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15
Mukhammadiev Davlat Mustafaevich, Doctor of Engineering Sciences, Leading Researcher, Institute of Mechanics and Seismic Stability of Structures of the Academy of Sciences of the Republic of Uzbekistan
E-mail: [email protected] Ibragimov Farkhod Khayrulloevich, basic doctoral student of the Institute of Mechanics and Seismic Stability of Structures of the Academy of Sciences
of the Republic of Uzbekistan E-mail: [email protected]
DETERMINATION OF CRITICAL ANGULAR SPEED OF SAW CYLINDER OF A COTTON GIN
Abstract: Results of calculations for determining the critical angular speed of saw cylinder are presented in the paper with account of a mass of raw material roll in the form of an ideal incompressible fluid. Comparison of the values of critical angular speed of a saw cylinder with an analytical method, not accounting the load from the raw material roll, made it possible to establish a difference in the calculations up to 3.5%.
Keywords: cotton gin, saw cylinder, raw material roll, critical angular speed, shaft, cushion, saw blade, distributed load.
Units of the working chamber and its main unit - a saw cylinder - are located in a saw gin under considerable force from external loads. To determine the critical speed of the saw cylinder shaft, the methods by A. N. Krylov, A. I. Makarov, I. Ya. Koritytskiy, M. Ya. Kushul, Rayleigh; the methods of successive approximations; initial parameters in the matrix form and others are used. In [1; 2] it was shown that the definition of critical speed is reduced to solving the problem of determining the natural frequencies of oscillations.
In [3], to reduce analytical computations when determining the critical speed of saw cylinder, the finite element method is used.
In Kh. K. Tursunov's monograph, results of theoretical and experimental studies of the dynamics of saw shafts and grates of cotton gin are given and their critical speeds are evaluated [4].
Determination of critical speeds of saw cylinder makes it possible to assess the danger of the operating speed approaching the critical speed.
At COp = (Okp the deflections of the shaft and the load on the bearings can increase without limit, although this does not happen due to restrain of deflections of shaft pinching in the bearings, the presence of frictional forces and external loads. However, when designing high-speed shafts, it is necessary to take into account the danger of their operating speeds approaching the critical ones, therefore at shaft speeds below the first critical one the following ratio is recommended: (O < (0.75 — 0.8) • (dkp. The interval between the first and the second critical speeds is 1.4 -W1kp ^ 0.8
2kp *
When operating in the first pre-critical zone of angular speeds the shafts are called rigid ones, and when operating in the zone after the first critical speed, they are flexible.
The goal is to determine the critical angular speed of saw cylinder Okp (D = 320 mm - a diameter of the saw blade, d = 100 mm -a diameter of the saw cylinder shaft under load from the raw material roll (ideal incompressible fluid, Figure 1).
The weight of the raw material roll consists of seeds and cotton per unit length of the saw cylinder m^ the weight of the length unit of saw cylinder is m0, (shaft, spacers, saw blades), the bending rigidity of the shaft is EJx. Coriolis forces of inertia of seeds and raw cotton are neglected due to their smallness.
Uniformly distributed loads are: m1 = 110.0 kg/m; m2 = = 114.54 kg/m; m3 = 27.27 kg/m.
At the state, deviated from the rectilinear equilibrium position, the inertial force due to rotation of saw cylinder with a raw material roll (the seeds and raw cotton) per each length unit equals to (m0+m1)-u2-y, and the centrifugal force of inertia of rotating raw material roll caused by the curvature of saw cylinder is m1 • V2 / p = -m1 • V2 • y" (Figure 2) (the minus sign is determined by the curvature sign).
Differential equation of the curved axis of the shaft of saw cylinder is represented in the form
EJx ■ yIV = (m0 + m1 )co2 ■ y - m1 ■ v2 • y" (1)
or yIV + k2 • y" - k24 • y = 0. where kf = m1 • v2 / (EJx),
v 2
K24 = (m0 + m1 )-co2 /(EJx). (m0 + m1)-a2 ■ y m1 ■ —
0.260 2.900
1 - Shaft, 2 - Bearing body, 3 - Saw blades, 4 - Shaft spacers, 5 - Washer, 6 - Nut Figure 1. Design scheme of a saw cylinder
Figure 2.
Differential equation of the curved axis of the shaft of saw cylinder is represented in the form
EJx • yIV =(m0 + m1 )o2 • y -m1 • v2 • y" (1)
yIV + k2 • y" - k24 • y = 0. where k2 = m1 -v2 / (EJx), K24 =(m0 + m2)-co2 /(EJx). (m0 + m2)-a2 ■ y m2 •-
=±Qk4 + 4.k4 -k2)/2;
or
v
m, — p
Characteristic equation for equation (l) has the form
hence
A + k2-A2 -k24 = 0,
^2,2 = ±i ^ (k4 k4 + 4 • k4)/4;
Solution of equation (l) is obtained in the form y=C1-sin(X1-z)+ C2-cos(X2-z)+ C3-sh(X3-z)+ C4-ch(X4-z).
To determine the integration constants C1t C2, C3, C4, the following conditions should be met: at z = 0, y = 0 and y" = 0; at z = l y = 0 and y" = 0.
To determine the critical angular speed of saw cylinder ®kp, the following equation is obtained
sin(X1■ l) ■ sh(X3 ■ l) = 0. (2)
The left-hand side of equation (2) is zero in the following cases: X=0, at cc=0; (X^ l)= k
At k = 1, that is, (X ■ l)= n, the value of the critical angular speed is obtained
n2mv2
'l (mo + m2) l (mo + m2)
(3)
Taking into account the values l = 2.64m; (m0 + +m1) = 110 kg/m; (EJ)= 785398.16 Hm2 and substituting into (3), the law of change of the critical speed of saw cylinder depending on the weight of raw material roll m1 and on bending rigidity of the shaft of saw cylinder EJx is determined (Fig. 4).
Figure 3. Change of the critical speed of saw cylinder aKp depending on the weight of raw material roll m1
Analysis of equation (3) and (Fig. 3) has shown that the critical angular speed decreases from 119.66 to 101.79 rad./s (15.0%) with an increase in the weight of the raw material roll per length unit of saw cylinder from 0 to 42 kg/m. Analysis of equation (3) and Fig. 4 has shown that at m = =38 kg/m, with an increase in bending rigidity of the shaft of
saw cylinder EJx from 100,000 to 1,000,000 Hm2, the critical angular speed decreases from 135.02 to 42.70 rad./s (68.4%).
With load acting from the raw material roll, the angular speed of gin saw cylinder 5DP-130 is in the first pre-critical zone (wv < 0.75-w1jcp - rigid shaft) 76.44 rad/s < (76.43-77.3)
rad/s.
Figure 4. Change of the critical speed of saw cylinder oKp depending on bending rigidity of saw cylinder EJx
The difference of critical speeds of the gin saw cylinder the load of the raw material roll, which is 121.3 rad/s [5] and 5DP-130 is obtained between analytical calculations according proposed by the authors calculations with account of the load to the data presented by JSC "Pakhtagin KB" not accounting from the raw material roll which is 103.1 rad/s (15.0%).
References:
1. Miroshnichenko G. I. Basics of designing machines for primary cotton processing. - Moscow: Mechanical Engineering, 1972.- 486 p.
2. Biderman V. L. Theory of mechanical oscillations.- Moscow: High School publ., 1980.- 408 p.
3. Mukhammadiev D. M., Rakhmatkariyev Sh. U., Arifdzhanov A. Z. Analysis of static and dynamic characteristics of saw cylinder of cotton gin. "Problems of machine building and machine reliability".- Russia. - Moscow. 2009.- No. 2.- P. 13-17.
4. Tursunov H. K. Mechanics of the working organs of cotton gin machines.- Tashkent: Fan, 1997.- 128 p.
5. Mukhammadiev D. M. Dynamics of machine units of saw gin with seed retractor device and condenser with pulse air streem: Dis ... Doct. tech. sciences.- Tashkent: TITLP, 2014.- 211 p.
