CC BY
16
DOI: http://dx.doi.org/10.20534/AJT-16-9.10-58-60
Karimov Rasul Ishakovich, Tashkent State Technical University named after Abu Raykhan Beruni, professor of the department "Theoretical mechanic's and the theory of mechanisms and machines"
Nematov Erkinjon Hamroevich, Tashkent State Technical University named after Abu Raikhan Beruni, senior staff scientist of the Department "Theoretical mechanic's and the theory of mechanisms and machines"
Baratov Nortoji Baratovich, Tashkent State Technical University named after Abu Raikhan Beruni, associate professor of the Department "Theoretical mechanic's and the theory of mechanisms and machines"
Axmedov Azamat Xaitovich, Tashkent State Technical University named after Abu Raikhan Beruni, senior staff scientist of the Department "Theoretical mechanic's and the theory of mechanisms and machines"
Shaxobutdinov Rustam Erkinbayevich, Tashkent State Technical University named after Abu Raikhan Beruni, senior staff scientist of the Department "Theoretical mechanic's and the theory of mechanisms and machines"
E-mail: azam0602@mail.ru
Determination of kinematic parameters, the redused moment of inertia and its derivative of the planetary mechanism with the variable moment of inertia of the planet pinion
Abstract: The article presents the analytical expressions for the determination of kinematic parameters of planetary gear with variable moment of inertia of the PLANET PINION. The influence of the mass, the mean radius of the center of mass of the processed material at the given laws of the change of inertia and its derivative.
Keywords: The planetary mechanism, the planet pinion, carrier, kinematics, the given moment of inertia, working drum, the processed material.
The planetary mechanisms are widely used in In this regard, the authors considered the guiding
modern technological machines, in particular mixers, planetary gear in figure 1, which consists of a fixed central
grinding machines and raw materials of minerals etc. wheel 1, the planet pinion 2 carrier of h, working drum 3.
To determine the actual laws and loading of units here rnH - the angular speed of the carrier, the "c" — the necessary to investigate these mechanisms for quality center of mass of the material is being processed RM -
actuating mechanisms of machine aggregates. It is known variable radius center of mass of the material is being
that at probe of aggregate of machine with variable processed.
moment of inertia essential to define the given moment The planet pinion makes difficult rotary the
of inertia and its derivative [1]. movements around two axes:
Determination of kinematic parameters, the redused moment of inertia and its derivative of the planetary mechanism.
1. However, a carrier with the carrier around a rotational axis;
2. Around own axis of planet pinions.
As a rule, the working member is connected with the planet pinion and makes the same movement as the
planet pinion. Due to the fact that during the operation of the above-stated machines at rotation ofworking body the center of mass of the processed material changes in relation to a planet pinion axis, the planet pinion has variable moment of inertia.
Figure 1. The kinematic scheme of the planetary gear traim with the variable moment inertia of the planetarium pinion.
The importance of the study of machines of this class is the definition of the center of mass of the kinematic parameters of the material is being processed. To determine the center of mass of the kinematic parameters of the material is being processed by the authors, based on the results of the study presented in the paper [2], proposed the following analytical expressions xc(t) = Acos(aH ■ t) -Rjt)cos(u2HaH ■t),
(t) = Asm(©H ■t) -Ru(t)sin(u2HrnH ■t), (1)
where
RM(t) = Kl ■ R0 + K2 ■ R0sin(K3©2i), A = (R + R), RM(t) - Variable radius center of mass of the material is being processed;
R1,R2 - The radii of the pitch circle of gear wheels; R0 - The average radius of the center of mass of the processed material;
K j, K2, K3 - The constant coefficients whose values depend on the composition of the processed material and structural parameters of the drum;
Ri r
planet pinion;
x c(t), yc (t) - The coordinates of the center of mass of the processed material.
Sx(t) = u2H ■ G)H sin(u2H -a>H ■t)(Kl ■ R0 + K2 • R0 sin(K3©2t))-- A ■ a>H sin(fflHt) - K2K3R0m2 cos(u2Ha>Ht),
(t) = A ■ a>H cos(a>Ht) - u2H ■ a>H cos(u2H -a>H ■t) x x(KR + K2 ■ R0 sin(K3rn2t)) --K2K3R®2 cos(K3©2t)sin(u2H©Ht). &cx(t ),3cy (t) - projection of speed of the center of
a2 =aH (1 + = aHui21H
- The angular velocity of the
(2)
mass of the processed material onto the coordinate axes.
a a (t) = u22H ■ < cos(u2H <H ■ t)(K1 ■ R0 + K2 ■ R0 sin(K3®2t)) -
- AttHH cos<Ht) - K2K23R< sin(K<2t)cos(u2Ha>Ht) + + 2K2K3R0u2« ■ cos(K3 ■<) ■ sin(u2H®Ht),
ay(t) = u22H • a>2Hsin(u2H • • t)(K1 • R0 + K2 • R0sm(K3rn2t))-
- A •oH sin(oHt) + K2K23R0o22 sin(K3o2t)sm(u2Ha>Ht) --2K2K3R0U2HO2OH • cos(K3 •ot) • cos(u2H®Ht). (3)
a cx(t ),acy (t) - acceleration projection center of mass of the processed material onto the coordinate axes;
For the definition given to a shaft carrier moment of inertia of the planetary gear with variable moment of inertia is possible to use the following expression J„,H(t) = Jh + Jc ■ )2 + J6 ■ «Y + mc ■ (R + R2)2 + (4) + (m6 + mM) ■ (R + R2)2 + mM ■ (uf))2 ■ R2u(t),
where:
Jnp.n - given to the carrier shaft moment of inertia of the planetary gear;
JH - moment of inertia of the carrier relative to its axis of rotation;
Jc - moment of inertia of the planet pinion relative its own axis;
J6 - moment of inertia of the drum relative its own axis of planet pinions;
mM - the mass of the material is being processed;
m6 - the weight of drum;
mc - the mass of the planet pinion;
R
u2h = (1 +—L) - the transmission ratio between the R2
shaft of planet pinion and the shaft of carrier.
Derivative of the given inertia moment on time determined by the following expression
dJ
^ = mR (u?J
dRM dt
= 0(t ),
(5)
dt
d(t) = 2K2K3 ■ R0 ■ u2Hmca>2 cos(K3 ■ ©2 ■ t) x x(Kj ■ R0 + K2R0 sin(K3®2t). It should be noted that derivative of the given inertia moment on time and an angle of rotation carrier are connected among themselves by the following ratio
dJ,
пр.Н
dJ,
пр.Н
Analytical expressions (1-5) have been realized on the computer in the environment of MathCAD.
Study of the effect mM, R0, on the change in given moment of inertia H, , and its derivative HdJ , for this
Jnp .H UJnp .H
mechanism were determined under variation the mass of the processed material from 5 to 20 kg in steps of 5 kg and the average radius of the center of mass from 0.2 to 0.35 m in steps of 0.05 m. The results of calculations are shown in the table.
dt H
Table 1. - Results of calculations of scope of fluctuations of the given moment of inertia, his derivative for the planetary mechanism at a variation of mass and average radius of the processed material
Variation of mass of the processec material тм
№ mM, кг Jпр. Нmax ' кгм2 J пр.Н min ' кгм2 Hj , Jnp .Н кгм2 f dJnpH ) V dt , кгм max, 2 Г dJn,H > v dt кгм min, 2 Häl , ät кгм2
1 5 14,66 11,96 2,7 35,13 -35,13 70,26
2 10 18,72 13,32 5,4 70,26 -70,26 140,52
3 15 22,77 14,67 8,1 105,39 -105,39 210,78
4 20 26,83 16,03 10,8 140,52 -140,52 281,05
Variation of average radius of the center mass oi the processed material R0
№ Ro> м Jпр. Нmax ' кгм2 J пр.Н min ' кгм2 H Jnp .И' кгм2 f dJnp,H > v dt кгм max, 2 Г dJn,H > v dt кгм min, 2 Häl , ät кгм2
1 0,2 12,55 11,35 1,2 15,62 -15,62 31,23
2 0,25 13,5 11,63 1,87 24,4 -24,4 48,8
3 0,3 14,66 11,96 2,7 35,13 -35,13 70,26
4 0,35 16,04 12,36 3,67 47,81 -47,81 95,63
According to the results ofcalculations on a computer determined laws of change the oscillation scope given moment of inertia, its derivative, which is calculated by the following formula (6):
H = j _j
Jnp.H ' пр.Hmax ' пр.Hmin ,
H
f dl,
dlnpH dt
пр.Н
dt
f dl,
max -
пр .H
dt
min.
(6)
The analysis of results of calculations has shown that in this range of change of mass of the processed material scope of fluctuations of the given moment of inertia increases with 2,7 to 10,8 kzm 2 and scope of fluctuations, derivative of the given inertia moment, increases with 70,26 to 281,05 ^m 2.
Besides that, it was investigated the influence of value of the average radius of the center of mass the processed
material at the given laws of the change of the given moment of inertia and its derivative. The average radius of the center of mass the processed material was changed from 0.2 to 0.35 m in steps of 0.05 m
It was established that the scope of fluctuation of the given inertia moment increases from 1.2 to 3.67 ^m 2, and the scope of fluctuation derivative from given moment of inertia increases with 31.23 to 95.63 kzm2.
Thus, it is set that unlike the normal planetary mechanism this planetary mechanism with a variable inertia moment of the planet pinion has the variable given inertia moment. this feature of the researching mechanism needs to be considered in case of determination of the valid laws of movement of the mechanism.
References:
1. Артоболевский И. И. Теория механизмов и машин. - М.: «Наука», - 1988. - 640 с.
2. Каримов Р. И. Теоретические основы и конструкции планетарного и бипланетарного приводов рабочих органов тестомесильных машин. Монография, Таш ГТУ, - 2013-38,5 п. л.
