Mathematical simulation of operation of the weighing type filling machine for dry products/Biriuju produktu svorinio dozatoriaus darbo matematinis modelis.
Bendoraitis, K. ; Paulauskas, L. ; Lebedys, A. 等
1. Introduction
Filling machines are widely used in almost every industry.
Construction industry employs filling machines for filling cement and
various components. In agriculture filling machines are used for dosing
feed and fertilizers. Food industry is exceptional for wide variety of
applications of filling machines. Sugar, flour, spices and many other
products and ingredients are being dosed and filled both manually and
automatically during the production and especially packaging of foods.
Packaging technology and technique is among the most powerful stimulus
for development of filling machines [1-3]. It promotes the necessity for
permanent increase of operation speed and dosing accuracy [4-7]. Dry
powdery materials make a big part of products being dosed and packed
through employment of various types of filling machines. Volumetric
filling machines are among the most popular [6]. This is because the
volumetric batchers feature high operating speed. However their dosing
accuracy is dependent on properties of material and not always complies
with the requirements. Therefore, besides the volumetric other types of
filling machines are widely used too, first of all weighing type [6-9].
These can guarantee higher and more stable filling accuracy as they are
much less dependent on the material properties. Furthermore, weighing
machines allow for adjustment of accuracy through setting the right
operating speed. Weighing machines are easy to control, easy to
integrate into automation systems, including sophisticated ones, e.g.
self-trained [5, 9, 10].
Working cycle of real weighing filling machine is quite complex. In
this paper the results of theoretical research of weighing machine in
operation shall be presented. Research out based on specially developed
mathematical model presented in previous paper [5] was carried. The
model covers all the dynamic processes, which may occur during the real
operation of weighing machine for dry products. The sensitive element of
the machine is regarded as an elastic system with one degree of freedom.
It is presumed that a body having variable mass is operating in this
elastic system [6-9]. This paper only presents some final equations of
the model and the simulation results in a form of displacement curves of
10 different projects.
2. Structural scheme of the system containing weighing filling
machine
Weighing type filling machines can achieve high dosing accuracy.
However, investigation of the processes, which influence the accuracy is
not simple as the structure of weighing filling machine is quite
complex. All the components comprising the machine affect the dosing
accuracy [2, 3, 5-7, 9] as well as their interaction due to dynamic
processes, which take place during the operation. A typical structural
scheme of the weighing filling machine is presented in [4, 5]. It should
be noted that filing weighing machine itself is mainly used as an
integral part of entire packaging system. Therefore its analysis should
be carried out taking into account the complexity and features of whole
packaging machine. Its especially important in case of development of
self-learning control systems.
[FIGURE 1 OMITTED]
A typical structural scheme of packaging system with weighing
filing machine (Fig. 1), contains: 1--dry materials; 2--feeder;
3--weighing mechanism; 4--control unit of weighing machine; 5--dose of
dry materials; 6--packing of dose; 7--control system; 8--packing
mechanism; 9--packaging workpiece; 10--control unit of packaging
machine; 11--transfer mechanism; 12--package with dry product.
The system works as follows. Feeder 2 feeds the materials 1 to the
weighing mechanism 3, signal from the weighing mechanism 3 is
transferred to the control unit of weighing machine 4, which disables
the feeder 2, when the quantity of the product in weighing mechanism 2
equals the asked value. At the same time the packing mechanism 8 takes a
workpiece from the packaging stack 9 and produces the pack 6. The role
of transfer mechanism 11 is to guarantee the right positioning of
package 6 against the product 5 being fed. As a result we have a package
filled with dry product. Control unit 10 controls operation of packaging
machine. Control system 7 controls and adjusts operation of the whole
system. Weighing filling machine combines several processes. These are:
vibratory transport, weight measurement, preparation, assessment and
control technology. All these processes need to be investigated
individually including the interaction between them. Vibratory transport
and vibratory processes are widely used [11, 12]. The elastic systems
are paid a lot of attention too [13-15].
Working cycle of weighing filling machine comprises five stages
[5]. First step is filling at increased productivity rate. Step two is
filling at low productivity rate. Step three--suspension of filling
(pause before discharge). The fourth step is discharge of the product.
Step five--suspension of discharge (pause after discharge). Each of
these phases is associated with the characteristics of the product,
accuracy of dosing, dosing efficiency as well as with other parameters
of the system. Those are dosing rate, the system sensitivity and working
range, sensor type, the influence of technological regimes, and others.
3. Mathematical model for describing the operation of weighing
filling machine for dry products
This article deals with a movement of sensitive element at all five
stages. As already mentioned, in this case we deal with the system with
one degree of freedom and the attached body of variable mass.
First step. Filling at high productivity rate. Mass of the product
in the shovel increases by linear law m = [[kappa].sub.1]t. Where t is
time, [[kappa].sub.1] is constant coefficient (efficiency of feeder at
increased rate). The duration of operation of the feeder depends on the
dose size to be set up at this stage. The feeder working range in the
first stage t [member of] [0:[t.sub.1]], where [t.sub.1] =
[[mu].sub.1]/[[kappa].sub.1]-- duration of feeder work, related to the
size of dose [[mu].sub.1].
In the first step the body moves according to the Eqs. (1) and (2)
[4, 5, 7, 12, 16-18].
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1)
and
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2)
where
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3)
where [u.sub.1] is the absolute velocity of joining particle [5], v
is the absolute velocity of shovel, dv/dt is the absolute acceleration
of the shovel, g is due to gravity acceleration. With this assess it is
possible to write differential equation, which describes movement of the
shovel. [J.sub.n] is the first kind and [Y.sub.n] second kind of Bessel
functions [16-18].
[C.sub.1] and [C.sub.2] are constants. The initial conditions of
movement t = 0, x = e = [m.sub.1]g/k, [??] = 0.
[C.sub.1] = A[[tau].sub.1.sup.1+[n/1]][Y.sub.n](B) - [pi][square
root of [b.sub.1]](e - [a.sub.1])[[[tau].sub.1].sup.[n+1]/2]
[Y.sub.n+1](B) (4)
[C.sub.2] = -A[[tau].sub.1.sup.1+[n/2]][J.sub.n](B) + [pi] [square
root of [b.sub.1]](e -
[a.sub.1])[[[tau].sub.1].sup.n+1/2][J.sub.n+1]([B.sub.1]) (5)
where A = [pi]g/[b.sub.1]; B = 2[square root of
[b.sub.1][[tau].sub.1]].
Second step. The second step is filling at low productivity rate.
The mass of the dosing product in the shovel increases by linear law m =
[[kappa].sub.2]t. Where t is time, [[kappa].sub.2] is constant
coefficient (efficiency of feeder at low rate). The duration of work of
the feeder depends on the dose size to be set up at this stage. The
feeder working range in the second stage t [member of] [0:[t.sub.2]],
where [t.sub.2] = [[mu].sub.2]/[[kappa].sub.2]--duration of the feeder
work required to from a dose [[mu].sub.2]. It is necessary assess the
change mass of the shovel. At this stage reduced mass of the shovel [12]
[m.sub.2] = M + [[kappa].sub.1][t.sub.1] + 33/140 m' where
[m.sub.2] is reduced mass of the shovel in the second step. In the
second step the body moves according to the Eqs. (6) and (7) [4, 5, 7,
12, 16-18].
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (6)
and
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (7)
where [u.sub.2] is the absolute velocity of joining particle in the
second step, and
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (8)
[C.sub.1]' and [C.sub.2]' are constants. The initial
conditions of movement are: t = 0, x = [x.sub.1], [??] = [[??].sub.1].
Dimensions [x.sub.1] and [[??].sub.1] are displacement and speed of the
shovel at the end of step one. These values are derived from Eqs. (1)
and (2) when t = [t.sub.1], and
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (9)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (10)
Third step. The third step begins when filling is suspended (pause
before discharge). The shovel mass makes [mu] = [[kappa].sub.1][t.sub.1]
+ [[kappa].sub.2][[tau].sub.2]. In third step t [member of]
[0:[t.sub.3]], here [t.sub.3] is pause time.
In the third step the movement of the body is described by Eqs.
(11), (12) or (14), (15) [4, 5, 12, 16-18]:
When [[omega].sub.0] > [??]:
x = g/k ([m.sub.1] + [mu]) + [Me.sup.-[??]t] sin([omega]t + N) (11)
[??] = -M[??][e.sup.-[??]t] sin([omega]t + N) +
M[omega][e.sup.-[??]t] cos([omega]t + N) (12)
where
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (13)
When [[omega].sub.0] < [??]:
x = ([m.sub.1] + [mu])g/k + [e.sup.-[??]t] (M'[e.sup.[omega]t]
+ N'[e.sup.- [omega]t]) (14)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (15)
In this case
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (16)
Dimensions [x.sub.2] and [[??].sub.2] are accordingly displacement
and speed of the shovel at the end of the second stage filing. These
values are derived from Eqs. (6) and (7) when t = [t.sub.2].
Fourth step. The fourth step is discharge of the product. Mass of
the dosing product in the shovel is falling by linear law m =
-[[kappa].sub.3]t. Here [[kappa].sub.3] is constant coefficient
(efficiency of discharge). The duration of discharge depends on the size
of the dose [mu] = [[mu].sub.1] + [[mu].sub.2] and is expressed t
[member of] [0:[t.sub.4]], where [t.sub.4] = [mu]/[[kappa].sub.3] is
time needed to empty the dose [mu].
In the fourth step the body moves according to the Eqs. (17) and
(18) [4, 5, 7, 12, 16-18].
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (17)
and
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (18)
here [Z.sub.n] (x) = [C.sub.1]" [J.sub.n](x) + [C.sub.2]"
[Y.sub.n](x) (19)
where [C.sub.1]" and [C.sub.2]" are constants, i =
c/[[kappa].sub.3].
When
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (20)
At the time, the movement of shovel in third phase is described by
the equation at t = [t.sub.3]. When [[omega].sub.0] > [??], from
formulas (11) and (12) follows that
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (21)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (22)
When [[omega].sub.0] < [??] from Eqs. (14) and (15) follows that
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (23)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (24)
Finally we get
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (25)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (26)
Fifth step. In the fifth step the discharge is suspended (pause
after discharge). In this case the body moves according to the Eqs. (27)
and (28) or (30) and (31) [4, 5, 12, 16-18].
If condition is that [OMEGA] > [THETA], the body moves according
to Eqs. (27) and (28).
x = [m.sub.1]g/k + A[e.sup.-[THETA]t] sin (pt + B) (27)
[??] = -A[THETA][e.sup.-[THETA]t]sin(pt + B)+ A[p.sup.e-[THETA]t]
cos(pt + B) (28)
where [OMEGA] = [square root of k/[m.sub.1]; [THETA] = c/2m;
[p.sup.2] = [[OMEGA].sup.2] - [[THETA].sup.2], A and B is constant. When
t = 0, x = [x.sub.4], [??] = [[??].sub.4], receive
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (29)
When we have condition [OMEGA] < [THETA], the body moves
according to the Eqs. (30) and (31).
x = [m.sub.1]g/k + A'[e.sup.(p-[THETA]])t] +
B'[e.sup.-(p+[THETA])t] (30)
[??] = A'(p-[THETA])[e.sup.(p-[THETA])t] - B'(p +
[THETA])[e.sup.-(p+[THETA])t] (31)
where [p.sup.2] = [[THETA].sup.2] - [[OMEGA].sup.2], A' and
B' are constants.
When t = 0, x = [x.sub.4], [??] = [[??].sub.4], it is obtained
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (32)
Dimensions [x.sub.4] and [[??].sub.4] are the displacement and
speed of the shovel at the moment when step four ends. These values are
derived from Eqs. (17) and (18) when [t.sub.4] = [mu]/[[kappa].sub.3].
This mathematical model describes the weight batcher operation at
various working conditions and can be used for theoretical
considerations at various development stages of dosing process and the
batcher itself in order to eliminate its critical elements and find the
right setup of the system.
4. Results
The developed mathematical model was used to analyse and compare
the behaviour and efficiency of several projects, which are basically of
the same setup, but differ in dimensions, mass as well as operational
parameters: travel, speed, acceleration, productivity, etc. The
sensitive element of weight batcher is regarded as elastic system with
one degree of freedom. Performance of the system has been analysed based
on displacement curves of the shovel. Project details can be found in
Table. Results of mathematical modelling are shown in Figs. 2-11.
The results of mathematical modelling of the projects have been
compared and influence from various design and process parameters
investigated without deep quantitative analysis. Presented curves show
that both design and process parameters change the nature and size of
movement. The obtained results clearly show the need for further
research. Detailed analysis should be carried out in order to assess the
affect each of the parameters could make and how this would influence
the process of dosing, productivity and accuracy, etc. This information
is essential for establishing optimal filling procedure setup and a
proper design of weighing filling machine.
[FIGURE 2 OMITTED]
[FIGURE 3 OMITTED]
[FIGURE 4 OMITTED]
[FIGURE 5 OMITTED]
[FIGURE 6 OMITTED]
[FIGURE 7 OMITTED]
[FIGURE 8 OMITTED]
[FIGURE 9 OMITTED]
[FIGURE 10 OMITTED]
[FIGURE 11 OMITTED]
5. Conclusions
In the article a typical structural layout diagram and the
mathematical model of operation of filling system with weighing type
batcher is presented. Sensitive element of weight batcher is regarded as
an elastic system with one degree of freedom, operating together with
the variable mass body attached to it. The mathematical model describes
operation of weight batcher throughout its full working cycle, covering
all five stages: stage one--filling at increased productivity, stage
two- filling at low productivity, stage three--suspension of filling
(pause before discharge), stage four--discharge of product and stage
five-suspension of discharge (pause after discharge). The model
incorporates all key elements affecting performance of the batcher and
is fit for theoretical analysis of machine operation, influenced by
various design and process factors. Analysis results representing for
ten projects is presented.
Structural scheme of the system with weighing filling machine for
dry products is analyzed. Sensitive element of weighing filling machine
is regarded as elastic system with one degree of freedom, on which the
variable mass body operates. Mathematical model for describing the
operation of the weighing filling machine for dry products is shown.
Working cycle of weight batcher consists of five stages. Step one is
filling at increased productivity. Step two is filling at low
productivity (accurate filling). The third--is suspension of filling
(pause before discharge). Step four is discharge of product. Step
five--the discharge is suspended (pause after discharge).
The model describes operation of the batcher at various working
conditions and can be used for theoretical considerations at various
development stages of dosing process and the batcher itself in order to
eliminate its critical elements and find the right setup of the system.
Several projects are compared. The modelling results, representing
curves of body displacement, are presented. The curves show that both
the design and process parameters have an influence and change the
nature and size of movement.
The obtained results show the need for further research. Detailed
analysis is to carried out in order to assess the affect each of the
parameters makes and how this would influence the process of dosing,
productivity and accuracy, etc.
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K. Bendoraitis *, L. Paulauskas **, A. Lebedys ***, S. Paulauskas
****, E.Milcius *****
* Kaunas University of Technology, Karaliaus Mindaugo pr. 22, 44295
Kaunas, Lithuania, E-mail: kostas.bendoraitis@ktu.lt
** Kaunas University of Technology, Karaliaus Mindaugo pr. 22,
44295 Kaunas, Lithuania, E-mail: lionginas.paulauskas@ktu.lt
*** Kaunas University of Technology, Karaliaus Mindaugo pr. 22,
44295 Kaunas, Lithuania, E-mail: alis.lebedys@ktu.lt
**** Kaunas University of Technology, Karaliaus Mindaugo pr. 21,
44295 Kaunas, Lithuania, E-mail: saulius.paulauskas@ktu.lt
***** Kaunas University of Technology, Karaliaus Mindaugo pr. 22,
44295 Kaunas, Lithuania, E-mail: e.milcius@ktu.lt
doi: 10.5755/j01.mech.18.3.1883
Table
Projects details of weighing filling machine
project\details k, [m.sub.1], [[micro].sub.1],
N/m kg kg
project 1 2000 0.1 0.126
project 2 2000 0.145 0.126
project 3 2000 0.1 0.126
project 4 2000 0.1 0.126
project 5 2000 0.1 0.126
project 6 4000 0.1 0.126
project 7 2000 0.1 0.126
project 8 2000 0.063 0.126
project 9 2000 0.1 0.222
project 10 2000 0.1 0.222
project\details [[micro].sub.2] c, [[kappa].sub.1],
kg Ns/m kg/s
project 1 0.024 0 0.2
project 2 0.024 0 0.2
project 3 0.012 0 0.2
project 4 0.024 0 0.2
project 5 0.024 0.1 0.2
project 6 0.024 0 0.2
project 7 0.024 0 0.2
project 8 0.024 0 0.2
project 9 0.018 0 0.35
project 10 0.018 0.2 0.35
project\details [[kappa].sub.2], [[kappa].sub.3], h,
kg/s kg/s m
project 1 0.1 0.08 0.01
project 2 0.1 0.08 0.01
project 3 0.05 0.08 0.01
project 4 0.1 0.08 0.05
project 5 0.1 0.08 0.01
project 6 0.1 0.08 0.01
project 7 0.1 0.2 0.01
project 8 0.1 0.3 0.01
project 9 0.075 0.3 0.01
project 10 0.075 0.3 0.1
