза Абдираматович /Kayumov Juramirza - доктор философии, старший преподаватель, кафедра технологии и конструирования изделий легкой промышленности, Наманганский инженерно-технологический институт, г. Наманган, Республика Узбекистан; 2Кастелли Винчензо Паренти / Castelli Vincenzo - доктор технических наук, профессор, кафедра промышленной инженерии, Болонский университет, г. Болонья, Итальянская Республика; 3Исаханов Хамидулла /Isaxanov Hamidulla - кандидат технических наук, доцент, кафедра общей технической науки; 4Козокбоев Дониёр Хабибулло угли / Kozokboyev Doniyor - студент; 5Норинова Рахнома Одилжон кизи / Norinova Rahnoma - студент, факультет технологии легкой промышленности, Наманганский инженерно-технологический институт, г. Наманган, Республика Узбекистан

Abstract: this paper presents the scientific results of the kinematic calculation and the law of motion of a new wool transport mechanism to reduce fiber entanglement. Аннотация: в этой статье даны научньк рeзультаты кинематического расчёта и закона движeния боронного мeханизма в новой шeрстомоeчной машине с учётом снижeния запутанности шeрстяного волокна.

Keywords: four bar linkage, positions analysis, velocity analysis, acceleration analysis, wool transport mechanism, kinematic analysis

Ключевые слова: четырёхзвенный механизм, анализ позиций, анализ скоростей, анализ ускорения, боронный механизм, кинематический анализ.

Synthesis of a new harrow type wool transport mechanism has presented in [1]. This paper dedicated for the kinematic analysis of the proposed wool transport mechanism.

The four bar of the proposed mechanism shown in Figure 1. The crank O1A is fixed on pivot O1 that is located at the origin of the coordinate system. The crank rotates counterclockwise with angle about fixed pivot O1. The point A of the coupler ABC rotates follows to the crank rotation. The rocker or lever O2B move in oscillation motion with respect to the coupler motion. The output link of the mechanism is the harrow and it connects with its one end to the point C. The desired motion of the mechanism has been obtained by point C.

W2fj Oi

Fig. 1. Vector representation of the four bar linkage [2]

1. Position analysis

The position analysis of the four bar linkage starts from defining the coordinates of centres of the revolute joints in a coordinate system. A reference system S1 (x, y, O1) with x axis or horizontal and the origin of the system at point O1, centre of the revolute joint connecting links 1 and 2 are chosen. Coordinates of the pivots O1 and O2 can be written as follows:

Oi = K yOl] = [0 0] ; (1)

(2)

The point A rotates about the origin of the coordinate system S1; its coordinates can be written as:

A = [XA yA] = Oi + l2 [ cose2 sine2]; (3)

Coordinates of point B:

Referring to the Figure 10, if the coordinates of the points A and O2 are known, the coordinates of the point B can be found by using the equation for circle as follows:

(*b - xo2)2 + (yB ~ yo2f = if

B(x B yB);

(4)

(;xB - xA)2 + (yB- yA)2 = ll The Equation (4) should be developed, simplified and one subtracted to the other as follows:

x| - 2xBx0 + Xq + y| - 2yBy0 + yl - I

1 = 0

xB

2xbxa + %i + y% - 2yByA + yi - l£ = 0 - 2 x (x02 - Xa) + xl2 x A 2y(y 02 - yA) + yl2 - l\~yl + l2 = 0; Here is

,yo2-yA _ *Q2-*i+yo2-yj + '3-'I _ N(Q2)2-N(A)2 + ll-ll x02-xa 2 (x02-xa) 2 (x02-xa)

The Equation (7) can be simplified as:

(5)

x + y-

yo2~y A xq2 xa

= A

(8)

S

n(o2)2-n(a)2+li-l% _ b 2 (x o 2~ X a)

The following simple equation can be obtained for x:

x = - Ay + B; (10)

The Equation (10) can be inserted into the first line of the Equation system (5) and can be written as follows:

A2 y2 +B2- 2 A By + 2 x02Ay - 2 x02B + x2 2 + y2 - 2 yy 0 2 + y22 - l\ = 0 ; (11) (A2 + 1 ) y 2 + (2Xo2A - 2AB - 2yo2)y + (B2 - 2Xo2B + N( O2) 2 - l2) = 0 ; (12)

a = (A2 + 1); (13)

b = (2Xo2A - 2AB - 2yo2); (14)

c = B2 - 2xo2B+N(O2)2-12, (15)

ay2 + 2by + c = 0; (16)

y = -b±^1, (17)

Coordinates of point C can be found as: a[C = a[A + AC;

C = l2(cosd2sind2) + Z5(cos(03 sin(03 -/?)); Xc = l2cos82 + l5cos(83-p)\ Yc = I2S in 62 + ls S in (e3- p) (18)

Here is, p = cos~ 1 Vl^tl], (19)

L 2¿5¿з J

Fig. 2. The position analysis of the proposed mechanism [2]

Coordinates of the pivot O3 can be found as follow:

O3 = [xo3 y o3] = O2 + 111 [cos611 sin 61J; (20)

Coordinates of the point E:

E = B + l7 [co s67 sin67]; (21)

Coordinates of the point D:

(22)

Loop closure equations for four bar linkage can be written as follow:

h + h+h + l 1 = 0 ; (23)

Rewriting the Equation (23) in its x andy axis component equations:

(24)

l2s in 62 + l3s in 63 + l4s in 64 + ^s in 61 = 0 ; (25)

We know that 62 is known, and 61 is also known, constant. In order to eliminate 63, we first isolate it on one side of the Equations (24) and (25):

(26) (27)

Both sides of the equations (26) and (27) should be squared, added and the result simplified using the trigonometric identity

This gives:

q = q + l2 + l2+ 2lil4(cos9icos94 + sin91sin94) —

; (28)

The Equation (28) gives 94 in terms of the given angle 92 (and constant angle 91 ), but not explicitly. To obtain explicit expression, the Equation (28) should be simplified by combining the coefficients of and as follows:

A cos94 + Bs in 94 + C = 0; (29)

Where,

A = 21114cos01 — 21214cos02;

B = 2l^sinS 1 - 2l2l4sin0 2; l (30)

C = If + \\ + 1| - 1| - 21112(COS01COS02 + sinOiSinOz)J

To solve the Equation (29), standard trigonometric identities can be used for half angles given in the following:

s in 9 JZupVL; (31)

4 l+tan2(04/2) v '

c o s94=1^^ll; (32)

4 l+tan2(04/2) v '

After substitution and simplification, the following equation can be obtained:

(33)

Where, t = tan ;

Solving for t gives:

_ -2B+a^AB2-A{C-A){C+A) _ -B+t]B2-C2+A2 _ = 2 (C-A) = C-A ; ( )

And (35)

Equations (26) and (27) can now be solved for 0 3. Dividing the Equation (27) by (26) and solving for 0 3 gives:

„ f— l4sin64—l2sin62 — lisinGil

93 = tan 11-I; (36)

L— i4cos04 — l2cos62—licos01i

In equation (34) it is essential that the sign of the numerator and the denominator be maintained to determine the quadrant in which the angle lies. This can be done directly by using ATAN2 function. The form of this function is:

A T A N 2 (s i n 9 3, co s 9 3 ) = tan - 1 f^-1; (37)

Equations (35) - (37) give a complete and consistent solution to the position problem of the four bar linkage. For any values of 92, there are typically two values of 93 and 94, given the substituting c = + 1 and - 1 , respectively.

2. Velocity analysis

The velocity equations can be developed by differentiating the Equation (23) as:

h + h + U + h = 0 ; (38)

Rewriting the Equations (38) in its x and y axis component is the same as that differentiating the Equations (23) and (24). The resulting equations are:

- l2sin92co2 - l3sin93co3 - l4sin94co4 - ^sin9-¡^co 1 = 0 ; (39)

(40)

Since, 91 is constant, the angular velocity of link 2 cc2 is known, the only new unknowns are and which are the angular velocities of the link 3 and the link 4 respectively. In matrix form, Equations (39) and (40) can be rearranged and rewritten as: \-l3sin93 l4sin94 rw3| _ | l2sin92a)2 j l3cos93 -l4cos94\ I CO4J \-l2cos92a>2y ( )

Solving these two equations in two unknowns yield:

l2SLn820>2--2-1 ■ a-

C04 —--

¿4 ¿>¿3204 •

l^COSd^-l4Sind4

l2Sin02O)2

l3sin6 3

l^COS6^-l2Sin62C02

hsind 3

Z4sm04-

iOo = —-

¿3CQ503-Z45tn04 " l3Sin6 3

l^sinO^

Velocity equations of the coupler point C can be found by differentiating the Equation (18) as:

C = l2M2{-sin62 cos62) + l5oj3(-sin(d3 - /?) cos(03 - /?)); = -l2sin82a)2 - lssin(83 - P)oj3 ( Yc = 12cos82co2 + l5cos(63 - /3)co3 3. Acceleration analysis

Since Oi is constant, the acceleration equations can be developed by differentiating the Equation (38) as:

(44)

h + h ~ h = 0;

(45)

Rewriting the Equation (2.45) in its x andy axis component equations is the same as that differentiating the Equations (2.39) and (2.40). The resulting component equations are: -lysinOyay —lycosOyco2 — l3sinO3a3 — l3cosO3col+l4sinO4a4 + l4cosO4co2 = 0; (46)

l2coso2a2—i2suio2uj2 ~r l3coso3a3 — l3sinO3w3 — l4cosO4a4 + l4sinO4a)4 = 0; (47)

2ZUIV2U.2 L2UUCU2UJ2

l2cos62a2—l2sin62M2 + l3cos83a3 These equations can also be represented in matrix form, where the terms associated with the known crank acceleration and the quadratic velocity terms are moved to the right-hand side as:

■l3sind3 l4sind4 1 ra3| _ (l2(sin62a2 + cos82m2) + l3cosd3a>3 — l4cosd4a>4 ) :os93 - l4cos64\ Iaj ~ [-l2(cos62a2 - sin92(i>l) + l3sin93io% - l4sin94aj|J Solving first raw of these equations yield two unknowns as:

(48)

a, = ■

(l2(sin92 + --'

l±sin9.—

l3 t'osi)11 sin 01 l3sinB3

i3sin83

(49)

■ (50)

Acceleration equations of the coupler point C can be found by differentiating the Equation (44) as:

Xc = —12 (sin 62 a2 + cos 62 a>2) — l5 (sin 65 a3 + cos 65 a>3) Yc = l2(cos d2a2 — sin d2oj2) + l5(cos 9S a3 - sin 9S 4. Results

Results of the kinematic analysis are shown in Figures 3-10.

(51)

Fig. 3. Simulation of the proposed mechanism

Driving link angle, teta2(rad)

Fig. 4. Angular displacement, velocity and acceleration of the coupler

Fig. 5. Angular displacement, velocity and acceleration of the rocker (link 4)

47

Fig. 6. Linier displacement, velocity and acceleration of the point C

Fig. 7. Angular displacement, velocity and acceleration of the long connecting rod (link 5)

Angular displacement of ihe harrow

2

£ of-

to

CD

a. .

I I 1 1 1 1 1 1 1 . • 1 1 1 1 I I 1 1

1 '

D 1 2 3 4 5 Driung link angle, telaSirad) Angular velocity of the harrow 6 7

I III i i i i ■■—.. i i i 1

J 1 i 2 3 4 5 Drung link angle, lela2(rarf) Angular acceleration of fianow i .i — i ■ i 6 7

-2l

2 3 4 5

Drhing link angle, tcto2(rad)

Fig. 8. Angular displacement, velocity and acceleration of the harrow (link 6)

Driving link angle, tela2(rad)

Fig. 9. Angular displacement, velocity and acceleration of the short connecting rod (link 7)

Fig. 10. Angular displacement, velocity and acceleration of the rocker 2 (link 8)

References

1. Kayumov J. A., Castelli V. P, Isaxanov X., Kozokboev D. X, Norinova R. O. Synthesis of a new harrow type wool transport mechanism. X International Scientific and Practical Conference «International Scientific Review of the Problems and Prospects of Modern Science and Education» Boston. USA 7-8 February 2016. p. 33-39.

2. Sharma C. S. and Purohit K. Theory of Mechanisms and machines, Prentice-Hall of India Private Limited, New Delhi, 2006.

3. Kayumov J. A. «Design of a new harrow type wool transport mechanism to reduce fiber entanglement». Doctorate thesis. University of Bologna. 2015.