Mathematical Problems in Engineering

Volume 2018, Article ID 5756180, 14 pages

https://doi.org/10.1155/2018/5756180

## Nonlinear Vibration of Ladle Crane due to a Moving Trolley

^{1}School of Mechanical Engineering, Taiyuan University of Science and Technology, Taiyuan 030024, China^{2}College of Economics and Management, Taiyuan University of Technology, Taiyuan 030024, China

Correspondence should be addressed to Gening Xu; nc.ude.tsuyt@gninegux

Received 19 September 2017; Revised 24 December 2017; Accepted 15 January 2018; Published 13 March 2018

Academic Editor: Xiao-Qiao He

Copyright © 2018 Yunsheng Xin et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

The structural vibration of the main beam of a crane causes fatigue damage and discomfort to the driver. The swing of the payload has an effect on positioning precision, especially for a ladle crane, and this directly affects production safety. To study the influence of system parameters on the vibration of a crane’s main beam and the angle of the payload, a system consisting of the main beam, trolley, payload, and cabin was constructed. A rigid-flexible coupling dynamic model of a moving trolley with a hanging payload that moves on the flexible main beam with a concentrated cabin mass is established, and the direct integration method is used to solve the nonlinear differential equations of system vibration, which are obtained through Lagrange’s equation. Then, the time domain responses of the flexible main beam, payload angle, and cabin vibration are obtained. The influences of the trolley running speed, quality of the payload, and quality and position of the cabin on the vibration of the main beam and payload angle are analyzed. The results indicate that the amplitude of the main beam is directly proportional to the quality of the trolley, payload, and cab; the position of the cabin is closer to the mid-span; the amplitude of the main beam is larger; the structural damping has some influence on the vibration of the main beam; and the swing angle of the payload is related to the maximum running speed of the trolley, acceleration time, and length of the wire rope. In order to reduce the vibration of the main beam and cabin, the connection stiffness of the cabin should be ensured during installation.

#### 1. Introduction

The ladle crane is a type of overhead crane and carries out onerous work under extremely hostile environments. Its mechanism and structure can withstand the strong impact vibration of the trolley on the main beam. Many scholars have simplified the vibration system model to the moving load beam coupling system and have carried out a significant amount of research on the dynamic response of beams under moving mass.

Michaltsos et al. [1] dealt with the linear dynamic response of a simply supported uniform beam under a moving load of constant magnitude and velocity by including the effect of its mass. By using a series solution for the dynamic deflection in terms of normal modes, the individual and coupling effect of moving load mass, its velocity, and other parameters were fully assessed.

Şimşek [2] investigated the dynamic characteristics of the main beam when the moving mass was running on a simply supported beam, established the equilibrium equations of the system by using Lagrange’s equations, and discussed the effects of shear deformation, various material distributions, velocity of the moving mass, inertia, Coriolis and the centripetal effects of moving mass on the dynamic displacements, and the stresses of the beam, in detail. Kiani et al. [3] investigated the maximum deflection and bending moment of the beams under various boundary conditions due to a moving mass by employing Hamilton’s principle and using Euler–Bernoulli, Timoshenko, and third-order beam theories.

A comprehensive parametric investigation on the effects of moving mass weight and velocity on the dynamic behavior of a simply supported Euler–Bernoulli was carried out by Nikkhoo et al. [4], by employing the eigenfunction expansion method. They introduced a concept of critical velocity in terms of beam fundamental period, mid-span, and moving mass weight, in which the effect of convective accelerations in moving mass formulation was not negligible for masses moving with velocities greater than this critical one.

Esmailzadeh and Ghorashi [5] analyzed the dynamic behavior of beams with simply supported boundary conditions, which carried either uniform partially distributed moving masses or forces, and evaluated the critical speeds of the moving load, by considering the maximum deflection for the mid-span of the beam. Moreover, they verified that the length of the distributed moving mass affects the dynamic response considerably.

Lee [6] investigated the dynamic response of a Timoshenko beam, which is acted upon by a moving mass, by using the Lagrange approach and the assumed mode method. He verified the results of the model by comparing them with the results of an equivalent moving load for the mid-span deflection of a simply supported beam for a small number of moving mass weights and velocities as well as for different slenderness ratios of the base beam. Lou et al. [7] presented a finite element formulation of the Timoshenko beam subjected to a moving mass. Their results were in good agreement with those obtained by using the assumed mode method employed by Lee [6]. Many scholars have applied the model of mass moving on the beam to engineering practice. Lou et al. [8] and Cheng et al. [9] established a bridge-track-vehicle model in order to analyze the vibration of railway bridges under a moving train by taking into account the response of the track structure. The above-mentioned studies have only investigated the dynamic characteristics of the simply supported beam under a moving mass, or the quality of moving mass, running speed, acceleration, or other factors. There is no model of moving mass that accounts for swing quality.

The dynamic response of cranes has been studied by many scholars. Niu et al. [10] presented a general mathematical modeling approach for electric crane system dynamics during operation, which could be used to analyze the dynamic responses of electromagnetism, mechanism, and structure, during the operation of electric cranes. Wu et al. [11–14] established a mathematical model of the horizontal and vertical vibrations of a crane structure under the trolley running and lifting processes by using finite element theory and calculated the vibration response of the main beam. The correctness of the theoretical solution was verified by experimental verification, and the parameters were modified. Oguamanam et al. [15, 16] established an Euler–Bernoulli equation in order to investigate the vibration response of a fixed crane beam. Zrnić et al. [17] constructed multi-degree-of-freedom vibration models for crane structures in order to analyze the effects of wire rope length, damping ratio, and tilt angle of the sling load on structural vibration. Xin et al. [18] constructed a nine-degree-of-freedom mathematical model of a “human–crane–rail” system and an annoyance rate model for use in the optimization of the structural parameters of overhead traveling cranes and used the particle swarm optimization algorithm to optimize the structural design of overhead cranes. Some scholars predicted crane fatigue life by analyzing dynamics of vibration system [19, 20].

Although the dynamic characteristics of the main beam and the swing angle of the crane have been studied; these two aspects have not been discussed simultaneously. The models reported in the literature contain only the main beam, trolley, and crane quality, without considering factors such as cabin quality, position, and structural damping of the main beam, which also affect the vibration of the main beam. The vibration time response of the cabin can be used to analyze the vibration of the human body during operation and thereby reduce the possibility of occupational disease. Therefore, this paper considers these factors in order to establish a dynamic model of the ladle crane vibration system. Based on the principle of energy conservation, the system vibration equation was established through Lagrange’s equation. In combination with a direct integral method for solving the approximate solution of nonlinear vibration, the influence of factors such as the quality (trolley, payload, and cabin), trolley running speed, length of wire rope, position and connection stiffness of cabin on the vibration of the main beam, and the influence of the swing of payload was analyzed. In this study, the vibration acceleration and amplitude time domain responses of the cabin were obtained, which provide a theoretical reference for the optimization of crane design.

#### 2. Crane Vibration System Modeling

##### 2.1. Description of Vibration System

The ladle floating crane includes a frame, trolley movement organization, cabin, and suspension system, and the ladle crane system vibration dynamics model is established according to the size and structural characteristics, based on the following assumptions:(1)The main beam is regarded as a flexible body, and only the vibration in the vertical direction is considered. If all the damping in the elastic body is viscous damping, the damping of the beam has very little influence on the vibration of the structure [21]; the damping ratio is usually 0.1%–0.7%.(2)The cabin is simplified as a lumped mass, and only the vertical vibration is considered. The connection between the cabin and the main beam is simplified as a spring and damping model with greater stiffness.(3)According to the characteristics of the ladle crane, the rope is regarded as rigid without quality [22].(4)The payload of the crane is reduced to a swinging mass, which is suspended in mass blocks by a nonmass rigid rope, which moves with the mass in the plane.(5)The deflection of main beam is considered as linear, while the nonlinear effect of main beam deflection in the dynamic response analysis of the ladle crane is ignored.

The design method of the main beam consists of the allowable stress method and limit state method. The allowable stress method assumes that the deflection of the main beam is linear; however, the limit state method takes into consideration the nonlinear deflection of the main beam, while the fatigue analysis and the calculation of the crane beam structure’s dynamic characteristics by these two methods have been reported in the literature [23, 24], where it was found that the results were basically the same with the exception of jib crane. Therefore, the nonlinear deflection of the beam had little effect on the dynamic response analysis of ladle cranes. In addition, after the completion of design and manufacturing, the beam of the bridge crane was formed on the precambering curve; therefore, the crane beam vibration amplitude was relatively small. Thus, this study adopted the mentioned hypothesis .

A schematic of the problem is depicted in Figure 1. The beam is simply supported and is assumed to be adequately modeled by using the Euler–Bernoulli beam theory. Only the bending stiffness was calculated, and the shear stiffness and torsion stiffness of the beam were not considered. The properties of the beam are Young’s modulus , volume density , cross-sectional area , length , and second moment of area . is the quality of the trolley, is the quality of the payload, is the quality of unit length for the main beam, is the length of the wire rope, is the connection stiffness between the cabin and the main beam, is the damping between the cabin and the main beam, is the quality of the cabin, is the mounting position coordinates of the cabin, is the gravitational acceleration, is the maximum running speed of the trolley, is the th modal damping ratio of the main beam structure, is the extreme left position of the trolley, is the extreme right position of the trolley, and is the swinging angle of the payload.