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RESEARCH ADEQUACY OF THE MATHEMATICAL MODELS OF DEFORMATION SHELL ELEMENTS WITH
LARGE DISPLACEMENTS
Dr. of Techn. Sci. Dzyuba A. P., Safronova I. A.
Ukraine, m. Dnipro, Dnipropetrovskiy National University named after O. Honchar
Abstract. Results of numerical and experimental researches of adequacy of two mathematical models of behavior rotation shells and annular plates with large displacements are presented in this paper. The algorithm for acceleration of the iterative process convergence for solving of nonlinear boundary value problem and the results of the comparative analysis of numerical and experimental data are given.
Keywords: rotation shell, annular plate, large displacement, numerical algorithm, experimental research.
The development of effective methods of investigation and substantiation of adequacy of mathematical models for problems of mechanics shells with large displacements belong to very important problems of solid mechanics, as these designs are used in aerospace, chemical, oil and gas, equipment and other industries widely. For example, the bellows as a compensators thermal displacement pipelines, shell elements in the form of membranes as sensing elements measuring devices and others. The specified subject is still very important for research [1, 4, 6 - 9].
The various options for approach to building the equations of nonlinear theory and modeling stress-strain state annular plates and rotation shells that have large displacements with small strains [1, 2, 6] and based on different hypotheses and assumptions associated with additional (non-linear) components are exist. The essence of the known in modern literature approaches to calculating
algorithms such shells is appropriate linearization of nonlinear systems of equations and further construction of numerical iterative schemes refining the solution or in the construction of nonlinear boundary problem to the Cauchy problem with unknown initial conditions, which further refined using various methods (eg, Newton - Kantorovich) [1, 7].
Mathematical models are an approximate in nature, and ensure convergence of the algorithms is often a problem that is not always possible to get results that would be adequate to the real behavior of the relevant structural elements. So, establishing reliability of the results that was obtained by using different approaches to the construction of the nonlinear theory, their comparative analysis, and estimation of errors that in this case the calculation of the linear and nonlinear theories appropriate gives and coincidence of calculation results with parameters of behavior of real structural elements are very important.
In the paper author's variant of acceleration of convergence of the iterative process of solving nonlinear boundary value problem for the case of two different the mathematical models is presented, and experimental researches of behavior of a flexible annular plate under transverse load and corresponding comparative analysis are conducted.
The algorithm for calculating the rotation shells of arbitrary profile is based on the use of the nonlinear theory of shells moment. Nonlinear components (second-order smallness) rotation angle of normal middle surface and its impact on the rest of the components of the stress-strain state is taken into account in equations of state compared with linear formulation. The system of nonlinear differential equations taken as [2]
— = - — cos (9 + 3r)u r sin 9 + —cos2 (9 + 3r)(Nrr) + ds r Kr
+ -^sin (9 + 3 r) cos (9 + 3r) Fp^ + cos (9 + 3 r)- cos 9 + 3 r sin 9,
^ = — (Mrr )-3 r— cos 9 + —(- sin (9 + 3 r) + sin 9 + 3 r cos 9), ds Dr r r v '
diNA = KiL—2) u + — cos (9 + 3 r )(Nrr) + — sin (9 + 3 r) ^ -
ds r r V ' r V ' 2^ (1)
d (MrT ) =3 rD (l — ) cos (9 + 3 r) cos 9 + sin (9 + 3 r)(Nrr ) + — cos (9 + 3 r)(Mrr )-ds r r
F (s) D (l-—2) w / x
- cos (9 + 3 r + —-cos (9 + 3 r) (sin (9 + 3 r)- sin 9-3 r cos 9),
dW = - — sin (9 + 3 r) u + 3 r cos 9 + ^^sin (2 (9 + 3 r)) (Nrr) +
l F (s )
+ — sin2 (9 + 3r)—^ + sin(9 + 3r)-sin9-3r cos9, Kr 2^ V '
where s0 < s < s\ is the length of the meridian; r(s) - the radius of the circle parallel; 9(s),
9+ (s) = 9(s) + 3r (s) - angles between the spin axis and the normal to the undeformed and
deformed surface, respectively; 3r -rotation angle of normal to the middle surface during deformation; u, w - radial and axial movement; Nr - the radial efforts; Mr - bending moment; F(s),
qr(s) - combined axial and radial components of external distributed load; K = E^(l -—2) ,
D = Eh (l2 (l -—2 )) - bending and cylinder stiffness, respectively; E, — - elastic modulus and Poisson's ratio; h(s) - variable wall thickness along the meridian shell.
Boundary conditions for basic state variables {Nr, u, Mr, 3r, wj of shell for the system (1) are given into accordance with the terms of fixing ends of shell
a) rigid clamping: u = 3r = w = 0;
b) hinged: u =w = Mr = 0; (2)
c) free edge: Nr = Mr = 0 or Nr = No, Mr = Mo.
System (1) is made for movements and efforts regarding global coordinate system associated with the normal to the shell axis and the axle, this allows us to consider rotation shell with any form of meridian , including broken, without additional components [2].
Solution of the system (1) considerably complicated by the fact that the value of a variable is included in the coefficients system.
The solving the resulting nonlinear system of differential equations (1), with the given boundary conditions (2), is carried out by its linearization at each step of the iterative process of successive approximations [8] and then by using an analog of method variables elasticity, when the part of nonlinear components attributed to the coefficients system and additional loads, when the value of the external load stepwise increases with depth iterative process and with sharing method of complex iteration (relaxing factors) where the variable value at the z-th point of integration interval (n + 1)-th approximation is given by [11]
aSr1^-y^r?-9ir1}), (3)
ffensen's
calculated by the formula
and Aitken - Steffensen's method [11], where each z-th node of integration variable 9^
Q«q (n-2) /q(n-1A +1) _ *ni*ri n )
*ri _a(»-2) 2q(w-1) + *n ' (4)
* ~ 2* + *,
ri ri ri
when the value of 3 previous iterations are necessary, and calculation of 9^- , 9? , 9? is carried out by the method of relaxation (3) where coefficient y selected by the results of a separate numerical experiment and dependent on process parameters, particular the value of the load. The case when y = 0, as shown in (3), corresponds to the method of simple iteration.
Linear boundary problems is solved using by tridiagonal matrix algorithm of orthogonalization by S.K. Godunov [5] at every step approximations.
The plate is rigid at w/h<0,2; flexible - at 0,2<w/h<5and completely flexible at w/h>5 [3]. One and the same plate under the growing cross-load can be seen as rigid, flexible, or completely flexible consistently.
Two approaches were applied for research of the problem of large movements of such annular plate
In the case by using the method proposed by V.L. Byderman [2], changes in the deformation
angle 0 between the normal and the axis of rotation of the shell is taken into account 0+ = 0 + & r. And annular plate is considered as rotation shell, when its meridian coincides with the radius the plate, where 0= 0, 0+ = 9 r.
From equation (1) get a system of nonlinear differential equations for annular plates relative to vector variables Y = {Nr, u, Mr, & r, wj after simple transformations
mA = KM u + „ cos 9 r (Nr ) + „ sin 3 ;
dr r r r 2%
du „ n nil 2n/*r\ 1 • n F (r)
— =--cos9ru + cos9r -1 +--cos2 9r (Nrr)+--sin 29r—— ;
dr r Kr y ' 2 Kr 2 %
d (Mrr) D (l - „2) „ F (r)
v r ' =—^-sin 29r + sin9r (Nrr) + „cos9r (Mrr)-cos9^; (5)
dr 2 r r 2%
d9r = — (Mrr )-„ sin 9 r; dr Dr r
dw = -„ sin9ru +—— sin 29r (Nrr) + sin 9r + —sin2 9r——, dr r 2 Kr y ' Kr 2%
that is complemented by boundary conditions (2).
In the case by using the other method proposed by Ya.M. Grigorenko [6] second-order smallness are taken into account in the equations for equilibrium and strain. In this case basic equations that describe the deflected mode transversely loaded annular plate relative to vector
variables Y ={Nr, u, Qr, Mr, 3r, wj (r0 < r < r;) with large movements, looks as
dNr 1 - |Ar (l - |2 ) K
-- —--— AT --n-
Nr + --— u
dr r r2
du = - 1 Nr-1 u-13 2;
dr K r 2
= -1 Qr-(l |)Ku3r_1 NrMr + qn; dr r r2 D (6)
dMr = -Qr- 1-1 Mr .(L-l^ S r;
dr r r
^ =—m r-r;
dr D r dw = 3
- = - 3 r •
dr
where Qr - cross efforts; qn- intensity transverse loads.
The system (6) is complemented by boundary conditions:
a) rigid clamping: u = 3r = w = 0;
b) hinged: u =w = Mr = 0; (7)
c) free edge: Nr = Mr = Qr = 0 or Nr = No, Mr = Mo, Qr = Qo.
In the case when the two edges of annular plate rigid clamping or hinged is necessary disclosure of static uncertainty by any known method.
The numerical results have been received by using the developed in [8] iterative algorithm. Ensuring of convergence of the sequence linearized boundary value problems was carried out by combination of methods serial load and Aitken - Steffensen [8, 11] for annular steel plate with rigid clamping outside and loaded force Q0 free inner edge and parameters r0 = 31,5 mm, r1 = 100 mm, h = 0,35 mm, E = 2-105 MPa, — = 0,3.
The relevant boundary conditions are as follows:
for the system (5) at r = r0 u = 3r = 0; at r = ri u = 3r = w = 0;
for the system (6) at r = r0 u = 0, Qr = Q0, 3r = 0; at r = r1 u = 3r = w = 0.
The convergence of the process was estimated by calculating the relative error variable 3r in two consecutive steps approximations for all integration points or variable wmax, as a certain typical values of the maximum bending plate.
Fig. 1 shows the dependence of the number of iterations ngr and nbd, for which the convergence of the iterative process was achieved with relative error s= 10-3 on the value the external load Q0 for cases of system (5) (line 1) and system (6) (line 2).
♦ line 2 A. line 1
Fig. 1 - The dependence of the number of iterations n on the load Q0
The convergence of proposed in [8] algorithm for the case of system (5), (6) is sufficiently high and stable this follows from the numerical results obtained. The process can speed up in 4-10 times depending on the load step compared to a complex algorithm iterations combined with additional loads.
Fig. 2 shows a graph of maximum deflection plate wmax on the load Q0 for linear system (line 1) and obtained by solving systems of nonlinear equations (5) (line 2) and (6) (line 3).
The obtained numerical results show that for small values of external load the maximum deflections calculated by linear and nonlinear theories coincide. When the maximum deflection is half the length (wmax = h/2, Q0 = 2,5 H so, solution theoretically still within the linear theory) value deflections obtained by using the nonlinear equations [2], different from the linear values almost 6.7%. Discrepancy between the results obtained by using two different approaches to the construction of nonlinear systems (5) and (6) is 6.2%. When a value of the maximum deflection is greater than two thicknesses such difference reaches 18,5%.
w (mm)
— — — line 3 -line 1 • line 4 -line 2
Fig. 2 - Dependence of maximum deflection of the plate wmax on the load Q0
It should be noted, that iterative process uses the system (6) requires more iterations than uses the system (5). The number of steps n of the iterative process for solving systems (5) and (6) until the values w/h<1,3^1,8 almost the same and is n =10^15, if w/h>1,8^2.0 then the number of iterations for solving systems (5) and (6) is significantly different (particularly for Q0 = 2,0 N) is almost 4 times. Moreover, such a process for the system (6) loses stability and at wmax > 2h even begins to disperse. These features of iterative process very consistent with results of [6], where the iterative process using an algorithm based on a combination of the Newton's method and the method of continuation of the parameter load (as an initial approximation selected linear solution of the problem) was also divergent. Order to overcome these difficulties authors [6] (p. 107) suggest to build an iterative process with four unknowns at once. These led to the acceleration of convergence iterative process and increase the range of loads but to increased costs of computing resources. At the same time it should be noted that this approach to solving nonlinear boundary value problem (1), (2), (and others) was impossible because one of the major unknown for the iterative process is included to coefficients system as an argument trigonometric functions ie in an implicit form. So, the approach is proposed in the paper has
certain advantages, as is more versatile and provides a sufficiently high rate of convergence for the considered range of tasks.
Experimental research. Experimental researches of behavior of annular steel plate sandwiched rigidly on the outside and loaded evenly distributed transverse force on free inner edge were held in order to evaluate and verify the results obtained using different approaches to drawing up a system of nonlinear differential equations describing the stress strain state annular plate with large displacements and demonstrate the effectiveness the author's calculation algorithm [8]. To determine the maximum deflection annular plate 1 under axially symmetric load 3, uniformly distributed contour of the internal hole that consisted of three parts - fixing 2, 3, 5, load 4 and measuring 6 the special testing device was established (Fig. 3). Physical and geometrical parameters of the test plate coincide with accepted numerical calculations. The plate was loaded with discrete variable transverse load from 1,7 N to 20,7 N. The value of the maximum deflection of the plate amounted to about 2h.
Fig. 3. The device for determining the bend annular plates under axially symmetric load distributed on the inner hole
Measurements of plate deflections were made at four points that were at the ends of mutually perpendicular diameters of plate hole. Arithmetic mean of these measurements was selected as the value of deflection. The data of the testing results are indicated by line 4 in Fig. 2. As seen from the results, data of experimental research give good convergence with numerical calculations obtained by using nonlinear theory describing the behavior of thin-walled structures that has been proposed in [2]. Thus, the difference of experimental and numerical results for investigated values the load does not exceed 5.5% in this case and for the version of the theory has been proposed in [6], the difference reaches 16%.
In general, experimental data also confirm that the method of calculation of nonlinear boundary value problems was proposed in [8] is very effective because the algorithm has sufficiently high convergence, and the results are close to the actual behavior of the design
Summary. Results of comparative analysis of numerical calculation of stress strain state symmetrically loaded annular plates considering large displacements with small strains by using two different approaches to building a system of nonlinear equations of that conduct are presented in the submitted paper.
Experimental data was been obtained, comparison of the experimental and the numerical results was given this demonstrating its high coincidence and confirms the effectiveness of the algorithm and veracity of the numerical calculations.
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АЛГОРИТМ РАСЧЕТА ДИСКРЕТНЫХ КООРДИНАТ КРАТЧАЙШЕГО ПУТИ В УСЛОВИЯХ ПРОСТРАНСТВЕННЫХ ПРЕПЯТСТВИЙ
к. т. н., доцент Мелкумян Е. Ю.
Украина, г. Киев, Национальный технический университет Украины "КПИ им. Игоря
Сикорского ", кафедра технической кибернетики
Abstract. The article describes the research results the object's geometry on-line monitoring of techno-environmental incidents. This results confirmed by computer simulation. The proposed algorithm for finding the coordinates of the shortest path in a changing scene allows to obtain coordinates in real time and used directly for the synthesis of the trajectory of the robot arm movement.
Keywords: object geometry monitoring, the shortest path coordinates calculation, robotics.
Введение. В современной жизни робототехника занимает главенствующую роль в задачах обслуживания человека, особенно в задачах ликвидации последствий его деятельности. Уменьшить степень участия человека при проведении работ в опасных условиях можно, при использовании мобильных роботов (МР), как дистанционно, так и автономно с использованием элементов искусственного интеллекта. Для проведения мониторинга геометрии объекта с помощью МР необходимо иметь алгоритмы управления как самим МР, так и его манипулятором. В рамках задачи синтеза траектории движения манипуляционной части в реальном масштабе времени, по которой можно двигаться плавно и с заданной скоростью, предусматривая запас для безопасного движения исполнительного механизма предлагается алгоритм расчета дискретных координат кратчайшего пути в условиях пространственных препятствий.
Цель. Синтез траектории должен происходить в реальном времени на борту МР, который имеет ограниченные вычислительные ресурсы, алгоритм синтеза должен быть минимальным по вычислительным затратам (а именно, скорости вычислений, размерам требуемых оперативной и физической памяти бортового компьютера) и максимальным по гладкости траектории (т.е. по непрерывности управления) [1].
Процесс восстановления гладкой траектории состоит из двух основных этапов: интерполяции гладкой кривой и аппроксимации (уточнения) полученного набора сплайн функций, которые подробно описаны в работе [2].
Для оперативного мониторинга манипулятором тоннелей определяем следующую задачно-ориентированную инструкцию:
