Determination of maximal dynamic force of crane-bridge.

Ibishi, Ismet Maliq ; Galica, Melihat

1. INTRODUCTION

Studying this problem we base on crane-bridge because it is comparable with the other cranes.

We simplify the problem changing at system of movement with one flax stair although it is less exact but gives a prime model for theoretical analyze.

During the lifting up of load by cranes, noticed swings, which cause dynamic load.

Dynamic forces can be calculated according to swings but and with physical features of carrier construction and the mechanism of movement of crane, the load which lift up and the movement of carrier construction of crane present a swing system in vertical direction and have at least tow flax stairs of movement.

Our job is to simplify the problem turning at system with one flax stair of movement.

The problem can be solved considering carrier construction of crane, which leans in rail as a beam by continual load from weight, which can be written as Gb and has a specific stiffness Cb.

Supposed that between carrier constructions it is a chariot, which has weight Gk. The chariot has tumbler by which roll up rope that lift up load Gn.

This rope has stiffness Ce but its weight is neglected comparing to the other weights.

Ce is the module function of elasticity and the length of the rope.

Simplifying the problem, the stiffness of rope can be taken as constant operand but this swing system still is complicated so we need to continue with further simplifications.

The weight of carrier construction is concentrated on reduced load Gred. The weight of chariot and reduced weight of carrier construction can be taken as carrier load of crane Gb

[G.sub.b] = [G.sub.n] + [G.sub.red] ... (1)

Let suppose that the hanging load in the rope has the maximal high from the base, this ensure stability and the rope should be taken shorter so we can have swing system with higher stiffness. In this case during the shaking both two loads can be taken as one load that consist the system with one flax stair of movement.

And the specific stiff nesses can be add and gaining the equivalent stiffness C.

C = [C.sub.b] x [C.sub.e]/[C.sub.b] + [C.sub.e] ... (2)

The load, which lift up in swing system:

Q = Z x C ... (3)

Equivalent system:

C = Q/Z ... (4)

Z[mm]--The shift where the load is hanging.

2. DIFFERENTIAL EQUATION OF MOVEMENT

We form equivalent scheme from the real scheme for analyze of movement.

[FIGURE 1 OMITTED]

[FIGURE 2 OMITTED]

[FIGURE 3 OMITTED]

According to the dynamic's laws of swings base on principle of D'Alembert equations for equilibrium of action forces we find differential equation of movement of system.

Differential equations of movement can be found by Lag ranch's equations of second type.

Q/q x [??] + C x Z--Q = 0 ... (5)

If we replace (6) we find circular frequency

[[lambda].sup.2] = C x q/Q ... (6)

[[??] + [[lambda].sup.2] x Z--q = 0 ... (7)

This equation is done from the homogenous part:

Z = [Z.sub.n] + [Z.sub.p] ... (8)

After the solution from the theory of differential equation:

Z = [K.sub.1] sin [[lambda]t + [K.sub.2] cos [[lambda]t + [q/[[lambda].sup.2] ... (9)

From the (6) formula [[lambda].sup.2] = q/f we have the equation of circular frequency:

Z = [K.sub.1] sin [[lambda]t + [K.sub.2] cos [[lambda]t + f ... (10)

The final solution, fixing constants of integration [K.sub.1] & [K.sub.2], which can be fixed after integration of equation (10) and the application of elementary and limited conditions per

t = 0. The extension of the rope is: f=Q/C.

Applying the law of Seri of movement, the speed of lifting up from basement:

[Z.sub.(0)] = -v[G.sub.n]/Q] = [alpha] x V [[G.sub.n]/Q] ... (11)

v[m/s]--The speed of lifting up from basement.

V[m/s]--The nominal speed of lifting up. [alpha]--The factor of lifting up.

From the limited constants conditions of integration:

[K.sub.1] = v [[G.sub.n]/[lambda] x Q] = v[[G.sub.n]/Q] [square root of f/g] ... (12)

And [K.sub.2] = 0.

The value of maximal dynamic force [F.sub.max] = C x [Z.sub.max]

[F.sub.max] = [G.sub.n] [[alpha] x v/[square root of g x f]][kN] ... (13)

3. SUMMARY

Theoretical calculations of movement of cranes are done with the intention of achieving:

* Stabile and rational work of cranes

* Security in point of view of resistance of rope and carrier construction.

* Not to be large dynamic force to cause resonance of carrier construction.

* Not to come till the extension of rope.

* Theoretical calculations to be compared with practical conditions.

4. REFERENCE

Dedier, S. (1970) Dinamicki koeficient pri radu mostofski dizalica malle I srednje nosivosti, Disertacija Beograd

Mijajovici, R. (1972). Dinamicki Faktor pri dizanju tereta kod mostnih dizalica. Masinski Fakultet, Beograd

Serdar, J. (1975). Prenosila e dizala, Zagreb

Pejovi, T. (1948). Diferencijalne Jednacine Knjiga2, "Naucna Knjiga", Beograd

Pejovi, T. (1948). Diferencijalne Jednacine Knjiga2, "Naucna Knjiga", Beograd

Zemerich, G. (1968). Dynamische Beanspruchung von Brunschenkranen beim Haben von Lasten, T.H.Brunschweig