INTRODUCTION
The vibrational motion induced by the moving carriage hoisted an object is one of important vibration problems in variety of engineering systems. Such systems may be found in crane system such as overhead cranes, gantry cranes, rotary cranes and other types of crane. An example of a crane system which is most widely used in factories, warehouse, shipping yards and nuclear facilities is nonslewingluffing crane type, namely gantry crane system. Gantry crane is usually designed to have very strong structures and big dimension in order to lift and transfer heavy payloads.
However, the lifting capacities heavier and the size of cranes have increased
continuously due to the increase of productivity and capacity. This condition
leads to the consequence that the elastic deformability of all elements of the
system cannot be neglected (Ren et al., 2008;
Bhimani, 1999). The presence of elastic deformability
will induce the unwanted vibration when it is subjected to dynamic loads. It
may cause issues related with safety of crane system and its framework, operators
and surrounding environment.
Moving load as a total weight of trolley, lift system and lifted load according
to (Bhimani, 1999) is taken as a moving subsystem, then
this moving subsystem which is accompanied by swinging payload will induce the
crane framework and conversely. They will create bidirectional dynamic interaction
as reported by (Oguamanam et al., 2001) and constitute
nonlinear coupling terms between crane system and crane framework, affecting
the motion of gantry crane system. This dynamic characteristic cannot be described
by classical model of pendulum system with moving pivot point. Flexible gantry
crane, unlike their counterpart namely rigid gantry crane, has not received
much attention. The published papers and conferences relate with this case are
limited. Some cited references can be referred in (Oguamanam
et al., 2001; Yang et al., 2007; Zrnic
et al., 2006; Mitrev, 2007). Among those
references, (Wu, 2004) has studied such a case by introducing
the concept of equivalent moving mass matrix, but the dynamics of payload was
not introduced in his work, restricted to planar swing and rigid cable. He also
concerned only in dynamic responses of crane framework.
The aim of this paper is to generate the equations of motion of threedimensional gantry crane system by introducing the flexibility of crane framework and hoist cable into the model. Computational technique for solving the coupled equations of motion is proposed by which allow us to investigate the bidirectional dynamic interaction between the gantry crane and crane framework.
MATERIALS AND METHODS
Gantry crane system can be divided into two subsystems, namely gantry crane
and stationary crane framework. In practice, gantry crane incorporates the interaction
among trolley, hoist cable and payload under trolley and hoist mechanism. The
payload is attached using hook system, hoisted from the trolley through hoist
cable. For simplicity of the characteristics of the physical gantry crane, several
assumptions are put forward to the proposed dynamical model. The mass of trolley
and payload are modeled as lumped mass which is connected by an elastic cable.
Figure 1 explained Payload and cable behave as an elastic
pendulum system. The payload has two swing angles with respect to the inference
frame: is denoted as angle between the X_{T}axis and X_{T}
Y_{T} plane, while notation φ is the angle between the cable to
X_{T} Y_{T}plane as defined by Fig. 2. The
payload swings either small or large swing angles. Dynamics of hoist drive mechanism
is not considered.
Dynamics of swinging motion of payload on flexible crane framework: The equations of motion of swinging motion of payload on flexible crane framework can be derived by Lagranges equations, with the following form.
The first three terms of q = (u_{T},v_{T},w_{T},x_{T},θ,φ,δ)
are defined as the generalized coordinates to describe the elastic deformations
of crane framework and the rest is trolley, payload and hoist cable motion.
General velocity of the system is denoted by ,
where it is time derivative q. The position vector of trolley r_{T}
and payload r_{P} as in Fig. 1 can be expressed as:
where, I, j and k are unit vectors along the x, y and zaxis, respectively.
For convenience, elastic displacements in Eq. 2a2b
can be expressed in following terms.
The term x_{T} is position of trolley carrying a swinging payload at
central point c_{P} of the top beam of crane framework which is timeinvariant.
Referring to Eq. 2, the flexibility of crane framework (u_{T},
v_{T}, w_{T}) and hoist cable δ is considered in the position
vector of trolley and payload.

Fig. 1: 
Finite element model of flexible gantry crane system 

Fig. 2: 
Elastic deformation of crane framework and hoist cable 
The flexibility of hoist cable is modeled as one linear spring with stretched
length .
This is sufficient approach since the cable is assumed to be in tension during
normal crane operation (Masoud, 2009). The linear spring
force of hoist cable can be expressed as:
It is noted that notation k is cable stiffness, while
is unstreched hoist cable. The Lagrangian L is defined as L = k–P, where
k is kinetics energy and P is potential energy of the system. Generalized force
is denoted as f_{I}, where they are fx, fy and fz, applied input force
for the x, y and z motions respectively. Kinetics energy of the system is the kinetics energy of the trolley and the payload, defined as:
The total potential energy of the system P is the potential energy of the trolley, payload and cable as follows:
By deriving L with respect to generalized coordinates and considering that the structural members flexibility (u_{T}, v_{T}, w_{T})are based on position x_{T} and time t, then
The equation motions of the system can be derived and summarized as Eq.
7a7d and the right side of Eq.13a.
Dynamics of crane framework: The crane framework model is established by the finite element method by introducing the global mass, damping and stiffness matrices of the crane framework. Based on the finite element discretization, the equation of motion for MDoF structural system, geometrically and materially linear dynamic is represented as follows:
where, [M_{st}], [C_{st}], [K_{St}] are the mass, damping
and stiffness matrices of the crane framework, respectively. Terms {q_{St}(t)},{q_{St}(t)},{q_{St}(t)}
are the acceleration, velocity and displacement vectors for the whole framework,
respectively. Due to the trolley traverses along the top beam of crane framework,
and assumed that the moving trolley carrying a swinging payload is always in
contact with the framework girder, there will be transmitted force to the framework
from the swinging payload through the cable and contact force at contact point
between the trolley and the crane framework. This is exciting forces for the
crane framework and noted as {F_{St}(t)}. In order to tackle the position
and time variant,{F_{St}(t)} in Eq. 8 is modified
as below:
Term f_{0} is external force vector acting on the framework girder, which can be written in the following form:
where, f_{0x}, f_{0y} and f_{0Z} are the corresponding
external force components in the x, y and z direction. The magnitude of these
external forces is given by Eq.13a.
Due to finite element model of crane framework using space frame element, term N_{K}(k = 1–12) is the shape functions of space frame element, as well. External force vector in Eq. 9 can be rewritten in the following form.
It is noted that terms {N_{K}}u u = k = 1,7, {N_{K}}v v = k = 2,6,8,12 and {N_{K}}w w = k =3,5,9,11 are shape
functions associated with degree of freedoms in three directions axial (x), vertical (y) and lateral (z). According to the concept of equivalent node forces, Eq. 11 can be presented in Fig. 3.
The notation f_{SK} is the equivalent nodal forces and u_{SK} is the displacements for nodes where sk(k 1–12) are the numberings for the twelve degrees of freedom of the one space frame element on which the trolley located.
For more detail about concept of equivalent node forces, it can be referred
to (Wu, 2004). The axial (x), vertical (y) and lateral
(z) displacement of space frame element at position x, can be obtained as below:
By manipulating Eq. 12a12c into Eq.
9 and 11, the equations of motion of flexible gantry
crane system yield nonlinear coupled equations of motion and written in Eq.
13s. Terms {N_{K}’} and {N_{K}”}, indicate the partial
derivative of shape functions with respect to trolley position.
Notation Δ_{r} and its derivatives indicate vectors of displacements
and its derivatives, as well for the rest of the degrees of freedom of the crane
framework. Under assumption that the crane framework is to be rigid or called
rigid model, vibration in the Eq. 13 is vanished. Equations
of motion of the system can be reduced into classical 3Dpendulum system with
moving pivot and the results are the same with Newton’s motion law as presented
by Eq. 14a14b.

Fig. 3: 
Equivalent nodal forces of one space frame element 
where,
are given by
It is noted that because of very lengthy and tedious, equations for
are not presented here.
The damping matrix in the left side of Eq. 13a is assumed
proportional to the combination of mass and stiffness matrix. Under this assumption,
The Rayleigh damping theory is used. The damping matrix can be written as:
The proportionality factors is calculated by using damping ratio δ=δ_{1}=δ_{2} and natural frequency ω_{1} and ω_{2}.
Numerical approach: If the swing angles, θ and φ in the right side
of Eq. 7a7b and Eq. 13
are zero, the case would just be that of moving load case in crane framework,
which imposed by (m_{t}+m_{p}) load. In order to solve Eq.
7a7b and (13), the computational
technique under Newmarkβ and fourthorder RungeKutta method is proposed.
The crane framework displacements are calculated by Newmarkβ method of
direct integration
The two parameters are selected as β=0.25 and γ=0.5, which implies a constant average acceleration with unconditional numerical stability, while payload angular displacements are calculated by fourthorder Runge Kutta method. For each integration step, Newmarkβ and RungeKutta methods are combined simultaneously to obtain the structure and payload responses (Table 1). The computational procedures with a time step of Δ_{t} that performs the direct numerical integration can be summarized as follows:
Step 1: Set initial condition for velocity and acceleration:
Step 2: The initial external force vector {F}_{0} = {F(t =
0)} is calculated using right side of Eq. 13 by using initial
conditions
of payload.
Table 1: 
Newmar’s parameters 

Stap 3: The initial acceleration vector is calculated as:
Step 4: Evaluation of constants from α_{0} to α_{7} .The parameters α_{i} are below.
Step 5: For each time step:
• 
The overall mass matrix [M] and stiffness matrix [K] of the
system are generated by using Eq. 13 
• 
Calculation of the first and second natural frequencies (ω_{1}
and ω_{2}) of the overall crane framework and the overall damping
matrix [C] in Eq. 13 and 15 
• 
Eqs. 7a, are solved to obtain
and
using fourthorder RungeKutta and external force vector {F}_{t+Δt}
is then updated. The force vector {F}_{t+Δt} denotes the external
loads of the system at time t+Δt 
• 
Equation of motion of the system is represented as below 
The effective load vector
is below:
• 
The displacement, velocity and acceleration responses are
computed with satisfying the following relationship: 
RESULTS AND DISCUSSION
The crosssectional area of crane framework is uniform, isotropic and homogeneous material properties. The gravitational acceleration is g = 9.81 ms^{–2} and time interval is Δt = 0.005 s. Crane framework is discretized into 58 elements and 82 nodes. The issue of total number of elements and nodes will not be treated as a parameter that will be varied in the simulations. It is also noted that there is no damping either in dynamics of crane framework or dynamics of payload motion, unless particularly stated. This is expected to make it as a direct comparison with the pendulum model.

Fig. 4: 
Pendulum attached to a MSD system 
Simple case of a coupled dynamic system: The developed computer program
will be verified first by solving a simple coupled dynamic model. The model
and its equations of motion are taken from (Kyrychko et
al., 2006), who used a simple nonlinear system consist of pendulum attached
to a MassSpringDamper (MSD). The configuration and the parameters of the system
are depicted in Fig. 4. The system is excited by sinusoidal
driving force as shown in Fig. 5.
Under the action of pendulum motion, the equations of motion are solved by developed Newmarkβfourth order RungeKutta and ODE45 for nonlinear model. The displacements are shown in Fig. 6 and very good agreement between the solutions offered by both methods.
Openloop responses of flexible gantry crane: In this subsection, as
a test for the dynamic model in Eq.7a, 7c7d
and 13, openloop responses are performed under two types
of driving force f_{X} , namely bangbang input force and harmonic input
force. It is noted that other forms of driving force could have been chosen,
but here an arbitrary form is chosen to primarily attempt to model the real
driving force situation for the actual crane system.
The parameters for cranes are shown in Table 2 while, the
model and the dimensions of crane framework are taken from (http://www.spanco.com/literature)
and shown in Table 3.
Responses under bangbang input force: The simulation is performed where
the bangbang input force is applied to move the trolley of gantry crane. Magnitude
of bangbang input force is varied in order such that the trolley reaches 3,
6 and 12 m from the left end of the top beam of crane framework as depicted
in Fig. 7.

Fig. 5: 
Driving force for pendulum attached to a MSD system 

Fig. 6: 
Pendulum attached to a MSD displacement (a) Vertical displacements
(b) Swing angle responses 
Table 2: 
Crane parameters 

Table 3: 
Crane framework properties 

The differences between rigid and flexible model are noted with Δθ = θ_{flexible}–
θ_{rigid} and Δφ = φ _{flexible} –φ
_{rigid}.
When the trolley accelerates, payload swings behind the trolley. Vice versa,
the payload swings ahead of trolley when trolley decelerates. The swing angles
are identical when the trolley speed is constant or the trolley stops. The payload
will continue swing although the force is taken after 13 sec (Fig.
8). This is due to the dynamics of payload is without damping as it can
be seen from Eq. 7.

Fig. 7: 
Bangbang input forces and trolley positions 

Fig. 8: 
Time history of under bangbanginput force (—) flexible
model () rigid model (a) 3m, (b) 6m and (c) 12m 

Fig. 9: 
Time history of Δ under bangbang input force (a) 3 m,
(b) 6 m and (c) 12 m 
The accelerations and decelerations periods does not appear in swing angle
φ, as one can notice from Fig. 1011.
That is because the horizontal (X) inertia force induced by moving trolley carrying
a swinging payload significantly affects the payload swing in planar motion
than that in space motion due to input force is applied in the horizontal direction.
Further, amplitudes and frequencies of Δθ and Δφ are significantly
affected by the change of trolley position on the top beam of crane framework
as depicted in by Fig. 9 and 11.

Fig. 10: 
Time history of φ under bangbanginput force (—)
flexible model; () rigid model (a) 3 m, (b) 6 m and (c) 12 m 

Fig. 11: 
Time history of Δφ under bangbang input force (a)
3 m (b) 6 m (c) 12 m 

Fig. 12: 
Vertical displacements under bangbang input force (—)
3 m; (  ) 6 m; (….) 12 m; corresponding static displacement 
From Fig. 12, it can be found that the vertical displacements of the central point c_{p} of the top beam of crane framework damps to zero when the trolley stops at the right side of beam for position 12m and to corresponding static displacements for position 3 and 6 m.

Fig. 13: 
Harmonic input force 

Fig. 14: 
Flexible and rigid model responses of θ flexible model ( 
) rigid model (a) 0.1 ω_{n} (b) 0.5 ω_{n} and
(c) 0.95 ω_{n} 

Fig. 15: 
Flexible and rigid model responses of φ flexible model
(  ) rigid model (a) 0.1 ω_{n} (b) 0.5 ω_{n}
and (c) 0.95 ω_{n} 
The change of trolley position on the top beam of crane framework induces slighter vertical displacement.
Responses under harmonic input force: The effect of different harmonic
input force frequencies on the responses of payload swings and crane framework
is investigated. All the parameters are identical with Table 2
and 3. The harmonic input force frequencies are 0.1ω_{n},
0.5ω_{n} and 0.95ω_{n} as presented in Fig.
13. It is noted that term ω_{n} corresponds to natural frequency
of payload with hoist cable length, =1m.

Fig. 16: 
Vertical displacements under harmonic force input (—)
0.1 ω_{n}; (  ) 0.5 ω_{n}; () 0.95 ω_{n} 
The dynamic responses of payload swings and difference with rigid model are
shown in Fig. 1415. The results show
that time history θ of and φ depend on the harmonic frequency of input force.
As the harmonic frequency approaches the natural frequency of payload, the amplitudes
of swing angles increase erratically. It is seen also that swing angle, θ is larger
than φ.
Vertical displacements of the central point c_{p} of top beam of crane framework under variation of harmonic force input are shown in Fig. 16. It is observed that different harmonic input force frequencies significantly affect the vertical responses of crane framework. As the harmonic frequency approaches the natural frequency of payload, the amplitudes of swing angles increase erratically. The large amplitude of swing angles make the vibration amplitudes of axial, vertical and lateral responses getting larger because of coupled system. Only vertical displacement is presented because similar trends are noted for axial and lateral displacements.
CONCLUSION
The equations of motion show that the flexible gantry crane system can be modeled by moving trolley carrying a swinging payload on flexible crane framework. It behaves as an elastic pendulum system with flexible moving support which undergoes acceleration in three directions.
Numerical simulations are then carried out to investigate the responses of gantry crane system by accounting flexibility of crane framework and hoist cable. Simulations show that gantry crane and framework is a coupled dynamic system, where bidirectional dynamic interaction is contributed by the flexibility of the crane framework and hoist cable. A direct access to the system equations enable the applications for advanced dynamic analysis, structural design and controller design for vibration suppression.
ACKNOWLEDGMENT
The authors are thankful to Universiti Teknologi PETRONAS for providing the research facilities.