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APPLICATION OF LAGRANGE EQUATIONS OF SECOND KIND TO A MECHANISM WITH TWO DEGREES OF FREEDOM
Sobirov Xolxuja Abbozovich Andijan Machine-building Institute, dots. E-mail: [email protected], тел: +99(891) 6149591
Bekqulov Botirali Rahmonqulovich Andijan Machine-building Institute, PhD, head of a chair
Yuldashev Kozimjon Komiljonovich Andijan Machine-building Institute, PhD, senior lecturer.
Xakimov Mirolimjon Murodiljonovich Andijan Machine-building Institute, post-graduator E-mail: safina20171988@,gmail.com, тел:+99(890) 571 8180
Annotation: This article describes the principle of operation of the automatic control valve of a portable drum of a grain dryer and also shows the process of compiling a mathematical model of the movement of the mechanism for spilling rice from the drying drum.
Аннотация: В данной статье описан принцип работы клапана автоматического управления переносным барабаном зерносушилки, а также приведен процесс составления математической модели движения механизма для слива риса из сушильного барабана
Аnnototsiya: Maqolada ko'chma don quritish qurilmasining mahsulot to'kish mexanizmi uchun Lagranjning 2- tur tenglamalari qo'llanilgan, shuningdek ularni keltirib chiqarish jarayoni yoritilgan va olingan matematik model keltirilgan.
Key words: rice, drying drum, automatic adjusting cover, spring, mathematical model, mechanical energy
Ключевые слова: рис, сушильный барабан, автоматически регулируюмая крышка, пружина, математическая модель, механическая энергия.
Kalit so'zlar: quritish moslamasi, baraban, prujina, matematik model, avtomatik moslashuvchan qopqoq, mexanik energiya.
Purpose of the article. Create prerequisites for determining the main parameters of the product unloading mechanism using the Lagrange equations of the second kind.
Цель статьи. Создать предпосылки для определения основных параметров механизма разгрузки продукта применяя уравнения Лагранжа 2- рода.
Maqsadi. Lagranjning 2- tur tenglamalari qo'llab mahsulot to'kish mexanizmining asosiy parametrlarini aniqlash imkonini yaratish.
Introduction. To ensure the food security of the country, it is important not only to increase its production, but also to ensure the appropriate quality of grain, the production of which is seasonal and, therefore, it is necessary to ensure its long-term storage due to post-harvest drying. To date, the issue of increasing the yield of grain crops is still relevant. In this direction, selection work is being improved, various agro technical measures are being carried out. It is known that some shortcomings in the drying and processing of cereals adversely affect the yield of grain crops [1].
Currently, approximately 15% of the total energy consumption in the agro-industrial complex of developed countries is accounted for by the processes of heat treatment of
agricultural materials. The main purpose of heat treatment of agricultural materials is to increase their storage stability or temporary conservation.
One of the most necessary and energy-intensive type of heat treatment of agricultural materials in agricultural production is the drying process. Drying provides the possibility of long-term storage of finished products, as well as effective pre-treatment of raw materials for most technological operations of the production and processing of agricultural products [2].
The inconsistency of the existing technical base with the conditions of agricultural production (various forms of ownership, the operation of market mechanisms) necessitates fundamental changes in the technical support of the processes of thermal processing of agricultural materials. Currently, there are a number of grain dryers in agriculture, which differ in the drying method, the design of the drying chamber, the drying mode, the state of the grain layers, and many other design features [3].
Materials. In a portable drying device for the proposed grain products (Fig. 1), the supply hopper 1 is installed with a screw conveyor 2 at an angle to the horizon, which has its own supports. At a certain angle, the groove 3 is attached to the screw conveyor 2. The dryer drum 4, located at an angle to the horizon, is attached to the trailer 17 using supports. The proposed drying drum has automatic adjustable pouring mechanisms 5 for rice. At the top of the dryer drum there is a fixed section 8, in which a heater 9 and a fan 10 are mounted. The drive of the dryer drum consists of a front clutch 12, a gearbox 13 and a rear clutch 14. The screw conveyor drive consists of an electric motor 15 and a belt drive 16. The grain sorting mechanism is equipped with a V-belt drive 18 and a fan 19 [4].
10 8 3 16 15 4 5 и
Fig. 1. Structural diagram of a portable grain dryer
The principle of operation of the portable dryer mechanism is as follows: the dried grain product is fed in a given portion into the supply hopper 1, the screw conveyor 2, and then through the chute 3 into the dryer drum 4. As soon as the grain is poured into the dryer drum in a certain portion, the grain is fed from the feed hopper 1 stops. Drying of grain (rice) is carried out by a stream of hot air in a drum. Hot air is generated by a heater 9 located in the fixed part 8 and is directed to the dryer drum by means of a fan 10. The grain product in the dryer drum is supplied from the drum to the sorting mechanism using a spring-loaded hinged cover with automatic mechanical control.
When the drum is tilted to a certain angle, the process of pouring the product stops for a certain time, and the cover closes automatically. This ensures that the hot air flow inside the drum does not escape into the outside environment. As a result, a promising design scheme of a horizontal type portable drum dryer will be developed, which will reduce energy costs and downtime.
Methods. As noted, the discharge lids, which are hinged to the dryer drum, open and close during the rotational movement of the drum. Each lid closes in the up position and opens in the down position. The main reasons that positively or negatively affect the regular opening and closing of lids are the forces acting on them, the rotational movement of the dryer drum, the stiffness of the springs, the weight of the product to be dried, etc.
To determine the main parameters that affect the regular opening and closing of lids, we will build a dynamic model of a rotating drum and a complex movable lid (Fig. 2). In this case, we consider this mechanical system as a two-mass mechanical system with respect to a stationary engine. We show the vectors of forces acting in a mechanical system and the directions of their moments in a dynamic model. Assume that the direction of the airing dryer is clockwise rotation.
Fig.2. Dynamic model of the mechanism
In the diagram: D - electric motor; 1- dryer drum with grain; 2- cover; pi - angular displacement of the drum, deg.; (2 - angular displacement of the cover, deg.; Md - engine torque, N.m.; Mqi - torque relative to the axis of rotation of the sum of resistance forces (resistance forces and grain friction forces) to the rotation of the drum, N.m.; Mg2 - the moment of gravity on the drum (relative to its axis of rotation, i.e. relative to the movable axis), N.m.; Mq2 - the moment of grain pressure force acting on the lid relative to the drum, N.m.; Ci -stiffness coefficient of the belt and other gears between the electric motor and the dryer drum, Nm/rad; C2 - coefficient of stiffness of the spring that opens and closes the lid, N/m.
We will now draw the following diagram to show the force vectors acting on the lid (Fig. 3).
Active forces acting on the cover: Gi is the gravity force of the drum, N; Gd is the gravity force of rice, N; G2 is the force of gravity of the lid, N; Qd is the pressure force of rice on the lid, N; h2g is the lever of gravity of the lid relative to point B, m; a = pi - (2 - complementary angle, deg.
Y - the angle of deviation of the gravity vector of rice (grain) from the OY axis due to the rotation of the drum.
Consider the moments of the acting forces on the cover relative to the point B:
mb (g2 ) = G2 * h2g = 0,5 * m2 * g * h Sin(p - (2) (1)
where Mb (G2) is the moment of gravity of the lid relative to point B, m2 is the mass of the lid, kg; g is the free fall acceleration, m/s2; BK - half the length of the cover (BK = I2 /2), m.; I2 - cover length
The moment of rice pressure force relative to point B is determined in this way
MB Q ) = Qd ■ BK = 0,5 • S • PD • l2 • sin^ (2)
where, S - cover window area, m2; Pd - rice pressure on the lid, kg/m2.
Fig.3. Calculation scheme of the mechanism
Also, consider the moments of forces acting on the axis of rotation of the dryer
drum:
Mb - torque from the side of the engine, i.e.
MB = MD ■ iDB (3)
where Md is the engine torque, Idb is the gear ratio between the rotation axes of the engine and the dryer drum.
Mqi is the moment of the resistance force on the rotation of the drum, i.e.
Mqi = M DQ + M
ishl (4)
where Mdq - moment of gravity of rice
mdq = gd * rb * sinx
where rB - part of the drum radius, m.
Mishi - moment of frictional force during rotation of the drum.
mishl + G2 + G3 )■ rb ■ f (5)
where tb is the drum shaft radius, m; f is the coefficient of sliding friction in the drum shaft bearings.
In the two-mass mechanical system under consideration, according to the dynamic model, one can apply the Lagrange equations as in a system with two degrees of freedom and compose the equations in the following form.
d_
dt
d_
dt
r dT\
r дтл
дт дп „ — + — = Q1
д Px дРх
дт
\д®2
дП
+ —=Q2
(6)
p p - (7)
where T - kinetic energy of the mechanical system, n - potential energy of the mechanical system, Q1 and Q2 - generalized forces acting on coordinates.
The kinetic energy of the considered mechanical system is determined in this way:
T = TB + TK (8)
where Tb - kinetic energy of the dram; Tk - kinetic en ergy of th e cover.
We know that TB = 0,5 • JOB • (aB )2 = 0,5 • JOB • (< )2 (9)
where Job - moment of inertia of the drum about its axis, kg.m2; wb - drum angular speed (p1) , rad/s.
Likewise, we find
tk = 0,5 •Jok-(a>b +a>k )2 = 0,5 •Jok^Pi +<P2 )2 (10)
where Jok - moment of inertia of the lid about its axis, kg.m2; wk - relative angular velocity of the lid (p '2), rad/s. So,
Т = 0,5 ■JOB ■ (Pi )2 + 0,5 ■Jок ■ (Pi + P2 )2
(11)
The potential energy of the considered two-mass mechanical system is defined as follows: When determining the potential energy of the considered mechanical system, the gravity force of rice in the drum, the elastic force of the V-belt transmission between the engine and the drum, as well as the elastic force of the spring for the cover are taken into account and we write:
n = nD + nci + nc 2 (12)
where nD - potential energy of rice in the drum; nci - potential energy of the elastic force of the V-belt transmission; nc2 is the potential energy of the elastic force of the spring.
The potential energy of the system has the form:
/ , \2
П = 0,15 ■ Gd + 0,5 ■ С
(Pi - Pd
1
db у
+ 0,5 ■ С ■ sin P2 )2
(13)
Thus, the generalized forces (moments) of the considered mechanical system will take the form:
q = mb -mk1 = Md • *db -Gd • rb • smr-(G1 + G2 + Gd)• rB •f (14)
Q2 = MbG)-MbQd) = m • g • BK • stn(p - <2)- 0,5 • S • Pd • l2 • sm < (15)
Results. Now, substituting the found expressions into equations (6) and (7), we obtain the systems of equations of motion of the device mechanism:
=
md • D - Gd • rb ■ sinr-(Gi + G2 + Gd )■ rB ■ f - С1 ■
1
Pi-Pd-t
db
^ ob + jok )
[0,5 ■ I2 (^2 ■ g ■ Sln(Pi - P2 ) - S ■ pd ■ sln Pi) - С2 ■ l2 C0s P2 ]
(16)
J,
ок
^2 =
We accept the initial conditions:
at t = 0; (P01 = (P02 = 0; W1 = wd ; £1 = £2 = 0 border conditions:
at t = T; pti = 1800 , those. this corresponds to half a revolution of the drum, which is sufficient to analyze the process of opening and closing the lid during operation.
Discussions. Thus, we have obtained a system of equations (16), which takes into account almost all quantities (design parameters) that affect the mechanical system, that is, the mathematical model of the first approximation of this mechanical system. The parameters £1 and £2 in the mathematical model are the angular accelerations of the dryer drum and the lid, respectively. To calculate their values, it is recommended to use the "numerical method".
Using the obtained mathematical model, it is possible to construct the following dependency diagrams, which are necessary for the synthesis of design parameters: (2 = p1 (f) - dependence diagram of the angular displacements of the drum and cover; (2 = S2 (f) - diagram of the dependence of the angular displacement of the cover on the
coefficient of spring stiffness; (2 = pd (f) - diagram of the dependence of the angular displacement of the cover on the pressure of the grain;
p2 = Gd (f) - diagram of the dependence of the angular displacement of the cover on the weight of the grain;
p2 = Gq (f) - diagram of dependence of the angular displacement of the lid on the weight of the lid;
P2 = wb (f) - diagram of dependence of the angular displacement of the lid on the angular velocity of the drum;
P2 = I2 (f) - diagram of angular displacement versus cover length. Conclusions:
1. A system of equations has been obtained that takes into account almost all quantities that affect the movement of a mechanical system, that is, a mathematical model of the first approximation of this mechanical system.
2. Using a mathematical model, it is possible to construct the necessary diagrams not only of the angular accelerations £1 and £2 of drums and covers, but also of their angular velocities W1, W2 and angular displacements (1, (2. It is possible to carry out the synthesis of parameters based on the given values of the coefficient of elasticity of the lid spring (C2) and the area of the window for unloading dried rice (S), which is necessary for the proposed drying device.
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