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4. Математическое моделирование
УДК 66.621.928.13
О В.Д. Анохин
МОДЕЛИРОВАНИЕ И НЕЛИНЕЙНАЯ ДИНАМИКА ВИБРИРУЮЩИХ СИСТЕМ
Рассмотрено новое направление в теории колебаний применительно к совершенствованию процессов, актуальных в ряде современных технологий.
Разработаны модели процессов, предложен ряд формул скорости для их расчета и регулирования по теоретическим уравнениям и аналитическим зависимостям.
Ключевые слова: методы моделирования и теория колебаний, регулярная динамика механических систем и нелинейные задачи вибрационной динамики.
О V.D. Anakhin
MODELING AND NONLINEAR DYNAMICS OF OSCILLATION PROCESSING SYSTEMS
A new direction in the theory of vibrations has been considered, it is applied to improvement of the processes that are relevant in a number of modern technologies. Processing models have been developed, a number of formula for speed of their calculation and regulation on theoretical equations and analytical dependences has been proposed.
Keywords: modeling methods and theory of oscillation, regular dynamics of mechanical systems and nonlinear problems of vibration dynamics.
Advanced oscillation processing systems are involved in the primary mode of completely new methods of application of vibration technology for efficient transport-related separation of chemicals, minerals, pharmaceuticals, foods, metallic powders (lead, copper, zinc, steel) and all types of powder products common in many industries: abrasive, powder metallurgy, paint and varnish, diamond, construction, mining and chemical. These new screen-less methods can be effectively used in the abrasive industry to produce materials in which more than 90% of the grains are isometric. Grinding wheels made from such grains are twice as effective as those made from regular grains which are unclassified by shape. Powder products generally are separated by particle size or shape without forming dust. In the diamond tool industry the vibrating equipment is used for selecting isometric, plane and needle-shaped diamond grains. In the agriculture and food industry this advanced machinery can utilized for removing harmful inclusions in grains. In the mining and chemical
industries high efficiency processing machinery with accurate separations in sizes from 15 mm down to 20 (j,m can be used for many types of the ore. Separation is based on the velocity differences of the particles due to the existence of oscillating driving force. If components are to be separated, their relative friction coefficients on a vibrating deck may be pertinent. The friction coefficient is a function of the particle shape and the particle size. A common correlation of the friction coefficient / and of the adhesion force Fa to the particle
size D in (jm is that of the author [1]:
f = 0,9-10~°'001D (1)
FJmg = 7,46 D~0'24 - 1, 68. (2)
The above functionality is useful to understand the phenomenon of effective process.
There are two types of operating equipment: 1) harmonic motion of a deck, where direction of oscillation does not have a lateral tilt, and 2) vibration characterized by a lateral tilt with respect to the vertical direction. The direction of vibration is also inclined at the angle of vibration (3 with respect to the surface of the vibrating deck. A vibratory effect appears in the process so that particles with different sizes, shapes, or coefficients of friction, therefore with different angles of vibration separation a0 move with different velocities over a separating surface. A subset of the two types of operation conditions are the 1st and 2nd basic separation mechanisms common to the 1st and 2nd types of machinery. The acceleration W0 A ofsinfi g cosa is the operating condition factor for these two types of equipment, where A is the displacement amplitude of the deck, co is the angular frequency of the simple harmonic motion, /? is the angle of vibration, g is the acceleration of gravity, and a is the longitudinal tilt of the deck. The force driving the separation could be several orders of magnitude greater than of gravity. Particles will be intensively tossed upwards
after a brief contact with the vibrating surface at phase angles cos <?0* =
and cos<?o = + R C0S£ for machinery of the 1st and 2nd types, respectively,
W0\ + R
where 8 is the lateral tilt of the deck with respect to horizon, X (see below) and R are the coefficients of momentary friction and elasticity in accordance with Newton's theory of impact, the coefficient p defines regime of continuous rate of particle flight over vibrating deck. The particle velocities along the x and z axes are represented by
for the equipment of the first type
mgA-R _ 2-X .
K = —(——ctg/3cosa--—sina) (3)
a 1 + R X
npg .2-X 1 -R.
Vz = -^(---) cos a sins (4)
co X 1 + R
for the equipment of the second type
mgA-R _ 2-X .
-ctgp cos a cos e--sina) (5)
co \ + R X
ВЕСТНИК БУРЯТСКОГО ГОСУДАРСТВЕННОГО УНИВЕРСИТЕТА
1/2013
npg 2- X .
Vz =--cosasins. (6)
io X
To maximize the operation that must be dealt with developing an optimal process the phase angle 80 and parameterp can be determined as follows: cos S0 = ±1, p = W0 (1+R) / k (1-R).
X 1 — R
Converting Eq. (3) - Eq. (6) and setting q =--, where q is a par-
2 — X 1 + R
tide parameter of vibration separation (a table of the parameter q as a function of X & R is presented in [1], the solution are of the forms:
for the equipment of the first type
Vx = Aa(cos P - tga) (7)
q
Vz = Am sin¡5 sine(— -1) (8)
q
for the machinery of the second type
sin
qcose
Vx = (cos p - Sin^ tga) (9)
V=Aco^-. (10)
qcoss
Dividing equations (8), (10) by Eqs. (7), (9), respectively, the expressions for the trajectory of the particle movement for both types can be written dz Vz (l-g)sine
dx Vx qctgP - tga dz Vz sine
(11) (12)
dx Vx qctgP cos s - tga The XZ plane of the vibrating deck with its longitudinal and lateral axis in the X and Z directions, respectively, has the length I and the width b of the deck, respectively. Solving Eqs (11) - (12), separability is defined analytically as follows
for the machinery of the first type
dx dx = — 1 dq
z = b=j^çtgp-tga (13)
sine (1 -q)2
dz
d,_ = —
x = — = — sine ——2ÍL—-2—— (14)
2 2 (qctg/btg a)2
I I . ctgP - tga
X = —
dq
for the machinery of the second type
DX]] = dctgsctgp (15)
D =l- *mlSCtgP 2. (16)
2,1 4 (qcossctgjB-tga)
In the design and operation of separation processes, each process will give a maximum removal of a single component from the mixture, if qctg(3 / tga = 1 and coss ctg(3 / tga = 1. The particle velocity in its movement in the longitu-
dinal direction can be reduced to zero. It can be used to write the angles of vibration separation for the 1st and 2nd types of equipment, as а„л= arctg (qctgP), а„л = arctg (qcoss ctg|3).
Conclusions
The analytical and inventive aspects of design include: estimate the velocities of each components; evaluation its angles of separation; calculation the separability of the process; designing an efficient equipment and its control strategy.
References
1. Anakhin V. et al. Vibrational separators. - M.: Nedra, 1991. - 157 p.
Владимир Дмитриевич Анахин, доктор технических наук, профессор кафедры машиноведения Бурятского государственного университета, e-mail: anakhin@mail.ru
Vladimir Dmitrievich Anakhin, doctor of technical sciences, professor, department of engineering mechanics, Buryat State University, e-mail: anakhin@mail.ru
