Тархов Д.А.1, Каверзнева Т.Т.1, Терёшин В.А.1, Виноходов Т.В.1, Капицин Д.Р.1, Зулькарнай И.У.2

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NEW METHODS OF MULTILAYER SEMIEMPIRICAL MODELS IN NONLINEAR BENDING OF

THE CANTILEVER

Introduction

obtained the approximate model for which the accuracy of the description of the real terminal is higher than the exact solution of the original differential equation. Our research is important for longterm forecasting of the status and behavior of construction beams and compression elements of cranes and other real mechanisms.

New methods of building; multilayer models; nonlinear bending of the cantilever; bending of the metal rod.

Modeling of complex technical objects is often hampered by insufficient knowledge of the processes occurring in them. As a result, the structure and coefficients of the differential equations describing these processes are not known accurately, the boundary conditions are known even less accurately. The problem of identifying equations and boundary conditions from the results of observations (the inverse problem) is usually much more complicated than the direct problem of solving a differential equation with boundary conditions. One of the methods for solving such problems is the approach based on the construction of the neural network model of the object by differential equations and additional data [ 1-8 ].

However, the training of neural networks requires a fairly large computational cost. To accelerate these processes, a new class of multi-layer models was developed [ 9-12 ], with which it is possible to do it without a complicated training procedure. In this paper, this approach is tested on the task of modeling a real object using our experiments.

The essence of the approach is to apply the known recursive formulas for the numerical integration of differential equations to an interval with a variable upper limit. As a result, an approximate solution is obtained as a function of this upper limit. In this paper, this approach is applied to the problem of modeling the shape of the bend of a directly cantilevered metal rod. In this case, attempts to select the coefficients of the equation in such a way that its exact solution corresponds to the experimental data with an acceptable accuracy did not lead to success. The developed methods can be used for long-term forecasting of the behavior of building beams, various structural elements of load-lifting machines and mechanisms taking into account the real picture of wear, aging and corrosion of metal.

Material and methods

The measurements were performed with a straight rod made of an aluminum alloy 940 mm long with a circular cross-section with a diameter of 8 mm and a mass of 126 grams. One end of the rod was tightly fixed. At the other end of the tube we weighted weights 100 grams, 200 grams and so up to 1900 grams. The positions of the rod were fixed with increasing mass, which acted on the unattached end of the object under study. The tube was photographed after attaching and detaching of every weight.

As a mathematical model, the equation of a large static deflection of a thin homogeneous physically linear elastic rod is used under the action of distributed q and concentrated p forces [ 14 ].

d 2

 mgL2  mi  z  cos   0 (1) dz2 D  m 

Where D and L are the constant flexural rigidity and length of the rod; θ – is the angle of inclination of the tangent; z  1 s / L - s –the natural coordinate of the curved axis of the rod, measured from the seal, m – the mass of the rod, mi – the mass of the load. In the experiment performed, the distributed and concentrated forces were the weights of the rod and the load at the end.

The boundary conditions have the form: d dz z 0  0; 

 0 z 1 The angle is related to the coordinates of the points on the rod by the equalities: dx dy  Cos( );

 Sin( ); ds ds

To obtain the equation (1) we used the equation of small pure bending of an ideal straight rod of infinitesimal length in the projection on the tangent to the line of large deflection.

Equations (1) and (2) describe the object in question inaccurately – the rod has a yield zone near the seal, has geometric and physical errors. By our methods, an approximate model is constructed by the equation (in the (2) where a  mDgL2 , i  mmi .

As indicated earlier, equation (3) describes the process of bending a rod with a large error. This statement was confirmed by numerical experiments. To build a more adequate model we move from the system (2 – 3) to its approximate parametric solution x(s, 0 , a) and y(s, 0 , a) . Parameters  0 , a can be found by the method of least squares, by minimization of:

N  (x(si , 0 , a)  xi )2 ( y(si , 0 , a)  yi )2 j1 (3) (4) (5) problem under consideration, these are equations (1 and 2)), the parameters of which are refined from the measurement data.

We rewrite equation (1) in the form: where 0  (0) the angle of the rod at its end. Substitution (5) into Simpson formulas let us to obtain dependence x(s, 0 , a) и y(s, 0 , a) . There are no restrictions on the parameters  0 and a .

In fact, accuracy of the given solution is higher, when the parameter a is lower, but in case of an approximate type of equation (3) we are interested not in a small error in the solution of this equation, but in the accuracy of the compliance with measurement data.

Calculations. Let's give the results of calculations for three values of the mass of the cargo m1  0 grams, m2  700 grams и m3  1500 grams. Figure 1 compares the measurement data and the results of calculations using formula (5) for m1  0 grams.

Here N - number of points at which measurements were taken, {xi , yi} - the coordinates of the points at which measurements were taken corresponding to the marks on the rod at distance si from the embedding. In this case, si are not known in advance. To search for them the length of the rod is divided into 100 parts and as si we take a number corresponding to the minimum value of the corresponding summand in the sum (4). Due to the fact that the number si is not known beforehand minimization (4) was conducted using the random search method [ 16 ].

To define functions x(si , 0 , a), y(si , 0 , a) we used method [ 9-12 ]. Its essence with respect to equation (3) is that the known formulas for the numerical solution of differential equations should be applied not to the interval [ 0,1 ] but to an interval with a variable upper limit [0, z] . In this case, instead of a table of numbers, we get a function (z, 0 , a) ,, and the parameters of the problem  0 , a are among its arguments.

From  (z, 0 , a) we go over to the original Cartesian coordinates, integrating (2) according to the Simpson formula for a variable-length interval:

s  M  1 s / l x(s)  1 cos (1 s / l)  4 cos   2M 6M  i1

 M 1  1 s / l   2i 1  2 cos   i   i1   M   , y(s)  s  M  1 s / l

 sin  (1 s / l)  4 sin   2M 6M  i1

 M 1  1 s / l   2i 1  2 sin   i  .

 i1   M   We applied this formula in calculations for M  10 .

As a result of the substitution, we obtain the dependences x(s, 0 , a) and y(s, 0 , a) . Parameters  0 , a , as it was mentioned above, are found by minimization of expression (4).

We present the results of calculations for above mentioned modification [ 9-12 ] of implicit Euler method [ 15 ] with one step. As a result, we obtain an approximate equation  (z) 0  z2a(i  z) cos( (z)), from which we find an approximate solution:

, 200 400 600 800

X mm

The standard deviation of the measurement results from the theoretical curve {x(s, 0 , a), y(s, 0 , a)} is 2.25 mm. The error lies within the measurement error.

On the figure 2 there is a comparison of measurement data and calculation results by formula (5) for m2  700 grams.

200 400 600 800

X mm Y mm 20 40 60 80 100 120 Y mm 50 100 150 200 250 200 400 600 800

X mm The standard deviation of the measurement results from the theoretical curve {x(s, 0 , a), y(s, 0 , a)} is 2.05 mm. The error lies within the measurement error.

On the figure 3 there is a comparison of measurement data and calculation results by formula (5) for m3  1500 grams.

The standard deviation of the measurement results from the theoretical curve {x(s, 0 , a), y(s, 0 , a)} is 3.39 mm. The error lies within the measurement error.

We note that similar results were obtained for other values of the mass of the cargo. Interest is the possibility of predicting the deflection of the rod, depending on the weight of the load. For this we studied the dependence of the parameters  0 , a and l from i . For the formula (5) the following results were obtained:

The data from the graph show that parameters a and l practically do not change almost at all intervals of the change in the mass of the cargo and angle  0 varies linearly. Let’s compaerexpethrimental data with the results of calculations using formula (6) for a  0.03 , l  850 и  0 , calculated on the base of the linearly dependence, that is built by first two dots of the graph 4 for the mass of the cargo 100 gram.

X mm

It can be seen from the graph that the discrepancy between the theoretical curve and experimental data differs no more than 10 mm, which can be considered a satisfactory result.

Results and Discussion

The fundamental difference between our method and traditional numerical methods is the obtaining of a function instead of a table of numbers, while the parameters of the problem naturally enter into the number of arguments of this function. This circumstance makes it possible to use the obtained models as components of cognitive systems, as they are easily adapted when new information about the modeled object appears.

A typical situation for practice is the situation when the results of observations of a real object contradict the mathematical model obtained on the basis of an attempt to apply known physical laws. In this situation, man often seek to refine the physical model of the object and obtain differential equations that reflect the processes occurring in it more accurately.

In particular, we can try to refine the coefficients of the equations. To solve such problems, there are a number of approaches, one of which is the use of neural networks [ 4, 7 ]. Sometimes no choice of parameters allows us to reflect the experimental data with reasonable accuracy, then it is necessary to change the structure of the model, which can lead to a sharp complication of differential equations and does not always ends successfully in a reasonable time. These difficulties often lead to the fact that the model of the object is constructed empirically by interpolation from experimental data.

Our method allows us to apply an intermediate approach, which consists in obtaining approximate semiempirical formulas based on an inaccurate differential model and measurement results. Known theorems on the error of numerical methods [ 15 ] allow us to state that we can obtain an arbitrarily exact approximation to the solution of a differential equation by using a partition into a sufficiently large number of intervals. Our approach allows us to obtain formulas without the use of interpolation, which can be refined from the experimental data. Об авторах: Тархов Дмитрий Альбертович, доктор технических наук, псророфескафедры «Высшая математика», Санкт-Петербургский политехнический университПееттра Великогоd,[e-mail] Каверзнева Татьяна Тимофеевна, кандидат технических наук, д,оцСеаннткт-Петербургский политехнический университет Петра Великkоaгvоe,[e-mail] Терёшин Валерий Алексеевич, кнадидат техничесикх наук, доцентС, анкт-Петербургский политехнический университет Петра Великteоrгvоa, @mail.ru Виноходов Темир Васильевич, студент института прикладной математики и м,ехСаанникктиПетербургский политехнический унивиертест Петра Великогоte,[e-mail] Капицин Даниил Романович, студент института прикладной математики и м,ехСаанникктиПетербургский политехнический университПееттра Великогоk,[e-mail] Зулькарнай Ильдар Узбекович, доктор экономических наук, заведующий лабораторией исследований проблем социальн-оэкономического развития регио н,овБашкирский государственный университет, [e-mail]

Note on the authors:

Tarkhov Dmitriy А., Doctor of Engineering Sciences, Full Professor of the

Peter the Great Polytechnical University, [e-mail] Kaverzneva Tatyana Т., Candidate of Engineering Sciences, Docent, Peter the Great Polytechnical University, [e-mail] Tereshin Valeriy A., Candidate of Engineering Sciences, Docent, Peter the Great Polytechnical University, [e-mail] Vinokhodov Temir V., Student of the Institie of Applied Mathematic and Mechanic, Peter the Great Polytechnical

University, [e-mail] Kapitsin Daniil R., Student of the Institie of Applied Mathematic and Mechanic, Peter the Great Polytechnical

University, [e-mail] Zulkarnay Ildar U., Doctor of economics, head of the laboratory of research of problems of social and economic development of the regions, Bashkir State University, [e-mail]

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