TRANSPORT PROBLEMS 2012 PROBLEMY TRANSPORTU Volume 7 Issue 3 optimization, numerical procedure, Gradient Projection Method, Finite Element Method, bearing structure, body, track machine, deflected mode, durability, testing Bogdan TOVT Dnipropetrovsk National University of Railway Transport named after Ac. V. Lazaryan Academician Lazaryan st., 249010, Dnipropetrovsk, Ukraine Corresponding author. E-mail: tovt@ua.fm IMPROVEMENT OF DESIGN TECHNIQUE OF TRACK MACHINES BODIES BEARING STRUCTURES Summary. The paper is devoted to improvement of design technique of track machines bodies bearing structures. The review and analysis of state-of-the-art of the mathematical programming theory and the optimal designing are done. The numerical optimization procedure for track machines bodies bearing structures is proposed. The approbation of the proposed procedure is taken on structural optimization problems. Mathematical aspects of proposed procedure implementation are investigated, particularly the algorithm convergence and the sensitivity analysis of initial designs.

Deflected mode of the ballast leveling machine SPZ-5/UA body bearing structure was investigated analytically (FEM study) and experimentally (trial running inspection on durability). The necessity of optimization implementation for bodies bearing structures is substantiated. The rational design of track machine SPZ-5/UA bearing structure was obtained with the numerical optimization procedure proposed in the paper. The deflected mode of the body rational bearing structure was investigated.

УСОВЕРШЕНСТВОВАНИЕ ТЕХНОЛОГИИ ПРОЕКТИРОВАНИЯ НЕСУЩИХ КОНСТРУКЦИЙ КУЗОВОВ ПУТЕВЫХ МАШИН Аннотация. Статья посвящена усовершенствованию технологии проектирования несущих конструкций кузовов путевых машин. Выполнен обзор и анализ современного состояния теории математического программирования и оптимального проектирования. Предложена численная процедура оптимизации несущих конструкций кузовов путевых машин. Проведена апробация предложенной процедуры на ряде задач оптимизации конструкций. Исследован ряд математических аспектов использования предложенной процедуры, в частности сходимость алгоритма процедуры и анализ чувствительности начальных проектов.

Исследовано напряжённо-деформированное состояние (НДС) несущей конструкции кузова планировщика балластной призмы СПЗ-5/UA аналитическим и экспериментальным путём. Обоснована необходимость проведения оптимизации несущей конструкции кузова рассмотренной машины. При помощи предложенной в работе численной процедуры оптимизации получен рациональный проект несущей конструкции путевой машины СПЗ-5/UA. Исследовано НДС рациональной несущей конструкции кузова рассмотренной путевой машины.

96 B. Tovt 1. INTRODUCTION A railway transport plays a key role in development of Ukraine national economy and conception of development of railway transport and speed motion of passenger-trains is for today worked out and calculated. In accordance with conception of organization of speed motion on the tracks of Ukraine the stage-by-stage rev-up of motion is foreseen on existent lines to 200 km/h with next building of the special speed highways. High-quality and effective realization of the marked program depends on the state of rolling stock on the whole, and track machines stock which produce works from permanent repair and maintenance of railway tracks, in particular.

Therefore the theme of paper, sanctified to the improvement of technique of design of bearing structures of bodies of track machines, it follows to consider actual scientific and technical problem.

The government scientific and technical program of creation of speed highways foresees the substantial restructure of overhead structure of railway that in turn results a necessity for perfection of existing and creation of new structures of track machines with the improved techno-economic indexes, by the strength property increased in particular and lower materials consumption.

Solution of problems of the increase of strength property and decline of materials consumption of both existent bearing structures of track machines and those which are designed, requires the improvement of technique of design of bearing structures of bodies of track machines. One of possible ways of such improvement consists in bringing in modern methods of designing, among which it is possible to distinguish the optimal designing.

2. THE NUMERICAL OPTIMIZATION PROCEDURE FOR BEARING STRUCTURES OF TRACK MACHINES BODIES A structural optimization theory is that key trend of science on the base of achievements of which mechanical structures must be created. The structural optimization theory began actively to develop in 60th, last decades formed new directions, considerable results, both theoretical and applied, were made. Amount of the publications sanctified to the structural optimization theory grows. A considerable contribution to development of theory and development of methods of solution of structural optimization problems were brought in by such scientists, as Аrора, Haug, Haftka, Gill, Keller, Levi, Mruz, Niordson, Olhoff, Prager, Reklaitis, Rosen, Rozvany, Taylor, Terner, et al. Among home scientists have most homage such as Banichuk, Poschtman, Vinogradov, Goldstein, Smirnov et al.

The preponderant number of numerical optimization methods requires the calculation of state variables at the action of the certain loading and also gradients of functions which set constraints on the design and state variables. At consideration of composite engineering structures which the bearing structures of track machines bodies belong to determination of state variables most effective is Finite Element Method (FEM), but its use in optimization procedure causes some difficulties. Videlicet, absence of analytical dependence of coefficients of stiffness matrix of structure from the design variables is not possible in the explicit form gradients of functions, which set a constraint on state variables.

Iteration procedure is offered for the solution of optimization problems of track machines bodies bearing structures. This procedure is based on use of standard bundled software what will realize FEM and one of widespread methods of the constrained optimization – Gradient Projection Method.

2.1. Procedure statement j We use in the next follow table of symbols: initial design b0, allowable design b, intermediate j design w, admissible error in active constraints definition 1, admissible convergence error 2, active constraint set D, objective function, constraint functions i,i =1,...,n,...,m, objective Improvement of design technique of track machines bodies… j j function gradient in b – b, projective matrix P, matrix of the gradients of state variables ( ) constraint functions A, vector, which set iteration scheme course, normalize multiplier, step s parameter, state variables residual h, Lagrangian coefficients v,u, unity matrix I.

( ) The problem of nonlinear mathematical programming in formal statement will formulate thus:

Let’s find such vector of design variables (the design project) b Rk, which minimizes objective function (b) at the set constraint i (b) = 0, i =1,...,n, i (b) 0, i = n +1,...,m.

For the problem solution we use the Gradient Projection Method [1, 2] which algorithm is based on that each iteration objective function decreases, and constraints aren’t violate.

The basic complexity to use the Gradient Projection Method for real structures optimization is represented by the procedure of calculation of the matrix of gradients of the functions setting constraints on design variables and state variables of structure.

The matrix of gradients A looks like [3]:

1 1...

b1 b2 bj 2 2...

(B,Z)T b1 b2 bj, A = = (1) B............

m m m...

b1 b2 bj where: Z – state variables vector of dimension m ; B – design variables vector of dimension j.

(B,Z) Matrix elements can be defined derivation of analytical dependences of constraints from B design variables. But in optimization of real mechanical structures it is almost impossible to receive such dependences, therefore it is offered to receive making elements of the constraint matrix numerically. With that end in view we use known expression for calculation of partial derivatives of many variable functions [11]:

i i (b1,b2,...,bk +bk,...,bj ) i (b1,b2,...,bk,...,bj ) = -. (2) bk bk bk Definition of function values i is carried out by design calculation of Finite Elements Method (FEM). Namely, at first the stress i in an element for which constraint is set is defined, then, after a k -parameter increment, calculation is repeated and the stress i is defined at the changed value of k -parameter.

The elements of constraint matrix are defined so:

-i i i =. (3) bk bk For the purpose of quality standard of influence of variations of design variables on the objective function expression is used:

-i i i, (4) = bk bk where: – normalize multiplier which is calculated according to expression:

=. (5) m -i i bk j= 98 B. Tovt It is possible to name the necessity count of the structure the lack of such procedure by the FEM [4]. However it is leveled by the modern level of development of the computing engineering. The important feature of the offered procedure is a small dimension of matrix of constraints. For example to the problems of mechanics of the deformed body, structure durability at some type of loading is conditioned by few elements durability. It goes out from it, that a number of constraints functions by durability and dimension of matrix of constraints A will be small.

Under the offered procedure, state variables are determined directly with the use of FEM at every step, and gradients of functions, which set constraints on them, – indirectly, with the use of numerical approximation. Thus the use of such procedure allows avoiding foregoing difficulties during optimization of the real track machines structures.

2.2. Procedure algorithm The algorithm of the numerical optimization procedure for bearing structures of track machines bodies is formulated as follows:

Step 0. For active set definition j D = i :i b 1,i =1,..., J ( ) { } j calculate in b constraints in inequality view.

j Step 1. Calculate P, s = -P b and, matrix A is making by (1), and matrix elements are ( ) defined by (2), (3) and (4), calculated by (5). The weighting matrix W may be set with necessity.

-P = W I - AT AAT A.

( ) ( ) Step 2. If s > 2, go to step 3. Otherwise calculate Lagrangian coefficients -v,u = AAT A ( ) ( ) and search um = min ul :l D.

{ } If um 1, stop calculating. Otherwise except constraint m from active set D and go to step 1.

Step 3. Define such maximal step lengthmax, and l w 0 for all l D. For each ( ) ( ) function w is iteration result:

( ) j-wt = b + s -wt +1 = wt - AT AAT h.

( ) j b = wt+with compliance all constraints and return to Step 1.

If at least one of defined constraints is non-compliance repeat calculation by (6) until all constraints are not satisfied.

2.3. Procedure approving The numerical procedure was approved on the number of problems of optimization of structures, similar to the most widespread bearing structures track machines. For test optimization was select a truss, which does not contain a central longitudinal element (7-bar truss, Fig. 1, farther structure А) and a truss which contains a central longitudinal element (6-bar truss, Fig. 2, farther structure B). At both structures constituent elements had a rectangular cross-section. Structure А is under concentrated load (Fig. 1). Structure B is under distributed load (Fig. 2).

Improvement of design technique of track machines bodies… As design variables for both structures the parameters of cross-sections of constituent elements were selected, thus they were accepted by permanent under length of corresponding element.

Fig. 1. 7-bar truss structure Fig. 2. 6-bar truss structure Рис. 1. Конструкция 7-балочного ростверка Рис. 2. Конструкция 6-балочного ростверка T Vector of design variables for a structure A looks like a0 = a1 a2 a3 a4, where a1, a2, a3, [ ] a4 – are parameters of cross-sections of constituent elements of structure A (Fig. 1).

T Vector of design variables for a structure B looks like b0 = b1 b2 b3 b4, where b2 and b4 – [ ] are heights of cross-sections of constituent elements of structure B, b1 and b3 – widths of crosssections of beams of the truss B, permanent sizes within the framework of this problem b1 = b3 = 4 cm (Fig. 2).

As objective function for both structures a general volume was selected because the main circuit of these problems was decrease of structure mass. Objective function for a structure A has expression:

= 5 (50 a1 a2) + 2 (100 a3 a4), cm3.

0 A Objective function for a structure B has expression:

=100 3 b1 b2 + 200 3 b3 b4, cm3.

( ) ( ) 0B Constraints on state variables for both structures were set identically – as strength condition by possible stresses:

i ( ) -[ ] = 0, i ( ) where:, i =1,2 – real stresses in the corresponding elements of structure;

– allowable stress, accepted = 200 MPa.

[ ] [ ] The design variable constraints were also set identically – as a condition of inalienability of structure sizes:

ak > 0, k =1,...,4, bk > 0, k = 2,4.

0 T 0 T or a structure A two initial designs a1 = 3 4 3 5,5 and a2 = 3,5 3,5 4,5 4,5 were [ ] [ ] T T selected, for a structure B – four initial designs b10 = 4 6 4 8, b20 = 4 7,3 4 7,3, [ ] [ ] T T b30 = 4 10 4 8 and b40 = 4 4 4 4 thus the last was in an prohibitive zone in obedience to [ ] [ ] constraints on state variables.

Progress of optimization procedure for a structure A presented by the graph of change of objective function depending on an iteration (Fig. 3). Will mark that character of design variables change for this problem coincides with character of objective function change.

Analysis of results of weight optimization 7-bar truss showed that the choice of different initial designs did not influence on final result – receipt of only optimal design (Fig. 3). An optimal design was attained after 28 and 37 iterations depending on an initial design. As a result of weight optimization 7-bar truss it was succeeded to get a 56 % diminishing of structure weight (Tab. 1).

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