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Home / Issues / № 2, 2013

Materials of the conference "EDUCATION AND SCIENCE WITHOUT BORDERS"

DEFINITION OF RATIONAL FORM OF A DUMP IN TRANSVERSE SECTION
Surashov N.T., Sh.D.Ahmetova

The rational form of a landslip in longitudinal section should be defined taking into account a corner of a natural slope ρ, radius of curvature R and length of an arch l a landslip under a condition maintenance of the maximum area of cross section of a prism of drawing.

Picture 1 – The settlement scheme of interaction of a landslip of the bulldozer with a ground at the maximum set of a prism of drawing on longitudinal section

The area of longitudinal section of a prism of drawing is defined from the sum

(1)

where F1 – the area of a segment; F2 – the area of triangle; F3 – the area of a triangle.

The area of a segment can be defined under the following formula

(2)

The triangle area can be defined under the formula

(3)

where a – size of half of chord ав; Н – height of landslim, m.

The triangle of area can be found on the following expression

(4)

where ве – the maximum length of a prism of drawing in longitudinal section, m.

Summarizing all areas, formulas (2 – 4), we will define the area of longitudinal section of a prism of drawing

(5)

Substituting value of a corner φ depending on length of an arch l and radius сurvature R of a landslip

and having made transformations we will receive

(6)

The given expression represents function of the area of longitudinal section of a prism of drawing depending on design data of bulldozer’s landslip (R – radius of curvature and l – length of an arch of a landslip).

Believing in each specific case a constant height of a landslip and a corner of a natural slope, we will find optimum value of radius of the curvature, providing maximum value of the area of longitudinal section of a prism of drawing, therefore, and volume of a prism of drawing at a minimum of metal consumption of a landslip.

Let's present expression (6) as follows:

(7)

Having made corresponding transformations we will receive

(8)

Equating a derivative to zero and having made corresponding transformations, we will receive following expression

(9)

which expresses optimum values of radius of curvature of a landslip for concrete values of length of an arch of a landslip l and a corner of a natural slope ρ.

Solving the transcendental equation (9) relatively to R, we will receive

(10)

For simplification of expression (5) it is necessary to know optimum value of the central corner of curvature φ.

Believing also in each specific case a constant length of an arch of a landslip l and a corner of a natural slope ρ, we will find optimum value of an angle of slope of a landslip ε.

For this purpose expression (10) we will present as follows

(11)

further we’ll differentiate the right part of expression (11) on ε

(12)

Equating a derivative to zero and having made necessary transformations, we will receive a following equation

(13)

which expresses optimum values of an angle of slope of a landslip ε for concrete ρ and φ.

Solving the equation (13) relatively to ε we will receive the following

(14)

The given expression is the biquadrate equation, that is the equation of the fourth order, for its decision, we will make following substitutions.

Let's designate

(15)

then we will receive following quadrate concerning a new variable a:

(16)

where

Having made appropriate transformations, we will receive

(17)

Let's substitute the received expression (17) in the formula (15) then we will receive dependence for definition ε

(18)

The analysis of the formula (18) shows that optimum values of an angle of slope depend on a kind of the moved ground characterized by its corner of a natural slope ρ.

Believing constants ρ =30˚ and having substituted value tg30˚ = 0,57 in the formula (18), we will receive concrete optimum value of an angle of slope of a landslip

(19)

On experimental data of professor Hmara L.A. value of a corner ε0 is 85˚ … 90 ˚. It is necessary to notice as that value of a corner ε0 according to Zelenina A.N. is ε0=75 ˚. Analytical definition of optimum value of a corner ε0=75˚18 ΄ is in admissible limits, and the divergence with experimental data of professor Hmara L.A. is 11,75 … 16,7 %.

To define optimum value of the central corner of curvature of a landslip φ we differentiate the right part of expression (7) on φ.

(20)

Equating a derivative to zero and having made necessary transformations, we will receive a following equation

(21)

which expresses optimum values of the central corner of curvature φ for concrete ε and ρ.

Solving the equation (21) relating to φ:

We receive a quadrate equation cos φ, that is

(22)

Having designated:

Through A, we will receive a following equation

(23)

Having presented cos φ = a, we will receive a following quadrate equation

solving according to a, we will receive

(24)

or

(25)

From here we find optimum value φ

(26)

Let's define size φopt for constant values ρ = 30˚ and εopt = 75˚18΄.

In the beginning we find value

А=sin²(2∙75˚18΄)+4sin(2∙75˚18΄)ּsin²(75˚18΄) ∙ ctg30˚ + 4sin¹75˚18΄ctg²30˚

А=13,92668.

Substituting А=13,92668 in the formula (25), we will receive the following

We will receive two roots φ1=1 or φ1opt = 0˚ that not really and has no physical sense

where we will receive φ2opt = 150˚.

Size φopt for constant values ρ =45 ˚ and ε =75˚18΄ is equal 134˚06΄.

For parity definition

Also we’ll differentiate the right part of expression (7) on l.

(27)

Equating a derivative to zero and having made necessary transformations, we will receive a following equation

(28)

which expresses optimum values of length of an arch of a landslip lopt for concrete ε and ρ or the relation for the same concrete ε and ρ.

Solving the equation (28) according to we’ll receive a quadrate equation

(29)

where

Having substituted we’ll receive a following quadrate equation

(30)

Solving according to a, we will receive

оr

(31)

(32)

Then we’ll find A for concrete values φopt, ρ and εopt:

1) First case

φopt=134°06', ρ=45°, εopt=75°18', lopt=1,29Ropt

2) Second case

φopt=150°, ρ=30°, εopt=75°18', lopt=0,44Ropt

Conclusion

1. From the condition of maximum cross-sectional area was determined by drawing prism rational form of the blade in the longitudinal section, taking into account the angle of repose ρ, the radius of curvature R and the arc length l, as well as the optimal values ​​of the radius of curvature and angle εopt = 75°18'.

2. Analyzing the process of interaction with the environment bulldozer working bodies, achieved to obtain fundamentally new structures with lateral blade bevels and knives - wideners, which leads to the preservation of the prism drawing during transportation, such as the development of soil loose, unconsolidated, reaching thus the adaptation thus adapting the frontal shape of blade the form of movable prism drawing.



Bibliographic reference

Surashov N.T., Sh.D.Ahmetova DEFINITION OF RATIONAL FORM OF A DUMP IN TRANSVERSE SECTION. International Journal Of Applied And Fundamental Research. – 2013. – № 2 –
URL: www.science-sd.com/455-24454 (29.03.2024).