Introduction. The analysis of dynamic processes in the mechanical part plays an important role in the development of new overhead cranes and modernization of existing ones in order to reduce loads on control devices and extend their service life [1].

The bridge crane is subjected to dynamic loads during non-static operations, such as acceleration and braking. Analysis of these processes allows identifying hidden impacts on the dynamic behavior of the bridge crane. Therefore, it is paramount for the researcher to make an optimal design choice to reduce these loads, ensuring that the crane can meet the required operating conditions [2].

In [3], a dynamic model of a crane lifting system was developed, using which an accurate direct numerical integration method was proposed for calculating the dynamic loads of the system.

Papers [4][5] studied dynamic loads in a metal structure, taking into account fatigue of the metal material. However, the researcher neglected the impact of forces from the drive of the lifting mechanism operating with artificial parameters.

In [6], dynamic loads in a bridge crane were determined during the operation of a lifting mechanism when hoisting a load suspended on a rope. The most important case studied was the effect of dynamic loads on the crane when removing a load from a solid foundation, at the moment of lifting-off.

Steel structures of bridge cranes experience non-stationary loads with different stress amplitudes and asymmetry of the working cycle [7]. To study the real load of bridge cranes under typical operating conditions, constant recording of their stress state is required, which is labor-intensive [8]. Therefore, statistical processing of the obtained results is used to assess the loading elements of metal structures of bridge cranes [9]. This involves changing individual components of the total load of metal elements, such as the gravity of the load being lifted, the angle of rotation of the load, and weather loads, and then summing them according to the laws of probability theory [9][10]. This approach is less labor-intensive than a comprehensive study of loading under typical operating conditions. However, determining the probabilistic characteristics of individual random loads also takes time, so the method for calculating load combinations is widely used in crane construction [11].

In [12], it was found that during the stage of selecting the rope slack, the value of the stator current of the electric motor of the lifting mechanism did not depend on the mass of the suspended load. However, as the load increased, the time of its rise also increased, and at the stage of separating the load from the surface, the amplitude values of the current increased. Furthermore, a noticeable difference appeared after five periods of mains voltage from the beginning of the stage. However, the researcher neglected the significant influence of forces arising from the drive of the lifting mechanism working with artificial elements.

By using the developed mathematical model, the research aims at optimizing the design of bridge cranes through studying the dynamic loads that occur during lifting operations, ensuring that the crane has the ability to meet the required operating conditions.

Materials and Methods

Kinetic Model of the System under Study. When developing a kinetic model of the system under study, it can be represented that the construction of a bridge crane for a given motion form consists of three connecting blocks and two flexible joints, and has the form shown in Figure 1:

Derivation of Kinetic Equations Reflecting the Motion of the Dynamic Model

Mathematical equations representing the motion of the dynamic model are derived from the partial differential Lagrange equation, which is considered to be one of the best methods used specifically in cases where the system consists of more than one element and when it has several degrees of freedom.

(1)

(2)

(3)

where T — kinetic energy; U — potential energy; D — damping energy; Q — external forces affecting the whole system; qi — common system of coordinates; i — degrees of freedom of the model under study.

Kinetic energy equation:

(4)

Potential energy equation:

(5)

Damping equation:

(6)

System equations:

(7)

(8)

(9)

where Md, Mc — resistance momentum and lifting mechanism momentum; J — inertia of rotating mass of the lifting mechanism.

Numerical Solution of a Mathematical Model Using the Fourth-Degree Equation of Runge-Kutta

The numerical solution of equations (7–9) was obtained using the Runge-Kutta method in the MATLAB program. The dynamic equations were derived within the program, incorporating the input data and a set of commands to process these equations. The resulting graphs illustrate the interconnections between the various blocks and components of the crane structure under consideration.

The Conditioning of the Input Values Required to Solve the Model

The study was conducted on an ACE type bridge crane consisting of three parts (Fig. 2):

To align the operation of the bridge crane with a standard set of coordinate axes, the following assumption was made:

A study was conducted on a prototype bridge crane with a lifting capacity of 10 tons and a width of 21.5 meters. The following characteristics were considered:

Research Results

Study of the Model Operation Using a Computer

Structural Behavior of a Metal Bridge Crane under Momenta Loads

The structural behavior of a metal bridge crane during the process of lifting a load is depicted by three curves (Fig. 3).

In Figure 3, the first curve shows the coordinates of the bridge crane along the ordinate axis as a function of time. During lifting, some vibration of the crane's metal structure is observed for 5–10 seconds, which then stabilizes and does not affect its rigidity.

The second curve in this figure depicts the change in the speed of the center of gravity of the bridge crane over time. During lifting the load, the speed initially increases and then gradually decreases until stabilization. This indicates that the center of gravity of the bridge crane assumes a stable position within the specified time period, during which the vibration stabilizes.

The third curve reflects the change in the center of gravity of the bridge crane over time. It is noteworthy that the acceleration value at the moment of lifting the load is 0.07 m/s², with the dynamic load reaching its maximum.°

Load Curves

Figure 4 shows three curves that reflect motion of the load during the crane operation at the time of lifting.

In Figure 4, the first curve reflects the change in the position of the center of gravity of the load over time during its lifting to a certain height, calculated by the program, by rotating the drum by 180 degrees and then stopping. The height of the load suspension indicated on the graph is 0.78 m.

The second curve on this graph shows the change in the lifting speed of the load over time. When the drum rotates, the lifting speed of the load initially increases, then gradually decreases until it stabilizes.

The third curve represents the change in the acceleration of the load lifting over time. At the moment of lifting the load, the acceleration reaches a maximum value of 0.26 m/s² in 0.8 seconds, and then stabilizes.

From the analysis of these three curves, it can be noticed that the stabilization time of the crane operation is almost constant.

Coil Angle Curve

Figure 5 shows the dependence of the winding rotation on time in degrees.

Figure 5 shows the change in the angle of rotation of the drum over time. The rotation angle stabilizes when reaching a value of 180 degrees, after which it remains stable, meaning it repeats.

Discussion and Conclusion. Analysis of the above graphs leads to the following conclusions.

The mathematical model and algorithms allow for a detailed study of the motion of a bridge crane at all stages. Adjusting the winch operation and improving the metal structure of the crane have many positive aspects, including reducing dynamic displacements in the metal frame of the crane and transmission elements (such as clutch, gearbox, motor, and pulleys), as well as in the hoisting ropes. This reduction in dynamic loads leads to a decrease in rapid wear of these elements. Additionally, cost savings on maintenance and the development of an optimal metal structure design are achieved, as well as an increase in the crane's service life. This is evident from changes in the center of gravity of the load, speed, and acceleration during its lifting over time. It has been established that the height of the load during lifting, which is directly related to the length of the hoisting ropes, as well as the mass of the lifted load and the design of the main bridge, including its shape and dimensions, significantly influence the dynamic loads experienced by the structure.

The mathematical model offers a new approach to improving the efficiency and safety of industrial operations by providing a more precise understanding and accounting for dynamic loads during the start-up of a bridge crane.