The system with three flax stairs of moving process of Crane Bridge.

Krasniqi, Fehmi ; Ibishi, Ismet Maliq ; Halimi, Imer 等

1. INTRODUCTION

Cranes in general and Cranes Bridge privately are machinery, which serves for the process of loading and unloading in transport. Cranes Bridge serves for internal transport in factories. With perfecting and increasing of productivity and industrial plants of different fields and needs for circulation of goods, moreover we need to use Cranes. It's necessary that movements of Cranes to be safe and without consequences from dynamic loads, which spring up in these machinery during their process of work. During the work of cranes springs up dynamic force which are unwanted. The time of movement during the work of cranes is separated in two periods:

The first period include the time when the load starts to lift up but it's still on the base. And the second period is when the load starts to move from the base.

These parameters are necessary to fix elementary conditions of the movement.

2. MOVING EQUATION FOR SYSTEM WITH THREE FLAX STAIR

At issue of swivel measure of mechanism for lifting up, we approach to the real situation of studying of Crane Bridge that means Crane Bridge present a system with three flax stairs. In contrast to system with two flax stairs, at the system with three flax stairs take into consideration during the calculation and the impact of swivel measure of lever mechanism. Differential equations of movement are more complicated but the solution is so exact. Differential equation of movement is won from the equivalent system according to the figure 1.

Extending of the rope in the second period is:

z = r x [phi]/[i.sub.1]--[y.sub.1]--[y.sub.2] (1)

Swivel measure of mechanism for lifting up, of reduction in axis of tumbler has a great impact in number of motor rotations in the first period of movement. Researches and measures have shown that the moment of motor is changeable, but it will take the simple case of researches with, M = const. In the first part of movement takes part the mass of mechanism for lifting up and the mass of carrier construction. From the equilibrium forces of system are defined differential equations of movement in the first period.

[J.sub.T] [??] + [F.sub.1]/[i.sub.1] r - [M.sub.T] = 0 (2)

[m.sub.b] [[??].sub.1] - [F.sub.1] +[F.sub.0] = 0 (3)

[FIGURE 1 OMITTED]

Where are:

[J.sub.T]--The moment of inertia of swivel mass of mechanism for lifting up, of reduction in axis of tumbler;

[F.sub.1] = [c.sub.1] z--Force in rope;

[F.sub.b] = [c.sub.b] [y.sub.1]--Force which action in carrier construction of crane bridge;

r--Radius of tumbler;

[i.sub.1]--The report of transmission of under wheel. Extending of rope in the first period.

z = r[phi]/[i.sub.1] (3')

Movement equations (3) and (4) after replacement of [F.sub.1] and [F.sub.2] can be written in this form:

[J.sub.T] [??]+[c.sub.1]/[i.sub.u](r [phi]/[i.sub.u]--[y.sub.1])-[M.sub.T] = 0 (4)

[m.sub.b] [??] + [c.sub.b][y.sub.1]-[c.sub.1](r [phi]/[i.sub.u]--[y.sub.1] = 0 (5)

[??] = [c.sub.1]r/[i.sub.u][J.sub.T] [y.sub.1]-- [c.sub.1][r.sup.2]/[i.sup.2.sub.u][J.sub.T] [phi]+[M.sub.T]/[J.sub.T] (6)

[[??].sub.1] =-[c.sub.b] + [c.sub.1]/[m.sub.b] [y.sub.1] + [c.sub.1]r/[i.sub.u][m.sub.b] [phi] (7)

After differentiation equations (5) and (6) is gained characteristic equation of homogenous part of differential equation:

[r.sup.4] + [Ar.sup.2] + B = 0

Roots of characteristic equations are (Pejovic, 1978):

[r.sub.1,2] = [+ or -] [[bar.[omega].sub.1]i; [r.sub.3,4] = [+ or -][bar.[omega].sub.2]i

General solution of homogenous part of equation (5)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (8)

Whereas the particular part of equation (5) is (Luteroth, i962):

[y.sub.p] = [M.sub.T][i.sub.u]/[rc.sub.b] (9)

The whole solution of differential equation of movement,

y = [A.sub.1] sin [[bar.[omega]].sub.1]t + [[bar.A].sub.1] cos [[bar.[omega]].sub.1]t + [A.sub.2] sin [[bar.[omega]].sub.2]t + [[bar.A].sub.2] cos [[bar.[omega]].sub.2]t +

+ [M.sub.T][i.sub.u]/[rc.sub.b] (10)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (11)

For constants of integration [A.sub.1], [[bar.A].sub.1], [A.sub.2], dhe [[bar.A.sub.2] can be arranged from the elementary conditions, whereas B and C constants are:

B = [i.sub.u]([c.sub.1] + [c.sub.b])/[rc.sub.1] and C = [i.sub.u][m.sub.b]/[rc.sub.1] (12)

Equations of movement for the second period are (Mijajlovic, i979):

[J.sub.T] [??]+[F.sub.1]/[i.sub.u] r--[M.sub.T] = 0 (13)

[m.sub.b][[??].sub.1]--[F.sub.l] + [F.sub.b] = 0 (14)

[m.sub.Q][[??].sub.2]--[F.sub.1] + [m.sub.Q]g=0 (15)

Where is:

[y.sub.2]--Coordinate of displacement of load.

In the second period of movement the extending of the rope is:

z = r[phi]/[i.sub.1]--[y.sub.1]--[y.sub.2] (16)

If we replacement [F.sub.b] = [c.sub.b] [y.sub.1] and [F.sub.1] = [c.sub.1]z in (13) and (15) equations, differential equations of movement takes form (Sedar, i975):

[m.sub.b] [[??].sub.1] + [c.sub.b][y.sub.1]--[c.sub.1](r[phi]/[i.sub.1]- [y.sub.1]--[y.sub.2]) = 0 (17)

[J.sub.T] [??]+ [c.sub.1]r/[i.sub.1](r[phi]/[i.sub.1]--[y.sub.1]--[y.sub.2])- [M.sub.T] = 0 (18)

[m.sub.Q] [[??].sub.2]--[c.sub.1](r[phi]/[i.sub.1]-[y.sub.1]--[y.sub.2]) + [m.sub.Q]g = 0 (19)

Corresponding precipitation will be:

[[??].sub.1]=-[c.sub.b] + [c.sub.1]/[m.sub.b] [y.sub.1] + [c.sub.1]r/[i.sub.1][m.sub.b] [phi] [c.sub.1]/[m.sub.b] [y.sub.2] (20)

[??] = [c.sub.1]r/[i.sub.1][J.sub.T] [y.sub.1]-- [c.sub.1][r.sup.2]/[i.sup.2.sub.1][J.sub.T] [phi]+ [c.sub.1]r/[i.sub.1][J.sub.T] [y.sub.2] + [M.sub.T]/[J.sub.T] (21)

[[??].sub.2] = [c.sub.1]/[m.sub.Q] [y.sub.1] + [c.sub.1]r/[i.sub.1][m.sub.Q] [phi]-[c.sub.1]/[m.sub.Q] [y.sub.2] + g (22)

The solution could be if we eliminate [y.sub.2] and [phi]

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (23)

Characteristic equation of homogenous part is:

[r.sup.4] + [Ar.sup.2] + B = 0

[r.sub.1,2] = [+ or -] [[bar.[omega]].sub.1]i

[r.sub.3,4] = [+ or -] [[bar.[omega]].sub.2]i

Where are:

A = [c.sub.1] + [c.sub.b]/[m.sub.b] + [c.sub.1]/[m.sub.Q] + [c.sub.1] r/[J.sub.T] [i.sup.2.sub.1]

B = [c.sub.1] + [c.sub.b]/[m.sub.b] [m.sub.Q] + [r.sup.2] [c.sub.b]/[m.sub.b][m.sub.Q] + [r.sup.2][c.sub.1][c.sub.b] /[J.sub.T] [i.sup.2.sub.1][m.sub.b]

Particular solution is:

[y.sub.1p] = [m.sub.Q] g[J.sub.T][i.sup.2.sub.1] + [m.sub.Q][M.sub.T]r/[c.sub.1] ([J.sub.T][i.sup.2.sub.1] + [m.sub.Q] [r.sup.2]) (24)

The whole solution of differential equation is (Pajer & Ustetingforderer, 1987):

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

3. CONCLUSION

Crane bridge present equivalent system of balance masses, in case of issue, the system is with two masses and a swivel mass. Connection of masses has elastic character that enables formulation of differential equations of movement. This we express, at the moment when the force of rope is equal with weight of load.

4. REFERENCES

Luteroth, A (1962). Dinamiche Krafte an Kranen bein Heben and Senken ales Last, Hebezeuge und Fondermitel Teill und 2.8 und 9, Berlin

Mijajlovic, R. (1979). Dinamicka Ponasanje Mostni Dizalnih Sistema Pri Dizanju Tereta, Naucna Knjiga, Beograd

Pajer, K.F. & Ustetingforderer, G. (1987). VEB Verlag Technik, VEB, Berlin

Pejovic, T. (1978). Diferencialne jednacine, Naucna knjiga, Beograd

Sedar, J. (1975) Prenosila i Dizala, FSB, Zagreb