# Dynamic Responses of an Overhead Crane’s Beam Subjected to a Moving Trolley with a Pendulum Payload

Authors: Huasen Liu; Wenming Cheng; Yinqi Li
Journal: Shock and Vibration
 (2019)

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## Abstract

An overhead crane with a flexible cable is an underactuated system; the vibration of the crane’s beam and the residual swinging of the payloads cause fatigue in the crane and affect the precise positioning of the payloads. In this paper, the coupling system of an overhead crane was simplified to that of a moving mass with pendulum swing passing beam model. The differential equation motion of a coupled overhead crane system was derived based on the Lagrange equation. Mathematical solution was carried out by using the Newmark-β integral method. The influences of the trolley’s acceleration and the parameters of the payloads on the vibration of the beam and the payloads’ swing were, respectively, analyzed. A numerical analysis of the results indicates that increasing the mass of the payloads leads to a larger deflection of the beam, whereas increasing the speed and acceleration of the trolley does not obviously influence the maximum deflection of the central beam.


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## Body

## 1. Introduction

Overhead cranes play a significant role in the modern logistics engineering industry and movement of massive goods. However, cranes are typical of nonlinear underactuated mechanical systems; the motion of the trolley normally induces an undesirable swing of the payload. It is thus difficult to control the trolley position accurately while keeping the payload swing angle small [1, 2]. At the same time, as the span of cranes increases, the vibration of the cranes’ beams [3–5] should not be neglected when loading and unloading large tonnage payloads; an unexpected swing of payload could lead to accidents and cause crane damage or workforce injury.The dynamic response of a beam under the action of a moving mass is of great importance when analyzing the vibration of the beam. Euler–Bernoulli [6], Timoshenko [7], and third-order shear deformation beam theories [8] are usually used to investigate vibrations in a simply supported beam due to a moving mass. Michaltsos [9] investigated the linear dynamic response of a simply supported elastic single-span beam under a constant magnitude moving load and variable velocity. Karimi and Ziaei-Rad [10] dealt with and analyzed the nonlinear coupled vibration of a beam with moving supports under the action of a moving mass. The research on the moving mass and beam coupling system focused mainly on the beam’s dynamic response when moving mass at high speed. Yang et al. [11] investigated the dynamic behavior of a bridge-erecting machine that carried a moving mass suspended from a wire rope. The damping coefficient and the spring constant of the rope caused significant variations in the deflection. However, the payload swing was not considered in this dynamic model.He and Ge [12] proposed cooperative control laws for a gantry crane with a flexible cable. The spatiotemporally varying tensions of flexible cables were applied to a crane system. However, the crane’s beam was still regarded as a rigid body, and the swing of the payloads was not considered in this model. Meanwhile, flexible cables were researched sufficiently for cable elevator systems [13], but elevators just rose and descended vertically. Fatehi et al. [14] proposed an advanced control system for an overhead crane when there are transverse vibrations in the flexible cables and when considering large angles of cable swing. The payload swing and the transverse vibrations of the cables could be simultaneously reduced and suppressed, respectively, by applying a horizontal driving force to the trolley. In addition, the vibration of the beam was not taken into account in this model.It has been demonstrated that the dynamic behavior of the beam is affected by many parameters, such as the mass of the moving load [15], the load’s acceleration [16, 17], the load’s speed [5, 18], and many others. Extensive research has been done on the methods and efforts of limiting undesirable payload swing, such as an energy-based control scheme [19], input shaping technology [20], nonlinear coordination control [21], neural network control [22], fuzzy control [23], a trajectory planning method [24], and so forth. However, all these methods focus on how to reduce the swing of the payload and do not take into account the influence of the cable’s flexible or the vibrations of the crane’s beam. As far as we are aware, there have been far fewer research results reported for overhead crane systems which consider all three factors simultaneously: the swing of payload, the vibration of beam, and the flexible cable.It is especially important for large tonnage and large-span cranes to consider the coupling vibration between the crane’s beam and the trolley with the swing payload. Oguamanam et al. [25] analyzed the differences between the results for “rigid” and “flexible” beam assumptions. Oguamanam et al. [26] used a simply supported uniform Euler–Bernoulli beam carrying a crane model to analyze the deflection of the beam and the swing of the payload. The location and the value of the maximum beam deflection were dependent upon the speed of the carriage.Xin et al. [27] constructed a nine-degree-of-freedom mathematical model of a “human-crane-rail” system and used a particle swarm optimization algorithm to optimize the crane’s structural design. The vibrations associated with the crane’s structure and caused by defects in the rail were reduced.Although the dynamic characteristics of the main beam and the swing angle of the crane have been studied, the vibrations of a crane’s beam and the swing of its payloads have not been discussed simultaneously. In this paper, we only try to investigate the interaction between the payload swing and the vibration of the crane’s beam. A complex multibody coupling dynamic system is established with a flexible crane beam, trolley, and payload, and the coupling dynamic system is used to investigate the dynamic response of a beam’s deflection and the payload swing. The coupling system could basically be treated as a moving mass passing a beam problem. The main challenge for solving the dynamic equations would arise when the swing degree of freedom of the payload is added as this creates the problem of more coupling effects to solve.For the present paper, a dynamic model of a multibodied coupling overhead crane system composed of a flexible crane beam, trolley, and underactuated payload was first established. Secondly, the vibration equation for this system was derived from Lagrange’s equation. Finally, through using the Newmark-β method for solving the approximate solution of the beam’s vibration, the influence of factors such as payload mass, the running acceleration and speed of the trolley, cable length, and the influence of the payload swing were analyzed.


## 2. Mathematical Modeling of an Overhead Crane

A two-dimensional model of an underactuated overhead crane system is presented in Figure1. The main beam is regarded as a flexible body, and only the vertical vibrations are considered. The beam is simply supported and is assumed to be modeled on the Euler–Bernoulli beam theory. mb is the unit mass of the main beam, and the length of beam is L. The properties of the beam are defined by Young’s modulus E, a volume density ρ, a cross-sectional area A, and a second moment of area I. xc is the real-time position of the trolley and Vmax is the maximum running speed of the trolley. It is assumed that the payload swings around a point O on the XY plane and the swing angle θ are measured on the x-y plane. The trolley is modeled as a point mass with a mass of mc whilst the payload is modeled as a rigid body of mass mp. The suspension cable is simplified as a massless cable, whose length is l, and the rope length is kept constant during a specific transportation process. Gravitational acceleration is represented as g. The inertial frame with basis vectors is i and j.Figure 1
Sketch of an underactuated overhead crane structure.As the beam is assumed to be a flexible body, the vertical deflection of the beam can be expressed as follows:(1)wx,t=∑i=1Nϕixqit,ϕix=siniπxL,where ϕix is the ith mode shape of the beam and qit is the generalized coordinate which are unknown time functions and N is the total number of considered modes.The velocity vectorvb of an elemental mass for a beam at time t is given by(2)vb=∂wxb,t∂tj=∑i=1Nϕixbq˙itj,where xb is the coordinate of an elemental beam at time t in j direction. For a certain elemental beam, the horizontal coordinate xb is a constant. As usual, the dot and double dots above a variable represent its first and second derivatives to time, respectively.The velocity vectorvc for a trolley at time t is given by(3)vc=x˙ci+x˙c∑i=1Nϕi′xcqit+∑i=1Nϕixcq˙itj.As the trolley moves on the beam,xc is thus a variable.The velocity vectorvp for a payload at time is given by(4)vp=x˙c+θ˙lcosθi+x˙c∑i=1Nϕi′xcqit+∑i=1Nϕixcq˙it−θ˙lsinθj.The payload’s swing angleθ is a variable.For a flexible beam, the kinetic energyTb and the potential energy of beam Vb can be calculated as(5)Tb=12mb∫0L∂wx,t∂t2dx=12mb∑i=1Nq˙it∑j=1Nq˙jt∫0Lϕixϕjxdx=12mb∑i=1Nq˙i2t∫0Lϕi2xdx=14mbL∑i=1Nq˙i2,Vb=12EI∫0L∂2wx,t∂x22dx=12EI∑i=1Nqit∑j=1Nqjt∫0Ld2ϕixdx2d2ϕjxdx2dx=EIπ44L3∑i=1Ni4qi2.The kinetic energy of the coupled systemT can be written as(6)T=12mb∫0Lvb2dx+12mcvc2+12mpvp2=14mbL∑i=1Nq˙i2+12mcx˙ct2+x˙c∑i=1Nϕi′xcqit+∑i=1Nϕixcq˙it2+12mpx˙c+θ˙lcosθ2+x˙c∑i=1Nϕi′xcqit+∑i=1Nϕixcq˙it−θ˙lsinθ2.The three terms of equation (6) are the beam kinetic energy, the trolley kinetic energy, and the payload kinetic energy, respectively.y = 0 needs to be set as the zero point for the potential energy, so the total potential energy of the coupled crane system V is(7)V=EIπ44L3∑i=1Ni4qi2−mcgwxc,t−mpgwxc,t+lcosθ−l.The first term of equation (7) is the beam’s straining energy and the last two terms are the potential gravitational energy of the trolley and the payload.Lagrange’s equations of nonconservative system are(8)ddt∂T∂η˙k−∂T∂ηk+∂V∂ηk=Qk,k=1,2,…,n,where ηk is the generalized coordinate, η˙k is the generalized velocity, and Qk is the generalized excitation force of external action.Substituting equations (6) and (7) into equation (8) and applying the Lagrange equation with respect to the generalized coordinate x, θ, and qi, respectively, yields(9)mpx¨c+lθ¨cosθ−lθ˙2sinθ+mcx¨c−mplθ¨sinθ∑i=1Nϕi′xcqi−mplθ˙2cosθ∑i=1Nϕi′xcqi+mp+mc∑i=1Nϕi′xcqi∑i=1Nϕixcq¨i+mp+mcx¨c∑i=1Nϕi′xcqi2+x˙c2∑i=1Nϕi″xcqi∑i=1Nϕi′xcqi+2x˙c∑i=1Nϕi′xcqi∑i=1Nϕi′xcq˙i−mp+mcg∑i=1Nϕi′xcqi=Fc,(10)mpl2θ¨+x¨clcosθ+lgsinθ−lsinθx¨c∑i=1Nϕi′xcqi+x˙c2∑i=1Nϕi″xcqi+2x˙c∑i=1Nϕi′xcq˙i+∑i=1Nϕixcq¨i=0(11)mp+mcx¨cϕjxc∑i=1Nϕi′xcqi+x˙c2ϕjxc∑i=1Nϕi″xcqi+2x˙cϕjxc∑i=1Nϕi′xcq˙i+ϕjxc∑i=1Nϕixcq¨i+mp−lθ¨sinθϕjxc−lθ˙2cosθϕjxc+12mbLq¨j−mpgϕjxc−mcgϕjxc+EIπ4j42L3qj=0,j=1,2,…,N.When the acceleration of a trolley is a known parameter, equations (10) and (11) can be expressed as a nondimensional differential equation of motion in the following form:(12)Mu¨t+Cu˙t+Kut=Ft.In equation (12), M, C, and K are mass, damping, and the stiffness matrices of the system; u=θ,q1,q2,…,qNT, u¨, u˙, and u are the acceleration, velocity, and displacement vectors; and Ft is a time-dependent loading vector. More details are included in Appendix A.If the swing DOF of the payload is ignored, equation (11) degrades to describe the problem of a simply supported beam subjected to a moving mass. In this case, equation (11) degrades to(13)12mbLq¨j−mcgϕjxc+EIπ4j42L3qj+mcx¨cϕjxc∑i=1Nϕi′xcqi+x˙c2ϕjxc∑i=1Nϕi″xcqi+2x˙cϕjxc∑i=1Nϕi′xcq˙i+ϕjxc∑i=1Nϕixcq¨i=0.Equation (13) is same as the reference in reference [28], and this can be viewed as indirect proof that this paper’s dynamic model is correct.


### 3. Solution Steps Using the Newmark-Beta Method

As shown in equation (11), the trolley’s acceleration is a key factor that affects the vibration of a beam. The trolley only moves on the beam from one side to another side. The acceleration of a trolley is as a given input signal for the dynamic system. Three types of trolley acceleration curve, which are usually used to control the movement of a trolley, are discussed in Figure 2. The acceleration and deceleration are symmetrical, so ta=tc−tb.Figure 2
The motion acceleration curves of a trolley.For a constant acceleration curve in Figure2, the acceleration of a trolley is described as(14)a=a1,0≤t<ta,0,ta≤t<tb,−a1,tb≤t<tc.For a ramp acceleration curve in Figure2, the acceleration of trolley is described as(15)a=2a3tat,0≤t<ta2,2a3tata−t,ta2≤t<ta,0,ta≤t<tb,−2a3tat−tb,tb≤t<tb+tc2,−2a3tatc−t,tb+tc2≤t<tc.For a sine acceleration curve in Figure2, the acceleration of trolley is described as(16)a=a2sinπtat,0≤t<ta,0,ta≤t<tb,−a2sinπtat−tb,tb≤t<tc.In order to make the velocity values of the three input signals equal during uniform motion,(17)∫0taa1dt=∫0taa2sinπtatdt=∫0ta/22a3tatdt+∫ta/2ta2a3tata−tdt.After some calculation for equation (17), we obtain(18)a1=2πa2=a32.The velocity curve and the displacement curve for the three types of input signals are included in AppendixB. Appendix B also proves that the final displacement of the three types of input signals is equal to each other.It is noted that equation (12) is a second-order nonlinear time-dependent differential equation with (N + 1) variables. For a beam subjected to a moving load problem, the mass matrix, the damping matrix, and the stiffness matrix are time dependent. The Newmark-β method is used for time discretization. The solution for the equation of motion is obtained through taking the following steps:(1)
The values ofut, u˙t, and u¨t for the first step denote the initial conditions of the displacements, velocities, and accelerations for the structural system at a time t=t0=0. The parameters γ and β are adjusted according to the accuracy and stability requirements to enable integration. In this paper, β=0.5, γ=0.25, and Δt=0.01. Thus, the integration constants are calculated:(19)α0=1γΔt2,α1=βγΔt,α2=1γΔt,α3=12γ−1,α4=βγ−1,α5=Δt2βγ−2,α6=Δt1−β,α7=Δtβ.(2)
Equation (12) is used to calculate the mass matrices M, damping matrices C, and stiffness matrices K at a time t.(3)
The effective stiffness matrixK¯ at a time t+Δt is calculated:(20)K¯=K+α0M+α1C.(4)
The effective loadsF¯t+Δt, at a time t+Δt are calculated:(21)F¯t+Δt=Ft+Δt+Mα0ut+α2u˙t+α3u¨t+Cα1ut+α4u˙t+α5u¨t.The first, second, and third terms on the right hand side of equation (21), respectively, are the external forces, inertial forces, and damping forces at a time t+Δt.(5)
The displacements at a timet+Δt are solved:(22)ut+Δt=K¯−1F¯t+Δt.(6)
The accelerations and velocities at a timet+Δt are calculated:(23)u¨t+Δt=α0ut+Δt−ut−α2u˙t−α3u¨t,u˙t+Δt=u˙t+α6u¨t+α7u¨t+Δt.Steps (2)–(6) are repeated until the final displacementsutc, the final velocities u˙tc, and the final accelerations u¨tc for the structural system are obtained.


## 3. Solution Steps Using the Newmark-Beta Method

As shown in equation (11), the trolley’s acceleration is a key factor that affects the vibration of a beam. The trolley only moves on the beam from one side to another side. The acceleration of a trolley is as a given input signal for the dynamic system. Three types of trolley acceleration curve, which are usually used to control the movement of a trolley, are discussed in Figure 2. The acceleration and deceleration are symmetrical, so ta=tc−tb.Figure 2
The motion acceleration curves of a trolley.For a constant acceleration curve in Figure2, the acceleration of a trolley is described as(14)a=a1,0≤t<ta,0,ta≤t<tb,−a1,tb≤t<tc.For a ramp acceleration curve in Figure2, the acceleration of trolley is described as(15)a=2a3tat,0≤t<ta2,2a3tata−t,ta2≤t<ta,0,ta≤t<tb,−2a3tat−tb,tb≤t<tb+tc2,−2a3tatc−t,tb+tc2≤t<tc.For a sine acceleration curve in Figure2, the acceleration of trolley is described as(16)a=a2sinπtat,0≤t<ta,0,ta≤t<tb,−a2sinπtat−tb,tb≤t<tc.In order to make the velocity values of the three input signals equal during uniform motion,(17)∫0taa1dt=∫0taa2sinπtatdt=∫0ta/22a3tatdt+∫ta/2ta2a3tata−tdt.After some calculation for equation (17), we obtain(18)a1=2πa2=a32.The velocity curve and the displacement curve for the three types of input signals are included in AppendixB. Appendix B also proves that the final displacement of the three types of input signals is equal to each other.It is noted that equation (12) is a second-order nonlinear time-dependent differential equation with (N + 1) variables. For a beam subjected to a moving load problem, the mass matrix, the damping matrix, and the stiffness matrix are time dependent. The Newmark-β method is used for time discretization. The solution for the equation of motion is obtained through taking the following steps:(1)
The values ofut, u˙t, and u¨t for the first step denote the initial conditions of the displacements, velocities, and accelerations for the structural system at a time t=t0=0. The parameters γ and β are adjusted according to the accuracy and stability requirements to enable integration. In this paper, β=0.5, γ=0.25, and Δt=0.01. Thus, the integration constants are calculated:(19)α0=1γΔt2,α1=βγΔt,α2=1γΔt,α3=12γ−1,α4=βγ−1,α5=Δt2βγ−2,α6=Δt1−β,α7=Δtβ.(2)
Equation (12) is used to calculate the mass matrices M, damping matrices C, and stiffness matrices K at a time t.(3)
The effective stiffness matrixK¯ at a time t+Δt is calculated:(20)K¯=K+α0M+α1C.(4)
The effective loadsF¯t+Δt, at a time t+Δt are calculated:(21)F¯t+Δt=Ft+Δt+Mα0ut+α2u˙t+α3u¨t+Cα1ut+α4u˙t+α5u¨t.The first, second, and third terms on the right hand side of equation (21), respectively, are the external forces, inertial forces, and damping forces at a time t+Δt.(5)
The displacements at a timet+Δt are solved:(22)ut+Δt=K¯−1F¯t+Δt.(6)
The accelerations and velocities at a timet+Δt are calculated:(23)u¨t+Δt=α0ut+Δt−ut−α2u˙t−α3u¨t,u˙t+Δt=u˙t+α6u¨t+α7u¨t+Δt.Steps (2)–(6) are repeated until the final displacementsutc, the final velocities u˙tc, and the final accelerations u¨tc for the structural system are obtained.


## 4. Numerical Simulations and Discussion

In this section, a model QD-300t/31m overhead crane was used to verify the vibrations in the beam and the swing of an underactuated payload. All the structures illustrated in this paper were made of steel with a mass density ofρ=7820kg/m3 and Young’s modulus E=2.06×1011N/m2. The gravitational acceleration was g=9.8m/s2. The length of the suspension cable l varied between 6 m and 22 m. The physical parameters of the simulated overhead crane system are listed as follows:(24)mp=0∼150×103kg,mc=46×103kg,I=0.1668m4,A=0.1477m2,L=31m,vmax=2∼25m/min.


### 4.1. Validation of the Model System

#### 4.1.1. Deflection of the Beam When Subjected to a Moving Mass

As mentioned in Section2, when the swing degree of freedom of the payload is ignored, equation (13) describes the problem of a simply supported beam subjected to a moving mass. In this study, the effectiveness of the mathematical model developed in Section 2 is first examined through the use of a classical moving mass passing a bridge model. To further confirm the reliability of the presented formulae and the developed computer programs, the parameters of the model system were chosen as follows: a moving mass mc = 70 kg with constant speed of vmax = 3.34 m/s. The length of the beam was L = 10 m and the cross-sectional area was A = 9 × 10−4 m2, and the second moment of area was I = 1.04 × 10−6 m4. The above parameters for the beam were the same as in references [29, 30].Figure3 shows the graphical time histories for the vertical central deflections of the simply supported beam. In Case 1 [29], the dynamic response of the middle point of the main beam was solved using the equivalent moving finite element assumption. However, for this study, the domain time response for the midpoint vibration of the main beam was solved by using the Newmark-β method. As can be seen from Figure 3, the two calculation results were consistent with the underlying trend. A close match can also be observed between the results shown in Case 2 [30] and the model developed in this paper. Since the deviations between the three curves are quite small, the formulations and computer programs developed in this paper should be acceptable methods of calculating the dynamic responses of a beam when subjected to a moving trolley.Figure 3
The calculated results for the central deflections using references [29, 30].


#### 4.1.2. Payload Swing Angle

The reliability of the method presented in this paper is further confirmed by the payload swing angle. The payload’s swing is affected by the trolley’s acceleration and the length of the wire cable. A simply supported beam of lengthL = 6 m was subjected to a moving mass. In practice, the trolley starts at an initial velocity of zero and then accelerates to a particular speed, which it holds constant for some time before decelerating to a rest. In all our cases, the trolley’s acceleration lasted for 15 s, the constantly held speed lasted for 30 s, and the deceleration lasted for 15 s. The velocity during the constant speed phase was V = 0.1333 m/s. The initial swing angle of the payload was θ(0) = −0.01 (rad). These factors were considered to be consistent with those that were reported in Case 3 [25]. From Figure 4, it can be seen that the simulated results of this paper are similar to the data in reference [25], which proves the validity of the simulation methods proposed in this paper.Figure 4
A comparison of the calculated results to reference [25] for the payload swing angle.


### 4.2. Dynamic Response of the Crane System

According to the dynamic model of crane system, the factors affecting the dynamic response of the crane’s beam and the payload swing are not only related to the beam’s parameters, such as the cross-sectional properties and the dimensional parameters of the beam, but also to the mass of the trolley, the acceleration of the trolley, and the mass of the payload. This section analyzes the vibration of the beam and payload swing angle which are affected by these parameters.


#### 4.2.1. Effect of the Trolley’s Velocity on the Dynamic Response

High trolley acceleration may lead directly to an excessively large payload swing angle, which greatly reduces safety when using the crane. For the trolley acceleration curve that was shown in Figure2, the trolley’s acceleration, and deceleration, time was set at 5 s. The maximum running speed of the trolley was vmax = 0.4167 m/s. Therefore, the trolley’s acceleration was up to approximate 0.0833 m/s2.Figure5 shows the vertical central deflection of the crane’s beam for the three speed patterns of the trolley. The maximum running speeds of the trolley were 0.25 m/s, 0.3333 m/s, and 0.4167 m/s, whilst the maximum vertical central deflections were 34.65 mm, 34.68 mm, and 34.69 mm, respectively. Thus, the larger the trolley acceleration was, the greater the vertical central deflection was. However, the effects of the acceleration of the trolley were not that obvious because the accelerations of the trolley were very small.Figure 5
The effect of trolley acceleration on the center deflection of the crane beam.


#### 4.2.2. Effect of the Trolley’s Acceleration on the Dynamic Response

Figure6 shows the phase plane illustrations affected by the three different types of acceleration curves when the acceleration (deceleration) time is equal to the swing period of the payload, as shown in Figure 2. Figures 6(a)–6(c) are the response results of the constant, sine, and ramp accelerations of the trolley, respectively. The maximum trolley accelerations were 0.0464 m/s2, 0.0729 m/s2, and 0.0928 m/s2 for Figures 6(a)–6(c), respectively; keeping these three maximum trolley acceleration values satisfies equation (18).Figure 6
Phase plane illustrations for the three acceleration curves: (a) when trolley acceleration is constant; (b) when trolley acceleration is sine curve; (c) when trolley acceleration is ramp curve.
(a)
(b)
(c)In Figure6, the trolley moves start at point A; when the curve gets to the point B, the trolley is at uniform speed; and the trolley decelerates at point C. Under these three type acceleration curves, the maximum running speed of the trolley was equal to each other when trolley moved at uniform speed. In Figure 6, the curves for the sine acceleration and the ramp acceleration are far from the origin of the coordinates at the end of the curves, which means that the payload swing angle became larger at the end of the deceleration phase. This was because the payload still had an angular swing velocity at the beginning of the deceleration phase, and the direction of the velocity was opposite to the direction of the trolley’s deceleration most of the time.Figure6 indicates that the acceleration and deceleration times are the key parameters that affect the payload swing angle. Although the sine acceleration and ramp acceleration curves are smoother than those of the constant acceleration curves, the payload swing angles of the constant acceleration curve is smaller than that of the other two acceleration curves. In designing trolley movement trajectory, it is thus fundamentally necessary to investigate the swing period of the payloads as well as the acceleration times of the trolley so that the payload swing angle can be suppressed.


#### 4.2.3. Influence of Payload Swing on the Crane’s Dynamic Response

Figure7 shows the central deflection comparisons for a beam between the result for “with” and “without” a payload swing when the trolley is in constant acceleration. The dashed line shows the result for beam’s central deflection without a payload swing, and the solid line indicates the result for central deflection for a beam with a payload swing. In order to make a fair comparison, the mass of the trolley in the system without considering payload swing contained the mass of the payload. As shown in Figure 7, the different lengths of cable affect the beam’s central deflection. When the cable length is shorter, the effects on payload swing are much more obvious. This is because the shorter the cable length is, the larger the payload swing angle is.Figure 7
The deflection of a beam affected by payload swing: (a) when the cable length is 6 m; (b) when the cable length is 14 m; (c) when the cable length is 22 m.
(a)
(b)
(c)Figure8 shows the differences between the beam central deflections for with or without a payload swing, respectively. When the payload swing angle was very small, the length of the cable had little effect on the vibration and central deflection of the crane’s beam. From reference [31], we can see that when the natural frequency of the suspended payload is half the fundamental natural frequency of the beam, a resonance may occur. However, the length of cable that would cause beam resonance is 4L4mbg/π4EI≈0.0125m, which is far less than the minimum length of the cable used in this paper. This is the reason why the deflection of beam was not obviously affected by payload swing.Figure 8
The difference between the two models: (a) when the cable length is 6 m; (b) when the cable length is 14 m; (c) when the cable length is 22 m.
(a)
(b)
(c)Figure9 shows the central deflection comparisons and results for a beam “with” and “without” a payload swing when the trolley is at sine acceleration. Figure 10 indicates the central deflection of a beam when the trolley is at ramp acceleration. Figures 11 and 12 show the differences between the two cases for Figures 9 and 10, respectively.Figure 9
The deflection of a beam affected by payload swing: (a) when the cable length is 6 m; (b) when the cable length is 14 m; (c) when the cable length is 22 m.
(a)
(b)
(c)Figure 10
The deflection of a beam affected by payload swing: (a) when the cable length is 6 m; (b) when the cable length is 14 m; (c) when the cable length is 22 m.
(a)
(b)
(c)Figure 11
The difference between the two models: (a) when the cable length is 6 m; (b) when the cable length is 14 m; (c) when the cable length is 22 m.
(a)
(b)
(c)Figure 12
The difference between the two models: (a) when the cable length is 6 m; (b) when the cable length is 14 m; (c) when the cable length is 22 m.
(a)
(b)
(c)The only differences among Figures7, 9, and 10 are the trolley accelerations in these three figures which are at a constant acceleration, sine acceleration, and ramp acceleration, respectively. The central deflections when the beam is subject to a swing payload in Figures 9 and 10 are smoother when compared with Figure 7. However, the central deflections in Figures 9 and 10 are similar. One of the reasons for this is because the sine acceleration and ramp acceleration of the trolleys in Figures 9 and 10 are close to each other, as indicated in Figure 2.When we compare Figures11 and 12, we can see that the difference in the deflection peaks for two cases in Figure 11 is larger than that in Figure 12. Both Figures 11 and 12 indicate the shorter the cable length, the larger the impact on the central deflection when the crane is subjected to a swing payload.


## 4.1. Validation of the Model System

### 4.1.1. Deflection of the Beam When Subjected to a Moving Mass

As mentioned in Section2, when the swing degree of freedom of the payload is ignored, equation (13) describes the problem of a simply supported beam subjected to a moving mass. In this study, the effectiveness of the mathematical model developed in Section 2 is first examined through the use of a classical moving mass passing a bridge model. To further confirm the reliability of the presented formulae and the developed computer programs, the parameters of the model system were chosen as follows: a moving mass mc = 70 kg with constant speed of vmax = 3.34 m/s. The length of the beam was L = 10 m and the cross-sectional area was A = 9 × 10−4 m2, and the second moment of area was I = 1.04 × 10−6 m4. The above parameters for the beam were the same as in references [29, 30].Figure3 shows the graphical time histories for the vertical central deflections of the simply supported beam. In Case 1 [29], the dynamic response of the middle point of the main beam was solved using the equivalent moving finite element assumption. However, for this study, the domain time response for the midpoint vibration of the main beam was solved by using the Newmark-β method. As can be seen from Figure 3, the two calculation results were consistent with the underlying trend. A close match can also be observed between the results shown in Case 2 [30] and the model developed in this paper. Since the deviations between the three curves are quite small, the formulations and computer programs developed in this paper should be acceptable methods of calculating the dynamic responses of a beam when subjected to a moving trolley.Figure 3
The calculated results for the central deflections using references [29, 30].


### 4.1.2. Payload Swing Angle

The reliability of the method presented in this paper is further confirmed by the payload swing angle. The payload’s swing is affected by the trolley’s acceleration and the length of the wire cable. A simply supported beam of lengthL = 6 m was subjected to a moving mass. In practice, the trolley starts at an initial velocity of zero and then accelerates to a particular speed, which it holds constant for some time before decelerating to a rest. In all our cases, the trolley’s acceleration lasted for 15 s, the constantly held speed lasted for 30 s, and the deceleration lasted for 15 s. The velocity during the constant speed phase was V = 0.1333 m/s. The initial swing angle of the payload was θ(0) = −0.01 (rad). These factors were considered to be consistent with those that were reported in Case 3 [25]. From Figure 4, it can be seen that the simulated results of this paper are similar to the data in reference [25], which proves the validity of the simulation methods proposed in this paper.Figure 4
A comparison of the calculated results to reference [25] for the payload swing angle.


## 4.1.1. Deflection of the Beam When Subjected to a Moving Mass

As mentioned in Section2, when the swing degree of freedom of the payload is ignored, equation (13) describes the problem of a simply supported beam subjected to a moving mass. In this study, the effectiveness of the mathematical model developed in Section 2 is first examined through the use of a classical moving mass passing a bridge model. To further confirm the reliability of the presented formulae and the developed computer programs, the parameters of the model system were chosen as follows: a moving mass mc = 70 kg with constant speed of vmax = 3.34 m/s. The length of the beam was L = 10 m and the cross-sectional area was A = 9 × 10−4 m2, and the second moment of area was I = 1.04 × 10−6 m4. The above parameters for the beam were the same as in references [29, 30].Figure3 shows the graphical time histories for the vertical central deflections of the simply supported beam. In Case 1 [29], the dynamic response of the middle point of the main beam was solved using the equivalent moving finite element assumption. However, for this study, the domain time response for the midpoint vibration of the main beam was solved by using the Newmark-β method. As can be seen from Figure 3, the two calculation results were consistent with the underlying trend. A close match can also be observed between the results shown in Case 2 [30] and the model developed in this paper. Since the deviations between the three curves are quite small, the formulations and computer programs developed in this paper should be acceptable methods of calculating the dynamic responses of a beam when subjected to a moving trolley.Figure 3
The calculated results for the central deflections using references [29, 30].


## 4.1.2. Payload Swing Angle

The reliability of the method presented in this paper is further confirmed by the payload swing angle. The payload’s swing is affected by the trolley’s acceleration and the length of the wire cable. A simply supported beam of lengthL = 6 m was subjected to a moving mass. In practice, the trolley starts at an initial velocity of zero and then accelerates to a particular speed, which it holds constant for some time before decelerating to a rest. In all our cases, the trolley’s acceleration lasted for 15 s, the constantly held speed lasted for 30 s, and the deceleration lasted for 15 s. The velocity during the constant speed phase was V = 0.1333 m/s. The initial swing angle of the payload was θ(0) = −0.01 (rad). These factors were considered to be consistent with those that were reported in Case 3 [25]. From Figure 4, it can be seen that the simulated results of this paper are similar to the data in reference [25], which proves the validity of the simulation methods proposed in this paper.Figure 4
A comparison of the calculated results to reference [25] for the payload swing angle.


## 4.2. Dynamic Response of the Crane System

According to the dynamic model of crane system, the factors affecting the dynamic response of the crane’s beam and the payload swing are not only related to the beam’s parameters, such as the cross-sectional properties and the dimensional parameters of the beam, but also to the mass of the trolley, the acceleration of the trolley, and the mass of the payload. This section analyzes the vibration of the beam and payload swing angle which are affected by these parameters.


### 4.2.1. Effect of the Trolley’s Velocity on the Dynamic Response

High trolley acceleration may lead directly to an excessively large payload swing angle, which greatly reduces safety when using the crane. For the trolley acceleration curve that was shown in Figure2, the trolley’s acceleration, and deceleration, time was set at 5 s. The maximum running speed of the trolley was vmax = 0.4167 m/s. Therefore, the trolley’s acceleration was up to approximate 0.0833 m/s2.Figure5 shows the vertical central deflection of the crane’s beam for the three speed patterns of the trolley. The maximum running speeds of the trolley were 0.25 m/s, 0.3333 m/s, and 0.4167 m/s, whilst the maximum vertical central deflections were 34.65 mm, 34.68 mm, and 34.69 mm, respectively. Thus, the larger the trolley acceleration was, the greater the vertical central deflection was. However, the effects of the acceleration of the trolley were not that obvious because the accelerations of the trolley were very small.Figure 5
The effect of trolley acceleration on the center deflection of the crane beam.


### 4.2.2. Effect of the Trolley’s Acceleration on the Dynamic Response

Figure6 shows the phase plane illustrations affected by the three different types of acceleration curves when the acceleration (deceleration) time is equal to the swing period of the payload, as shown in Figure 2. Figures 6(a)–6(c) are the response results of the constant, sine, and ramp accelerations of the trolley, respectively. The maximum trolley accelerations were 0.0464 m/s2, 0.0729 m/s2, and 0.0928 m/s2 for Figures 6(a)–6(c), respectively; keeping these three maximum trolley acceleration values satisfies equation (18).Figure 6
Phase plane illustrations for the three acceleration curves: (a) when trolley acceleration is constant; (b) when trolley acceleration is sine curve; (c) when trolley acceleration is ramp curve.
(a)
(b)
(c)In Figure6, the trolley moves start at point A; when the curve gets to the point B, the trolley is at uniform speed; and the trolley decelerates at point C. Under these three type acceleration curves, the maximum running speed of the trolley was equal to each other when trolley moved at uniform speed. In Figure 6, the curves for the sine acceleration and the ramp acceleration are far from the origin of the coordinates at the end of the curves, which means that the payload swing angle became larger at the end of the deceleration phase. This was because the payload still had an angular swing velocity at the beginning of the deceleration phase, and the direction of the velocity was opposite to the direction of the trolley’s deceleration most of the time.Figure6 indicates that the acceleration and deceleration times are the key parameters that affect the payload swing angle. Although the sine acceleration and ramp acceleration curves are smoother than those of the constant acceleration curves, the payload swing angles of the constant acceleration curve is smaller than that of the other two acceleration curves. In designing trolley movement trajectory, it is thus fundamentally necessary to investigate the swing period of the payloads as well as the acceleration times of the trolley so that the payload swing angle can be suppressed.


### 4.2.3. Influence of Payload Swing on the Crane’s Dynamic Response

Figure7 shows the central deflection comparisons for a beam between the result for “with” and “without” a payload swing when the trolley is in constant acceleration. The dashed line shows the result for beam’s central deflection without a payload swing, and the solid line indicates the result for central deflection for a beam with a payload swing. In order to make a fair comparison, the mass of the trolley in the system without considering payload swing contained the mass of the payload. As shown in Figure 7, the different lengths of cable affect the beam’s central deflection. When the cable length is shorter, the effects on payload swing are much more obvious. This is because the shorter the cable length is, the larger the payload swing angle is.Figure 7
The deflection of a beam affected by payload swing: (a) when the cable length is 6 m; (b) when the cable length is 14 m; (c) when the cable length is 22 m.
(a)
(b)
(c)Figure8 shows the differences between the beam central deflections for with or without a payload swing, respectively. When the payload swing angle was very small, the length of the cable had little effect on the vibration and central deflection of the crane’s beam. From reference [31], we can see that when the natural frequency of the suspended payload is half the fundamental natural frequency of the beam, a resonance may occur. However, the length of cable that would cause beam resonance is 4L4mbg/π4EI≈0.0125m, which is far less than the minimum length of the cable used in this paper. This is the reason why the deflection of beam was not obviously affected by payload swing.Figure 8
The difference between the two models: (a) when the cable length is 6 m; (b) when the cable length is 14 m; (c) when the cable length is 22 m.
(a)
(b)
(c)Figure9 shows the central deflection comparisons and results for a beam “with” and “without” a payload swing when the trolley is at sine acceleration. Figure 10 indicates the central deflection of a beam when the trolley is at ramp acceleration. Figures 11 and 12 show the differences between the two cases for Figures 9 and 10, respectively.Figure 9
The deflection of a beam affected by payload swing: (a) when the cable length is 6 m; (b) when the cable length is 14 m; (c) when the cable length is 22 m.
(a)
(b)
(c)Figure 10
The deflection of a beam affected by payload swing: (a) when the cable length is 6 m; (b) when the cable length is 14 m; (c) when the cable length is 22 m.
(a)
(b)
(c)Figure 11
The difference between the two models: (a) when the cable length is 6 m; (b) when the cable length is 14 m; (c) when the cable length is 22 m.
(a)
(b)
(c)Figure 12
The difference between the two models: (a) when the cable length is 6 m; (b) when the cable length is 14 m; (c) when the cable length is 22 m.
(a)
(b)
(c)The only differences among Figures7, 9, and 10 are the trolley accelerations in these three figures which are at a constant acceleration, sine acceleration, and ramp acceleration, respectively. The central deflections when the beam is subject to a swing payload in Figures 9 and 10 are smoother when compared with Figure 7. However, the central deflections in Figures 9 and 10 are similar. One of the reasons for this is because the sine acceleration and ramp acceleration of the trolleys in Figures 9 and 10 are close to each other, as indicated in Figure 2.When we compare Figures11 and 12, we can see that the difference in the deflection peaks for two cases in Figure 11 is larger than that in Figure 12. Both Figures 11 and 12 indicate the shorter the cable length, the larger the impact on the central deflection when the crane is subjected to a swing payload.


## 4.2.1. Effect of the Trolley’s Velocity on the Dynamic Response

High trolley acceleration may lead directly to an excessively large payload swing angle, which greatly reduces safety when using the crane. For the trolley acceleration curve that was shown in Figure2, the trolley’s acceleration, and deceleration, time was set at 5 s. The maximum running speed of the trolley was vmax = 0.4167 m/s. Therefore, the trolley’s acceleration was up to approximate 0.0833 m/s2.Figure5 shows the vertical central deflection of the crane’s beam for the three speed patterns of the trolley. The maximum running speeds of the trolley were 0.25 m/s, 0.3333 m/s, and 0.4167 m/s, whilst the maximum vertical central deflections were 34.65 mm, 34.68 mm, and 34.69 mm, respectively. Thus, the larger the trolley acceleration was, the greater the vertical central deflection was. However, the effects of the acceleration of the trolley were not that obvious because the accelerations of the trolley were very small.Figure 5
The effect of trolley acceleration on the center deflection of the crane beam.


## 4.2.2. Effect of the Trolley’s Acceleration on the Dynamic Response

Figure6 shows the phase plane illustrations affected by the three different types of acceleration curves when the acceleration (deceleration) time is equal to the swing period of the payload, as shown in Figure 2. Figures 6(a)–6(c) are the response results of the constant, sine, and ramp accelerations of the trolley, respectively. The maximum trolley accelerations were 0.0464 m/s2, 0.0729 m/s2, and 0.0928 m/s2 for Figures 6(a)–6(c), respectively; keeping these three maximum trolley acceleration values satisfies equation (18).Figure 6
Phase plane illustrations for the three acceleration curves: (a) when trolley acceleration is constant; (b) when trolley acceleration is sine curve; (c) when trolley acceleration is ramp curve.
(a)
(b)
(c)In Figure6, the trolley moves start at point A; when the curve gets to the point B, the trolley is at uniform speed; and the trolley decelerates at point C. Under these three type acceleration curves, the maximum running speed of the trolley was equal to each other when trolley moved at uniform speed. In Figure 6, the curves for the sine acceleration and the ramp acceleration are far from the origin of the coordinates at the end of the curves, which means that the payload swing angle became larger at the end of the deceleration phase. This was because the payload still had an angular swing velocity at the beginning of the deceleration phase, and the direction of the velocity was opposite to the direction of the trolley’s deceleration most of the time.Figure6 indicates that the acceleration and deceleration times are the key parameters that affect the payload swing angle. Although the sine acceleration and ramp acceleration curves are smoother than those of the constant acceleration curves, the payload swing angles of the constant acceleration curve is smaller than that of the other two acceleration curves. In designing trolley movement trajectory, it is thus fundamentally necessary to investigate the swing period of the payloads as well as the acceleration times of the trolley so that the payload swing angle can be suppressed.


## 4.2.3. Influence of Payload Swing on the Crane’s Dynamic Response

Figure7 shows the central deflection comparisons for a beam between the result for “with” and “without” a payload swing when the trolley is in constant acceleration. The dashed line shows the result for beam’s central deflection without a payload swing, and the solid line indicates the result for central deflection for a beam with a payload swing. In order to make a fair comparison, the mass of the trolley in the system without considering payload swing contained the mass of the payload. As shown in Figure 7, the different lengths of cable affect the beam’s central deflection. When the cable length is shorter, the effects on payload swing are much more obvious. This is because the shorter the cable length is, the larger the payload swing angle is.Figure 7
The deflection of a beam affected by payload swing: (a) when the cable length is 6 m; (b) when the cable length is 14 m; (c) when the cable length is 22 m.
(a)
(b)
(c)Figure8 shows the differences between the beam central deflections for with or without a payload swing, respectively. When the payload swing angle was very small, the length of the cable had little effect on the vibration and central deflection of the crane’s beam. From reference [31], we can see that when the natural frequency of the suspended payload is half the fundamental natural frequency of the beam, a resonance may occur. However, the length of cable that would cause beam resonance is 4L4mbg/π4EI≈0.0125m, which is far less than the minimum length of the cable used in this paper. This is the reason why the deflection of beam was not obviously affected by payload swing.Figure 8
The difference between the two models: (a) when the cable length is 6 m; (b) when the cable length is 14 m; (c) when the cable length is 22 m.
(a)
(b)
(c)Figure9 shows the central deflection comparisons and results for a beam “with” and “without” a payload swing when the trolley is at sine acceleration. Figure 10 indicates the central deflection of a beam when the trolley is at ramp acceleration. Figures 11 and 12 show the differences between the two cases for Figures 9 and 10, respectively.Figure 9
The deflection of a beam affected by payload swing: (a) when the cable length is 6 m; (b) when the cable length is 14 m; (c) when the cable length is 22 m.
(a)
(b)
(c)Figure 10
The deflection of a beam affected by payload swing: (a) when the cable length is 6 m; (b) when the cable length is 14 m; (c) when the cable length is 22 m.
(a)
(b)
(c)Figure 11
The difference between the two models: (a) when the cable length is 6 m; (b) when the cable length is 14 m; (c) when the cable length is 22 m.
(a)
(b)
(c)Figure 12
The difference between the two models: (a) when the cable length is 6 m; (b) when the cable length is 14 m; (c) when the cable length is 22 m.
(a)
(b)
(c)The only differences among Figures7, 9, and 10 are the trolley accelerations in these three figures which are at a constant acceleration, sine acceleration, and ramp acceleration, respectively. The central deflections when the beam is subject to a swing payload in Figures 9 and 10 are smoother when compared with Figure 7. However, the central deflections in Figures 9 and 10 are similar. One of the reasons for this is because the sine acceleration and ramp acceleration of the trolleys in Figures 9 and 10 are close to each other, as indicated in Figure 2.When we compare Figures11 and 12, we can see that the difference in the deflection peaks for two cases in Figure 11 is larger than that in Figure 12. Both Figures 11 and 12 indicate the shorter the cable length, the larger the impact on the central deflection when the crane is subjected to a swing payload.


## 5. Conclusion

In this paper, a dynamic model of the coupling system of a flexible crane beam—a moving trolley—and a swing payload was established to analyze the deflection of the crane’s beam and payload swing, under different trolley acceleration and payload mass conditions. The dynamic responses of the crane’s beam and payloads were obtained through a numerical solution that used the Newmark-β gradual integration method. The following conclusions were obtained:(1)
When the speed and acceleration of the trolley was small, increasing the speed and acceleration of the trolley did not obviously influence the maximum deflection of the central beam, but it did change the curve shape of the crane’s beam deflection in some instances.(2)
The vertical deflection of crane’s beam was directly proportional to the mass of the trolley and payload. When the payload’s angle was small, the vertical deflection of crane’s beam was not obviously affected by the payload’s swing.(3)
Resonance can be avoided by keeping the natural frequency of the payload away from half the beam’s fundamental natural frequency. The closer these two frequencies are, the larger the payload’s swing is.



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*Source: 1291652-2019-06-24.xml*