Construction and practical application of hybrid statistically-determined models of multistage mechanical systems/Hibridiniu statistiskai apibreztu modeliu daugiapakopese mechaninese sistemose kurimas ir praktinis taikymas.
Goroshko, A. ; Royzman, V. ; Pietraszek, J. 等
1. Introduction
It is possible to construct a relatively simple structural model of
complex mechanical systems, adequate to real object by identifying such
primary factors which have a significant impact on the output
characteristics of the object material, simplifying the model by
discarding unimportant factors. Meanwhile, it is disputable whether the
criterion of substantiality of the primary factors considered is valid.
Although methods of parametric sensitivity theory [1, 2] allow to
evaluate the degree of influence of each of the primary factors on
baseline characteristics of the object, it does not provide an answer to
this question.
At the same active methods of design of experiments (DOE) for
building statistical models provided a reasonable opportunity to screen
out non-essential primary factors [3]. However, DOE can be performed not
only on the real object, but also on its determined mathematical model.
In this case, using the principles of DOE will also enable the specified
determined model simplification.
For this use of DOE as a method of building statistical models to
analyze and simplify deterministic models demonstrates the close
relationship of these species examined of structural model of mechanical
systems and processes for their manufacture. Indeed, the abovementioned
division of mathematical models is rather relative, since both
approaches are organically interrelated and complementary. Thus,
clarification and authentication of determined models should be
implemented by the statistical data of experiments on the real object or
physical models, and statistical models--should be constructed with
regard to well-known theoretical relationships. In addition, the
identification of reliable statistical distribution models of structural
model of complex technical products parameters that can reasonably judge
the actual spread of values, increases the effectiveness of the use of
mathematical
The following describes some of the ways that reduce the complexity
and duration of the operations that are necessary to implement a full
factorial design to create mathematical models of many cascading of
structural model of mechanical systems. This can be achieved by
constructing hybrid statistically determined models, and through
informed choice of the minimum number of n overlap each experiment with
the given reliability P.
2. Problem statement
Let's consider an object that contains such independent
cascades where the variation of the primary factors of any of them
changes the output characteristics of this cascade only. The problem
through the use of DOE simulated output characteristic of all products
in cases where mutual influence cascades is known in advance. Thus we
consider multistage products, whose design and traditional manufacturing
techniques cannot and (or) does not provide for an interim control of
individual stages. At the same time, the possible values of the
simulated measurement function for an arbitrary set of values of the
primary factors of all cascades.
3. The results of theoretical studies
Further, we'll provide the solution method of the considered
problem.
The desired model can be represented as a known function:
y = f ([[phi].sub.1], [[phi].sub.2], ..., [[phi].sub.k]), [phi]
[member of] [R.sup.k], (1)
where
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2)
are unknown functions which model i -th cascade, and [x.sub.ij] is
coded primary factors.
We denote the set of factors {[MATHEMATICAL EXPRESSION NOT
REPRODUCIBLE IN ASCII]} vector [x.sub.i], i = 1,2, ..., k, then
relationship (1) and (2) takes the form of
y = f ([[phi].sub.1]([x.sub.1]), [[phi].sub.2]([x.sub.2]), ...,
[[phi].sub.k]([x.sub.k])). (3)
Thus, the task is to construct a method of DOE polynomial
representation of the function (3), specified by the primary factors.
The method for solution of this problem is suggested, which is based on
modeling the dependence of output of the product of the primary factors
of each stage separately (the value of the primary factors other stages
at this time are fixed at a certain level) and subsequently combining
them within a common object model in accordance with the known
theoretical dependence.
To implement the proposed methods it is necessary to carry out the
following sequence of actions.
Formation at the beginning k support functions:
[y.sub.i] = [f.sub.i] ([[phi].sub.10], [[phi].sub.20], ...,
[[phi].sub.(i-1)0], [[phi].sub.i]([x.sub.i]), [[phi].sub.(i+1)0], ...,
[[phi].sub.k0]), i = 1,2, ..., k, (4)
where [[phi].sub.j0] is fixed but unknown value function
[[phi].sub.j] the unknown fixed set of primary factors [x.sub.j0].
Applying k times full factorial design, we obtain the polynomial
representation of each function [y.sub.i] in the form:
[y.sub.i] = [b.sub.0] + [[l.sub.i].summation over (i=1)]
[b.sub.i][x.sub.i] + [summation over (i,j)] [b.sub.ij][x.sub.i][x.sub.j]
+ ..., (5)
where [b.sub.0], [b.sub.i], [b.sub.ij] are coefficients of the
model, written in coded values of factors. Thus, by implementing
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] experiments
determine the dependence [y.sub.i] = [y.sub.i]([x.sub.i]). Next,
substituting the obtained function alternately to the left side (4),
expression all [[phi].sub.i]([x.sub.i]) through [y.sub.i]([x.sub.i]) and
k - 1 constant [[phi].sub.j0]:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (6)
Measuring the value of simulated output characteristics for fixed
values of the primary factors that record value:
[y.sub.0] = f[[[phi].sub.1]([x.sub.10]), [[phi].sub.2]([x.sub.20]),
[[phi].sub.3]([x.sub.30]), ..., [[phi].sub.k]([x.sub.k0])] = = f
([[phi].sub.l0], [[phi].sub.20], ..., [[phi].sub.k0]).
As can be seen from (4),
[y.sub.i]([x.sub.i0]) = f([[phi].sub.10], [[phi].sub.20], ...,
[[phi].sub.(i-1)0], [[phi].sub.i], [[phi].sub.(i+1)0], ...,
[[phi].sub.k0]) = [y.sub.0].
Substituting Eq. (6) by Eq. (1), after transformation we obtain the
function y, depends now on the primary factors [x.sub.1], [x.sub.2],
..., [x.sub.k] fixed and unknown numbers [[phi].sub.10], [[phi].sub.20],
..., [[phi].sub.k0], i.e.:
y = [PSI]([x.sub.1], [x.sub.2], ..., [x.sub.k], [[phi].sub.10],
[[phi].sub.20], ..., [[phi].sub.k0]). (7)
Because [x.sub.i] = [x.sub.i0], i = 1,2, ..., k occurs y =
[y.sub.0], it is true inequality:
[y.sub.0] = [PSI]([x.sub.10], [x.sub.20], ..., [x.sub.k0],
[[phi].sub.10], [[phi].sub.20], ..., [[phi].sub.k0]), (8)
that allows you to opt in Eq. (8) from unknown [[phi].sub.i0],
expressing them through the measured value [y.sub.0], and thereby obtain
the desired appearance modeling function.
At the same time, necessary for the implementation of the described
methods the number of experiments [MATHEMATICAL EXPRESSION NOT
REPRODUCIBLE IN ASCII] significantly smaller number of experiments
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], required to
implement the standard full factorial design.
For the informed choice of the minimum number of repetitions of
each experiment by measuring the value of the function y will use the
method of confidence intervals for assessing the expected value and
standard deviation of normally distributed random variables with given
probability [P.sub.1], [P.sub.2].
It is easy to show that the reliability [P.sub.1], [P.sub.2] number
n can be found from the expression n = [T.sup.2][[sigma].sup.2] [(1 -
q).sup.2]/[[delta].sup.2], where a is selective standard deviation is
corrected, T is such value of argument of function of Laplace [PHI](T),
at which [PHI](T) = [P.sub.1]/2, q is number which is determined from a
table [4], [delta] is set number that determines possible deviation of
mean value of the realized experiments from the truth value of measuring
size.
Thus, the offered method of construction of the hybrid
statistically-determined models of multistage objects, that allows to
form statistical models taking into account the known theoretical
dependences, gives the considerable winning in an amount experiments at
raising of DOE in comparing to the construction of clean statistical
models.
This circumstance acquires the special importance in those cases,
when amount of primary structurally-technological factors that influence
on the value of initial description of object and amount of cascades
great enough, and also, when quality of work of good is characterized
not one, but by a few initial parameters.
4. Implementation outcomes of the method
The method of creation of the hybrid statistically-determined
models was applied for providing the power stability of initial signal
of microwave-range device. The signal generated by mechanical factors.
The device being examined is the transmitter of airplane defendant
(AD). AD SO-69 (Fig. 1) is used to work with the air traffic control
systems and installed on civil and military aircraft, such as for
example the MiG-23, MiG-25, MiG-27, MiG-29, MiG-31, Tu -142, Tu-95,
Yak-40, L-410, AN-26 and transport helicopter Mi-26. In its manufacture,
testing and operation there were multiple cases, when AD were accepted
during initial tests during that correspondence of all initial
descriptions was set to the technical requirements, and farther in
exploitation there were refuses of these wares on completion of some
time. The analysis of such refuses showed that most percent of
non-fulfillment (23.7%) AD the functional setting took place in a kind
the impermissible on technical requirements decline of power of initial
high-frequency signal.
[FIGURE 1 OMITTED]
Previous experiments, carried out by the manufacturing plant,
showed that further treatment of electric parameters of defendant cannot
result in stabilizing the size of power of its outgoing high-frequency
signal, as these parameters on the basis of existent element base are
well-proven to some optimal values. However, the indicated power is
formed under the action of some primary factors of mechanical nature.
Thus, we put task to identify primary factors of mechanical nature
that influence on the size of power of initial high-frequency AD signal
and to find the great number of values of these factors that provide
implementation of technical requirements on this initial description
during all term of exploitation.
5. Construction of mathematical model of assembly of AD
Power of initial high-frequency signal is formed in the transceiver
of basic block of SO-69, and more precisely in intended for generating
of high-frequency impulses in a decimetric range generator of microwave
transmitter.
Microwave oscillator contains two independent cascades. The first
cascade is an oscillator (master clock MC) second--power-amplifier (PA).
Thus it is known that power of weekend of high-frequency signal U
appears as a product of two functions U = [[phi].sub.1] x [[phi].sub.2]
that design each of cascades of MC ([[phi].sub.1]) and PA
([[phi].sub.2]), i.e.:
U = [[phi].sub.1][[phi].sub.2]. (9)
Thus construction and technologies of stowage of defendant are not
envisaged by intermediate control of these initial descriptions of
separate cascades. Measuring of values of initial description of
generator is possible only at the arbitrary sets of values of primary
factors both MC and PA. Taking into account such multistageness of
microwave oscillator for the design of power U was used described higher
method of construction of the hybrid statistically-determined models.
Will consider application of the worked out method for the design
of work of microwave generator, for that a transmission function looks
like Eq. (9), where, [[phi].sub.1](x), [[phi].sub.2](x) are unknown
functions that design MC and PA accordingly. Then the measured value of
the designed power equals at some fixed values of primary factors of
MC([x.sub.10]) and PA ([x.sub.20]):
[U.sub.0] = [[phi].sub.10][[phi].sub.20], (10)
where [[phi].sub.i0] = [[phi].sub.i]([x.sub.i0]), i = 1,2 is fixed
but unknown value function [[phi].sub.i].
Will form auxiliary functions:
[U.sub.1] = [[phi].sub.1]([x.sub.1]) [[phi].sub.20], [U.sub.2] =
[[phi].sub.10][[phi].sub.2] ([x.sub.2]), (11)
from that
[[phi].sub.1]([x.sub.1]) = [U.sub.1]([x.sub.1])/[[phi].sub.20],
[[phi].sub.2]([x.sub.2]) = [U.sub.2]([x.sub.2])/ [[phi].sub.10]. (12)
Putting Eq. (12) in (9), will get U = [U.sub.1]([x.sub.1]) x
[U.sub.2]([x.sub.2])/([[phi].sub.10] x [[phi].sub.20]), where with
taking Eq. (10) into account find:
U = [U.sub.1]([x.sub.1])[U.sub.2]([x.sub.2])/[U.sub.0]. (13)
To get the statistically-determined model of kind Eq. (13) it is
necessary to build polynomial models for functions [U.sub.i]([x.sub.i]),
i = 1,2 by realization [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN
ASCII] of experiments on the chart of full factorial design ([l.sub.1],
[l.sub.2] are amount of primary factors for MC and PA accordingly). To
that end it was first of all necessary to define those primary factors
of MC and PA, that must be taken into account in these models.
On the basis of expert survey of specialists who have experience in
previous research of defendant construction and their assembly
technology, it was found that the following parameters of details and
knots of each of the cascades of oscillator transmitter can
significantly influence on initial description U--power of initial
high-frequency signal.
Parameters of details and knots of MC: pressing force of the anode
plug to the anode collet [q.sub.1] x [10.sup.2], N; body diameter
cathode-grid circuit [q.sub.2] x [10.sup.-3], m; diameter glass
cathode-grid circuit x [10.sup.-3], m; arithmetic average roughness
glass cathode-grid circuit [q.sub.4] x [10.sup.-6], m; pressing force of
the large plunger [q.sub.5] x [10.sup.2], N; the arithmetic mean
deviation of the anode surface profile case circuit [q.sub.6] x
[10.sup.-6], m.
Parameters of details and knots of PA: diameter of glass
cathode-grid circuit [Q.sub.1] x [10.sup.-3], m; diameter of body
cathode-grid circuit [Q.sub.2] x [10.sup.3], m; the arithmetic mean
deviation of body surface profile of the anode circuit [Q.sub.3] x
[10.sup.-6], m; pressing force of the large plunger to the wall circuit
[Q.sub.4] x [10.sup.-2], N.
Farther in correspondence with worked out before by the method of
construction of the hybrid statistically-determined model full factorial
design was conducted, as a result of that expression is got for power
through absolute variables:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].
6. Identification, analysis and ways of removal of influence of
basic destabilizing primary factors
Undertaken studies educed two priority factors that more
substantial in all influence on the size of power of initial
high-frequency signal of AD: AV arithmetic rejections of profile of
surface of corps anodic to the contour and effort of pinning of great
[TEXT NOT REPRODUCIBLE IN ASCII] to the wall to the contour.
In order to study the possibility of change in the values of these
factors in the process of exploitation and their destabilizing influence
on the size of initial power conference of specialists on the
manufacturing plant was held, where the following statements were made.
In the microwave-range endovibrators are used as oscillating
systems where the transitional resistance of contacts of piston
influences with the corps of resonator (Fig. 2) and with an anodic
collet has a significant impact on the power. For providing of reliable
contact it is necessary to have an internal surface of corps of
resonator and external--anodic collet, enough high class of cleanness.
However at the frequent tuning to the contour of resonator there is a
heterogeneous wear of surface of contacts through the wipe of layer of
silver, that covered plungers, that results in large variation of size
of transitional resistance of contacts on length of their moving. The
considerable rejections of electric parameters of resonator take place
from it, in particular, initial power in the process of exploitation.
Thus, it was set that priority factors are considered and can
destabilize the size of the indicated power. Research of possibility of
support of values of the indicated factors at certain level during all
term of exploitation showed that for the first from them implementation
of such requirement was arrived at difficult and expensive, and
adjusting of effort of pinning of large plunger to the wall of corps in
general is not envisaged in a construction. In addition, even providing
of the indicated requirements does not remove the wipe of thin level of
silver on the surface of imyHKcepa. In this connection further measures
from providing of the indicated requirements on the considered
destabilizing factors were confessed by ineffective, and was made
decision about the necessity of revision of AD construction.
The construction of resonator was simplified by introduction of
dielectric retort (Fig. 3), dielectric basis and silver segments, as a
necessity of centring of piston falls off at the use of dielectric
glass, requirements go down to the class of exactness of making of
metallic retort and corps of resonator a surface, and the exception of
direct contact of piston with the corps of resonator allows to bring
down requirements to the cleanness of internal surface of corps of
resonator and metallic glass. Thus, in the new pin-noncontact
construction of AD it was succeeded to remove destabilizing factors that
influence on the size of power of weekend high-frequency.
[FIGURE 2 OMITTED]
[FIGURE 3 OMITTED]
For the modified construction of AD the hybrid
statistically-determined model was also built:
U = (8.858 + 0.109[q.sub.1] + 27 x [10.sup.3] [q.sub.2]) x x
(22509.043 + 3.12[Q.sub.1] + 124.857[Q.sub.2] - 812-103[Q.sub.1]).
The set dependences of power of initial highfrequency signal on the
primary factors of defendant it is allowed to put plural reverse problem
of determination of such admittances on these factors, at that
implementation of condition would be provided on the considered initial
description:
U [greater than or equal to] [U.sub.0], (14)
where [U.sub.0] is minimum value of power, that answers
requirements to the technical requirements.
The considered problem can be set forth as follows. At the set
nominal values of primary factors [x.sub.0] = {[q.sub.20], [q.sub.40],
[Q.sub.10], [Q.sub.20], [Q.sub.30]} it is necessary to define such their
possible rejections [[delta].sub.i], i = 1, 2, ..., 5 from basic values,
that in the got parallelepiped [x.sub.i0] - [[delta].sub.i]/2 [less than
or equal to] [x.sub.i] [less than or equal to] [x.sub.i0] +
[[delta].sub.i]/2 a condition (14) was executed on initial description.
As coming from reasoning of economy admittances it is desirable
maximally to extend on all parameters, then the considered problem is a
multicriterion, and the plural of criteria of part can be set forth in a
kind
[F.sub.i] = -[[delta].sub.i] [right arrow] min, i = 1,2, ..., 5.
For erection of this multicriterion problem to the onecriterion
linear convolution of criteria of kind was used [??] = [5.summation over
(i=1)][c.sub.i][[delta].sub.i] [right arrow] max [5, 6], where [c.sub.i]
[greater than or equal to] 0, [5.summation over (i=1)] [c.sub.i] = 1 is
rationed positive numbers that is determined from the productive or
economic reasoning.
Thus, a problem is taken to determination of such values of
admittances [[delta].sub.i], i = 1, 2, ..., 5, that achieves the
objective function maximum [??] on condition of implementation of limits
on the primary factors taken from technical requirements and AD output
description. Limitations were set by the set of inequalities (Table).
For the new construction of AD determination of admittances was
conducted on the primary factors of defendant, the basic values of that
are driven to the Table.
The values of primary factors of AD are got at [[delta].sub.i],
that maximize a function [??] for [U.sub.0] = 600 BT, driven to the
Table.
7. Conclusions
1. Developed a method for constructing a hybrid statistical and
deterministic models of multistage structurally complex of mechanical
systems, which can reduce the amount of experimental research. The
surgical creation of a statistically-determined model to determine
reasonable assumptions on the value of the primary structural and
technological factors of a stabilization of output parameters to set
limits, achieved the study of functions of several variables obtained
from a known mathematical methods.
2. Constructed hybrid statistical-deterministic model of power
output high-frequency signal of AD under primary factors of mechanical
nature. Identified priority factors that destabilize the value of
output.
3. Defined tolerances for primary factors of AD in which compliance
is guaranteed power output high-frequency signal with specifications
Received June 17, 2014
Accepted October 15, 2014
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Mathematical Modeling Of Electronic Means in Mechanical Actions. Moscow:
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[4.] Wentzel, E.S. 1969. Probability theory [Russian translation].
Moscow: Nauka. 576p.
[5.] Nakayama, H., Yun, Y., Yoon, M. 2009. Sequential Approximate
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A. Goroshko *, V. Royzman *, J. Pietraszek **
* Khmelnytskyi National University, Instytutska 11, 29016
Khmelnytskyi, Ukraine, E-mail:
[email protected]
** Cracow University of Technology, Krakow, Poland, E-mail:
[email protected]
http://dx.doi.org/10.5755/j01.mech.20.5.8221
Table
Values of the primary factors and constraints in the optimization
problem
A limit is on Value of primary factors of AD
primary factors and
initial description
of AD
U [greater than or Nominal Optimal
equal to] 600
50 [less than or [q.sub.20] = 250 211 [less than or
equal to] [q.sub.1] [10.sup.2], N equal to] [q.sub.1]
[less than or equal [less than or equal
to] 450 to] 289
5 x [10.sup.-6] [q.sub.40] = 6.25 6.25 x [10.sup.-6]
[less than or equal [10.sup.-6], m [less than or equal
to] [q.sub.4] [less to] [q.sub.4] [less
than or equal to] 2 than or equal to] 2
x [10.sup.-5] x [10.sup.-5]
0.5 [less than or [Q.sub.10] = 2.50 N 0.5 [less than or
equal to] [Q.sub.1] equal to] [Q.sub.1]
[less than or equal [less than or equal
to] 4.5 to] 4.5
0.33 [less than or [Q.sub.20] = 0.33065 0.0330 [less than or
equal to] [Q.sub.2] m equal to] [Q.sub.2]
[less than or equal [less than or equal
to] 0.033 to] 0.0331
0.0320 [less than or [Q.sub.30] = 0.03201 0.03191 [less than
equal to] [Q.sub.3] m or equal to]
[less than or equal [Q.sub.3] [less than
to] 0.032 or equal to] 0.03209