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1. Математическое моделирование
УДК 66.621.928.13
© В. Д. Анохин
МОДЕЛИРОВАНИЕ КОЛЕБАНИЙ НЕЛИНЕЙНОЙ ДИНАМИЧЕСКОЙ СИСТЕМЫ
Разработана математическая модель инновационного процесса, имеющего перспективу использования в ряде отраслей промышленного производства в целях создания и совершенствования новых аппаратов и технологий. Основное условие возникновения нового эффекта -асимметрия колебаний нелинейной динамической системы.
Ключевые слова: асимметрия колебаний, эффекты в системах с трением, математический анализ динамики процессов, мониторинг параметров.
© V. D. Anakhin
MODELING OSCILLATIONS OF NONLINEAR DYNAMIC SYSTEM
A mathematical model of innovative process has been developed, it has the prospect for use in a set of branches of industrial production for creation and improvement of new machine and technologies. The main condition of the new effect concludes in the asymmetry of oscillation of the nonlinear dynamic system.
Keywords: asymmetry of oscillation, effects in the systems with friction; mathematical analysis of the dynamics of processes, monitoring of parameters.
Completely new equipment design of VS distinguished from a vibrating screen in that it is complete with an un perforated deck (separating surface) specified with respect to the ground surface (horizon). Many variations of parallel-deck VS are now available: some have flat decks, some use concave decks. Most of the decks are placed on two tilt angles to allow the components of the mix to disengage as they pass through the deck section. Parallel decks added to VS can greatly improve separation quality and increase capacity. The parameter factor of the solids present in the mix is also essential with the screen less method motivated by the acceleration of vibration. The design of equipment for this particular case is less developed aspects of vibration technology. Selection of the separation technique or techniques to be used for a particular system can be also broken down into the task to use the translational vibration motion in the direction of the Y axis of the deck of VS (longitudinal vibration). The technical feasibility and engineering perspectives
of a given method must be attractive. Some comments on the selection are included to provide some perspective for discussion in works. Mixtures of dry solids can be separated by the specific thickness differences of the components. The proper introduction to feed to a VS is one of the keys to its performance. The batching and removal of solids require a good control. For some critical designs, the performance cannot be predicted theoretically and such systems require experimental work to determine concentration of feed, sizing (spacing for equipment), material of construction, operating conditions and costs, quality required, etc. For consistence performance dynamic models are useful in evaluating optimal performance.
The design shown below which is common for the process depended on the differences of the solids present in a multicomponent mixture. A schematic diagram of a typical motion is shown in Fig. 1. The gap may be calculated by the expressions:
8 y.max = 8 + (Aj + A2)Sin p 8 y.min = 8 -(Aj + A2)Sin p
where 8ymax is the maximum linear shift in the direction of Y- axis;
y. max "
8 y min is the minimum linear shift in the direction of Y- axis;
8 is a gap between the bar (part 1) and the deck (part 2) as shown in
Fig. 1.;
Ai is a shift amplitude of the deck;
A2 is a shift amplitude of the bar, and
P is the angle of vibration
The schematic representation of the displacement amplitudes of the mechanical parts of VS used in the design and operation of separation process arising from differences in particles illustrates possibility of passage of thin particles through opening S between the bar and deck plane. The device has been tried experimentally but are not in use commercially. This horizontal device can be used to estimate the performance of VS of various sizes and it can be used to predict the effect of specific difference of solids, and allows prediction of capacities at various flows of the mixtures as a function of dynamic parameters. This information will be adequate to determine the final design of a VS, provided the solids of the mix are readily characterized and the solids concentration in the feed is steady. For applications where the solids widely in thickness, a test program should be undertaken. The VS of this type is applicable where separated solids are expected to be very dry. In horizontal VS mathematical modeling is useful in evaluating their performance.
Figure 1. Diagram depicting shift of mechanical parts of the machine used to illustrate the principle of process by particle parameters.
Theoretical calculations are recommended for potential design of VS with the deck generally placed horizontally and acted upon by the rectangular step excitation.
The mathematical basis for process motivated by the rectangular acceleration of translational vibration motion and steps in exact solution of transcendental equations of particle movement are presented below. Motion of single degree-of-freedom system of the VS deck acted upon by the rectangular step excitation (the rectangular acceleration pulse of magnitude w and duration t) is discussed. The corresponding velocities of time histories are also for various conditions. The magnitude of the velocity u change defines the intensity of the process. The longitudinal displacement of the deck during the vibration is characterized by three steps (for purposes of illustration in the following examples the primary time history is that of acceleration, time-histories of velocities may be derived there from by integration). If the velocity u is zero at time t=0, then the velocity time-history is a line of constant slope, the corresponding acceleration time-history is the acceleration step of constant value as was shown]. The first step is defined as a forward motion which has value u of zero and a value of w slightly greater than zero {wx > 0, u1H = 0); the second step describes forth and back motion for the conditions (w2 < 0, u2H = u1H > 0, u2K < 0); that is the acceleration step has a value less than zero significantly; the third step is defined as backwards motion which has a value of acceleration somewhat greater than zero (w3 > 0, u3H = u2K < 0, u3K = 0), where wi is the acceleration step , u
iH
uiK are initial and finite deck velocity steps.
The motivated particle movement is defined mathematically as a function of
w1, w2, w3,u1K,u2K . If the accelerations are w1 = w3 , w1 <-w2 , with the simplest representation of the Coulomb friction force and the effect of air resistance (Fc = 3nfjDv) the mathematical expressions describing the motion of a particle of mass m are
dv r 3nuDv
m— = %• f ■ g--,
dt m
dv r 3nuDv
or -¡T = X'f" g--,
dt m
where f is the coefficient of friction, D is the particle diameter , fj. is the air viscosity, g is the acceleration of gravity. The algebraic sign of the friction term changes when the velocity changes sign (-1 1). For
forward sliding when v < u it must have a positive sign % = +1 ; for backwards sliding : v> u, 1; at rest v = u it is X^+I.
By performing transformation of the latter equation the following differential equation of the particle motion is obtained
dv ,
~r = x-f • g - nv; dt
where n = 3rc^D/m. Rewriting,
dv - %■ f ■ g ■ dt - nv- dt The solution for the latter equation is of the form
" = * — + — I exp[- n{t - tH )J
n ^ n J
I = X — (t - tH)-1 (x^ -Vh^{l - exp[- nit - tH)]}; n n ^ n J
where the required values of particle displacement l and its velocity v are found.
Consider the following operating conditions:
1. w1 < -w2 « fg is a stationary rate.
2. w1 < fg < -w2. A brief review of the complete solution evaluated from a knowledge of these starting conditions is given as follows: the particle size is a somewhat factor with the moving and separation motivated by the rectangular vibration. They occur at values of velocities not greatly different from each other; hence, attention is devoted to the next case.
3. fg < w1 < w2 . For the forward sliding mode (v < u) the highest possible value of the velocity is described by the following equation:
Ц* =
ä. J fjL _U2к 1 exp[- n(T - r)l
n V n
and the expression for the response particle displacement is
l- = fgT - T')~ -ff -v2K!{l - exp[- n(T - T n n ^ n J
For the backwards sliding mode (v> u) the terminal velocity and displacement function are defined by
v2 k =-— + {— ~»ik 1 exp (-nf); n I n I v 7
L =-fgr+1 f fg -
n n V n
ViK I1 - exp (~nT')\,
where T = 12 - tj.
The foregoing equations are alike, mathematically, and a solution may be applied to any of the others by making simple substitutions. Therefore, the equations may be expressed in the general form:
T Y1 - exp (~nT)\ - + ( exp {-nT)~2exp [-n (T - T')])
T -
fg n
w.
1 + exp {-nT) - 2 exp {-nT )
w
fg
г[1 - exp (-/
Tyj • j1+exp {~nT) - 2exp \_~n (T - T)]} = - |w21 • (t1 -T);
fg
n [1 - exp {-nT)]
- exp {-nT) + 2exp (-nT')J = w1 (T2 -12);
l = l +1 = fg (T - 2T ) 12 n '
V = —.
y t
If the air resistance is negligible (n = 0), the equations reduces to the form
= fgT;
V2 K = fg
^y = fg
T
т--
2
T
т--
4
Within each case there are variations and differences of effects. For the third case 3, separation depends essentially on the size differences of the particles
l
present in the mix. The maximum value of the velocities will hold in this case: w1 > fg. w1 > fg (n ^ 0) w2 .The particle velocity forward the deck plane directly related to the friction coefficient which is a function of the particle shape and size. Rectangular pulse excitation: the excitation function given by x and T includes the natural period of the responding system and a significant period of the excitation. The excitation may be defined in terms of various physical quantities, and the response factor may depict various characteristics of the response. The purpose is to compare vibration motions, to design equipment and to obtain useful information. Care must be taken to assure that the same VS and its performance can be predicted theoretically but such VS require experimental work. The mechanical design, however, may be used commercially by the application of vibration considered to be sinusoidal or simple harmonic in form. Alternate form of the excitation may be applied after making simple substitutions of vibration exciters.
CONCLUSIONS
1. In the design and operation of processes depended essentially on the parameter differences of the solids present in the mix a problem is approached logically by first preparing an initial design. A subsequent analysis point to desirable modifications.
2. An initial prototype equipment has been designed and is then constructed in which actual operating conditions preferably are determined and considered from practical point of view.
3. From the analytical point of view and to provide some perspective for later improvement of existing processes the single degree-of-freedom system of the horizontal VS model acted upon by rectangular step excitation is considered with mathematical method of analysis to obtain useful information. The technical feasibility of a given separation method might be essentially attractive.
4. The optimum design must arise from careful consideration of all feasible alternatives and represents the further inventive aspect of process design.
Владимир Дмитриевич Анахин, доктор технических наук, профессор, e-mail: anakhin@mail.ru
Vladimir Dmitrievich Anakhin, Doctor of Technical Sciences, Professor.
