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23
Section 7. Mechanics
Matmurodov Farkhod Matkurbonovich, the head of the fundamental project of the Tashkent branch of the Russian state university of oil and gas of I. M. Gubkin E-mail: matmurodovfarhod@yandex.com
Abdullaev Abdukhakim Nigmatovich, associate professor of the Tashkent state agricultural university E-mail: abdullaevan@mail.ru
Dynamic model of the inertial hydrodifferential transformer working process of the rotating moment in various automatic machine drives
Abstract: Working process of the inertial hydrodifferential transformer of the rotating moment of infinitely variable transmission is investigated. Dynamic mathematical model for the 4 mass system with gidrodifferentsialyis prepared. The equations are solved using the method of final differences, and the calculation algorithm is proposed.
Keywords: mechanical infinitely variable transmission; inertia moment; power drive; inertial hydrodifferential transformer; dynamic mathematical model.
The development of contemporary mobile technics is based on the aspiration to create and apply the infinitely variable transmission that could provide machineswith specific advantages, namely increase in average movement speed and decrease in fuel consumption. On application of automatic infinitely variable transmission, the work of the driver could therefore be also considerably facilitated.
To our knowledge, currently the mechanical infinitely variable speed transmission within inertial-pulse regulationis promising for a number of inherent advantages such as automaticity and infinite speed and torque control to the driving wheels of the machine with high lifting efficiency, the ability to protect the motor against overload, simplicity and ease of operation [1, 21-22].
However, this type of transmission has not been widely used in the industry due to the failure of the rectifier inertial torque via depreciation and fracture of freewheel mechanisms. In order to improve the reliability of transmission, there has been implemented the development of original scheme of hydrodifferential rectifier moment, where hydraulic fluid is used for wedging, and two hydraulic machines are permanently fixed by the use of differential series, which simplifies the design of the hydraulic valves.
We investigate the workflow of hydraulic differential transformer of infinitely variable speed transmissioninertial torque.
The dynamic mathematical model for 4-mass system with hydraulic differential is modelled as:
- - A (( -(Pv )2 + Ae\ = Ma;
Al(Pll - A&23 + Ai(P21 = -J1MT! ± M1;
(P21 (1+k )- (P23 k
= -J2MT2 ±M
(1)
J(& =-Mc ± M1 ± Mr From equation (1) we see that the links between the first equation to the second and the links between third equations of the fourth equations are independent. They will be solvedindepen-dently. While solving (1) within the second two equations, we find Herewe solve simultaneous equations based on (1) the first two ones.
The solutions of other two equations are a little bit complicated. We remove squares.
¡AcPn - A2<Pi3 - A,<p^ + 2A,<pll<p1,- A,<p+ A<<\3 = M;
Wil - A Pv + Ai<Pl1 = -J1MT1 ± MV
From the square of the member system for writing a multiplication and consider the first term permanent.
While considering the first term we receive the following issues:
\A1<Pl1 - Al<Pl3 - A3<Pl1<Pl1 + lA3<Pl1<Pl3 - A3(Pl3(Pl3 + A// = Ma ;
IA2V21 - A3(Pi3 + A/Pu = -JMt 1 ± Ml. For the zero-order approximation: (ptj = 1, (ptj = 1:
\A1<P21 - A2<P23 - A3<PH + 2A3(P23 - A^V + A<P23 = Ma {A2V21 - As(P23 + A^ = - JM 1 ± M1. We describe the vector matrix:
A = A — a2 , B = - A3 A3(2
A2 — A3 At0
,C--
Ma , ^ = V21
-J mt 1 ± M1 V23
0 A4 00
F =
The equation approximationare the following: + BY" + C = F, where k — approximation coefficient.
We estimate the equations for the zero approximation, within k = 0.
+ BY" + C = F. (2)
While considering zero approximation, within k = 1 :
Awn - Av* - Avhvh+--^vlvv + AwV^ = Ma ; A&i- AV + AVu = -jimtI ± Mi. Term of the equation is considered to be permanent:
The matrix is the following:
B = _ AtfjzAfa - AtiO , c = 0 A^
Ai0 00
Thus:
AY1 + BY1 + C Y1 = F. Y0-Y1! <e;
(Y1 -Y 2| <e.
If the condition (4) is satisfied, then the problem is solved.
(3)
(4)
Y0 obtain a solution for q = 1, y = 1, Y° =
V021
Considering the power load in 3 modes: engine acceleration F1 = Fm, engine at maximum loadings F2 = F<2) and engine braking F1= F(3). J = 1, 2, 3.
Dynamic model of the inertial hydrodifferential transformer working process of the rotating moment.
We investigate the force loading the engine acceleration, starting at 0 and extending to k:
+ B 0vF + C 0¥= F; + Blx¥ + C F;
Under k = k-1, №k-¥"1 <e,
i = N:
Bk =
V N ) [t^ - Bk ^ ! - CÎ ^ 2 ] Under zero approximation we assume
V = 1, $0 = 1
'-Atâ1 ( 1 1))
; Ck 1 =
A4 0
v 4
Using (8) we estimate the vector equation:
A¥k + Bk 1x¥k + Ck = F.
0 AiÇkls
0
0
(5) (51)
A¥ + Bk ¥ + Ck ¥ = Fv
The equation should meet the conditions:
— ^ < £k
This equation can be solved by finite differences of the second order of accuracy. Approximating Y :
*=«i); ^( -2^i).
2t T
4 (+1 - 2^ + Vtl ) + f (+1 -Vtl ) + Ck = F(I). T 2t
This equation is multiplied by time steps z2 :
^ A + B- t j '- (-2 A + CkT ) ) + ^ A - B- t j ' 1 = t2F< (I). Considering: Ak = ^ A + y Tj ; Bk = (-2A + CkT ); Ck = ^ A - B- t
The total equation takes the form:
Ak +1 - Bk + Ck 1 = r2. Solving equation (5)relative to x¥i 1 , 1 < i < 3 :
¥. 1 =(Ak ) 1 [t2Fi(i) + Bk- Ckl ]. Under the initial conditions i = 0 :
= (A,k ) 1 [r2F0(i) - Bk - Ck ¥ 1 ].
Considering:
0 =a; = 0; — ( 1 ) = a; ^ = 0; < 2t
1 ) = 2ra; - 2ra.
While considering the initial conditions (6):
^ = ( Ak ) 1 [t2F0№) + Bkp - Ck ( - 2t« )] ;
(F + (Ak ) 1 Ck ) = (Ajk ) 1 [t2F0<1) + B[p+ 2xaCk ], where: E — matrixunit.
^ = (E + (A1k ) 1 Ck ) 1 (A1k ) 1 [t2F0<1) + Bk p- 2aCk ],
.=1:
^0=( Ak )1 [t2F1<1) - Bk^ - Ckp]. Solvingthe equation (5) (beginning i > 1): Under k = 1, — is determined, Under k = 2, Y 2 — is determined; <e,
Under k = 3,Iy2<e,
(6)
(52)
(7)
(8)
Equation (9) is solved by finite differences of the second ac-curacyorder:
4 (( - + ^k i) +—( - i ) + +Ck 1 Yk = F(1). (10)
t ' ' ' 2T ' ' '
This equation is multiplied by r2 and similar summands are introduced:
Bk
A + —T J^L +(2A + Ck1r2 ))k +
(11)
+| A — —t |^k1 = F(I)T2.
We introduce the designations:
Bk i Bk-i
Ak 1 = 4 +-T; Bk. 1 = -2A + Ck V; ck 1 = A - — t. (12)
1 2 1 1 2
Taking into account the designations (12) we rewrite the equation (11):
Ak +Bk 1^k+ck ^ 1 =t2F(i). (13)
Equation (13) is solved under the function ¥k+1:
=(Ak1)1 [t 2F (i) - Bk 1^k - ck 1^k l j. (14)
Under i = 0, equation (14) obtains the form:
=(Ak 1) 1 [t2F0(1)-Bk 1^0k - Ck 1^k1 ]. (15)
We formulate the initial conditions^0 =a ; ^0 = Pk; ^0 — and approximate within the second order of accuracy.
^o=^^(^k-^'1)=«. (16)
This equation is solved for ^k1:
-^k1 = 2t«; vi1k1 - 2Tak. (17)
Phrases ^k1 and TO are substituted into the equation (15):
< = (4 1) 11Vfo(1)-B'k1pk - Ck1 2r«k . (18)
We introduce similar summands:
[F + (41" 1) 1 Ck 1 =(Ak 1) 1 [t2F0(1)-Bk 1pk + 2rCk ak]. (19)
From equation (19) we define T : ^k =[f + (Ak 1) 1 Ck 1 ] 1 (Ak 1) 1 [t2F0(1)-Bk 1pk + 2TCk a]. (20)
Under i=1and the entry conditions ^O = fr from the equation (14) we obtain:
= (Ak 1)1 [t2f1(1)-Bk 1^1k - ck 1pk ]. (21)
Computer realization:
1. Basic data;
2. i = 0;
3. The equation is solved (20);
4. i = 1;
5. The equation is solved (21);
6. i = i + 1;
7. The equation is solved (14);
8. i < N, if the condition is satisfied, then move to step 6. Otherwise to step 9;
9. End.
Thus, within the dynamic motel we investigated working process of the inertial hydrodifferential transformer of the rotating moment of infinitely variable speed transmission. The equation (1) is solved by the method of final differences; the algorithm of calculation is also formulated. Thus, using the equations (20) and (21) we can find the values of the movement in the first and second shaft of the inertial hydrodifferential transformer.
(9)
References:
1. Grebennikov D. V. Method of forecasting technical resource inertial hydrodifferential automatic mobile carstransmission. The abstract
on the thesis for the degree of Candidate of Technical Sciences. - M., 2005.
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