Kadyrov A.S., Kurmasheva B.K.

Karaganda State Technical University

Development and research of mathematical model of

optimization of trench machines purpose parameters

 

Process of trench foundations arrangement is characterized by various organizational and technological parameters which gang defines conditions of construction. These parameters form general economic parameters according to which efficiency of building process is estimated. Establishing interrelations among parameters of building process is the main task of analytical research.

After formation of set of works manufacture technological variants, dividing them into subsets depending on types of machines carrying out the leading operation, function of reduced expenses Ïç.î was investigated. As a result of mathematical transformations dependence (1) [1] is received

,                                         (1)

where  a is a parameter describing conditionally constant part of reduced expenses for works amount;

b is a value defining change of reduced expenses depending on cost expression of digging speed;

c is a parameter describing influence of specific energy intensity on reduced expenses;

N is realizable capacity of the digging machine;

V is driven element axis velocity.

At the stage of hypothetical (or projected) driven element optimization, for which it is impossible to define capital investment into the base machine, criterion “reduced expenses” decreases to the part of the first cost independent of capital investments.

.                                              (2)

The general view of reduced expenses and the first price functions (1) and (2) remains constant. An attempt to define absolute minimum of function Ïç.î from two variables N and V was unsuccessful as first derivatives are not equal to 0. In this connection establishment of minimum of function Ïç.î was made by Lagrangian method.

Research of criterion function consists in finding of its extreme points and revealing of their character (maximum or minimum). The analysis of the goal function can be executed by various mathematical methods. If between variables the interrelation is formally established research of criterion function is reduced to definition of relative extremum. If such relation misses, the optimum variant is defined following search and comparison of all ways of works execution.

Let us assume there is following interrelation between arguments N and V:

 

.                                              (3)

 

The relative extremum of criterion function can be found with a method of Lagrangian multiplier that allows investigating its several variables. For this purpose auxiliary Lagrangian function is made:

                             (4)

where  f (N, V) is goal function;

λi is Lagrangian multiplier for i-equation of relation;

m is a number of the relation equations between variables.

Then we calculate and equate to zero partial derivatives on N, V and λ:

 

                                   

Received equations are united in a system and are solved concerning variables N, V, λ. The equation system represents only necessary conditions of the first order, so N and V will refer to as conditional stationary points. For definition of their character according to work [2] there is a condition that allows establishing extreme values of function:

,                                          (8)

where                  xi = N;

ε1 = ΔN;

ï = 2;

x2 = V;

ε2 = ΔV.

The quadratic form second order condition (4) is sufficient to define relative extremum character. If in the received point the goal function is minimal, the quadratic form is positive, and, on the contrary, if it is maximal, the quadratic form is negative. Thus equality should be carried out

,                        (9)

where k = 1, 2... m.

Let us consider a group of driven elements where drilling, milling, bar and other machines that have one rotary and linear and translational and relative motion (or on the contrary) are included.

Capacity is connected with parameters of process by dependence

,                                                     (10)

where Q is haulage pull to the cutter;

          M is the moment from cutting force;

         ώ is angular speed.

Opening values of the torque and haulage pull by expressions of specific forces of resistance to approach and rotation of a chisel, we shall receive:

,                            (11)

Transforming equation 11 we used dependence, connecting angular speed, thickness of cut off scrap and axis velocity

.

Let's present the equation of communication in the form of

,                                          (12)

where m=2πRB, ..

Making an auxiliary function of Lagrange introducing Lagrangian coefficient into it we shall receive

.                 (13)

Let us define private derivatives on N, V, λ and we shall equate them to zero

.

Solving the system of three equations allowed to define extreme values of driven element axis velocity and realized capacity

,                                         (14)

 

.                             (15)

When solving the system of the equations we received coordinates of a conditional stationary point. Character of relative extremum in this point is defined according to the equations (8) and (9).

If minimum in a conditional stationary point exists the inequality should be

.        (15)

Let us make the analysis of expression (15)

.                                      (16)

In the second equation of system (16) having used substitution from expression of first derivative function L on N having replaced λ we shall receive

.

Thus, all the second derivative functions L are positive, as they include economic and physical parameters which inherently cannot be negative. For the same reason increments ΔN and ΔV are positive. Thus, the goal function under arguments N and V defined from equations (14) and (15) will be minimal.

Analyzing function of the first cost (2) solving algorithm will be same and the minimal goal function will be defined by coordinates

 ,                                              (17)

.                                         (18)

The analysis of operating modes researches results of machines and the mechanisms participating in the process of making ditch foundations shows that formal interrelation between parameters N and V is established for digging ditches by rotary [2, 3] drilling. For digging ditches by churn drilling there are available statistical data. As similar data are not received from the review for digging ditches without excavation they were established during these researches.

 

The list of the used sources

 

1. Kadyrov A.S. Theory and calculation of milling and boring driven elements of digging machines applied at construction by the method of “walls in ground”., ÌICI named after Kuibyshev, dissertation for scientific degree of doctor of technical sciences, 1989 – p.273.

2. Kadyrov A.S., Korkin A.A. Establishment of technological parameters of drilling machines and mechanisms with screw driven elements. Mechanization of labour-intensive processes in road construction industry. Karaganda, 1983. – pp. 102-104.

3. Kadyrov A.S. Researching loading of screw driven elements of big diameter (with reference to drilled-in caissons): Autoreport of dissertation of candidate of technical science.- M, 1979. – p.22.