Nonlinear Dynamics 27: 255–269, 2002.
© 2002 Kluwer Academic Publishers. Printed in the Netherlands.
Pendulation Reduction in Boom Cranes Using Cable
Length Manipulation
E. M. ABDEL-RAHMAN and A. H. NAYFEH
Department of Engineering Science and Mechanics, MC 0219, Virginia Polytechnic Institute and State
University, Blacksburg, VA 24061, U.S.A.
(Received: 17 August 2001; accepted: 2 October 2001)
Abstract. A technique is proposed to reduce payload pendulations using the reeling and unreeling of the hoisting
cable. The payload is modeled as a point mass, the cable is modeled as a rigid link, and the assembly, a spherical
pendulum, is attached to the boom tip. An excitation is applied to the assembly at the boom tip. The motion of the
payload is described using two-dimensional and three-dimensional models. Our results demonstrate that cablelength manipulation can be used to reduce payload pendulations due to near-resonance excitations. Significant
reductions can be obtained via an appropriate choice of the reeling/unreeling speed. We also demonstrate the
limitations inherent in two-dimensional modelings of a crane.
Keywords: Boom crane, control, stability, three-dimensional.
1. Introduction
Cranes are widely used in construction and transportation industries. Boom cranes are a particularly important class of cranes. They are more stable structurally and more compact in
comparison to similar capacity gantry or rotary cranes. Consequently, all mobile cranes use
boom cranes. A common factor among all types of cranes is the high compliance of the cablepayload assembly, rendering the payload susceptible to significant pendulations due to base
excitations, inertial forces, and wind gusts. As the excitation amplitude increases and/or as its
frequency approaches the natural frequency of the cable-payload assembly, the base excitation
becomes the dominant source of payload pendulations.
A particularly important case is that of ship-mounted cranes. These are boom cranes mounted on see-going vessels and used to transfer cargo from larger cargo ships to lighter port-going
vessels. Wave-induced motions of the ship produce base excitations at the boom tip, thereby
subjecting the payload to pendulations. As the sea state rises, these pendulations increase. In
high seas, cargo transfer has to be suspended due to payload motions reaching unsafe levels.
This constraint on ship-mounted crane operations leads to time delays in shipping flow and
cost overruns [1].
Several control strategies were proposed to reduce payload pendulations in the absence of
base excitations. Sakawa et al. [2] proposed an optimization scheme to generate the torque
profile necessary to transfer a load along a predefined path while minimizing the payload
pendulations during transfer and at the target point. The transfer time is minimized by iteration. The technique was applied to a simulated model of a boom crane slewing at a constant
luff angle while reeling in the cable and linearized around the payload equilibrium position. Simulation results showed that the payload is free of residual pendulations at the target
256 E. M. Abdel-Rahman and A. H. Nayfeh
point, however payload pendulations develop along the path and increase as the slewing angle
increases.
Sakawa and Nakazumi [3] proposed a two-tier control strategy for a crane traversing a
predefined trajectory. An open-loop controller tracks the trajectory the boom travels, while an
LQR optimized linear state feedback controller employs the slew, luff, and hoisting to eliminate inertia-induced residual pendulations at the end of the maneuver. Computer simulations
showed pendulation angles during the maneuver of as high as 21.6◦ . Sato and Sakawa [4]
extended the application of this control strategy to a boom with a flexible jib at the boom tip.
Takeuchi et al. [5] proposed an input-shaping strategy to achieve a time-optimal slew
motion only while minimizing the residual pendulations. The strategy uses a slew angle acceleration profile similar to that generated by Yamada et al. [6], for gantry cranes, to perform
the slew motion and control the pendulations. Numerical simulation showed that the strategy
can suppress out-of-plane pendulations but not in-plane pendulations, which persist well after
the boom comes to a stop. Takeuchi et al. [5] also proposed a fuzzy logic strategy to achieve
a time-optimal slew motion only while minimizing the residual pendulations. The strategy
imitates the input-shaping strategy proposed in the same work. Numerical simulation showed
that the fuzzy logic controller is unsuccessful in dampening in-plane pendulations. This is
expected since the control effort applied, the slew motion, is an out-of-plane motion.
Hara et al. [7] proposed an LQR optimized state feedback controller using the boom telescopic motion as a control input to control planar payload pendulations due to the telescopic
motion of the boom. A saturation condition is applied to the controller input to the plant to
keep it within available control authority, thus producing a suboptimal controller. In computer
simulations and testing on an actual crane, the control strategy was successful in suppressing
pendulation.
Nguyen et al. [8] proposed a state feedback control strategy, based on a linear planar
model, to hoist and stabilize the payload and position the boom. Two independent controllers
are employed, one to control the boom luff angle and payload pendulation and another to
control the payload hoisting. Experimental verification on a scaled model showed oscillations
of the boom around the reference path and steady-state errors in the boom angle and cable
length. On the other hand, transient pendulations were contained to less than 4◦ .
Souissi and Koivo [9] proposed a two-tier control strategy to stabilize a boom crane against
inertia-induced payload pendulations. A PID controller tracks a reference trajectory using
the slew and luff of the boom and the reeling/unreeling of the cable, while a PD controller
dampens the payload pendulations. Numerical simulation of the boom performing a predefined luffing-slewing-luffing maneuver at a constant cable length showed significant payload pendulations, as much as 15◦ , indicating that the PD controller was not effective in
damping the pendulations.
Gustafsson [10] proposed a control strategy employing two independent, in-plane and
out-of-plane, linear position feedback controllers designed based on a partial linearization of
the spherical pendulum model to suppress inertia-induced payload pendulations. Computersimulation results showed stable responses for operator commanded slewing rates away from
the natural frequency of the cable-payload assembly and small pendulation angles.
Lewis et al. [11] applied quasi-static notch filters to the operator’s input commands to
avoid exciting the cable-payload assembly at its natural frequency. The notch location varies
with the length of the cable to filter out excitations at the current natural frequency of the
cable-payload assembly. The roll-off coefficient for the notch filter is held constant and thus
is optimum only at a single cable length, and the filter characteristics change with changes in
Pendulation Reduction in Boom Cranes 257
the cable length. Simulation results showed reductions in both the in-plane and out-of-plane
payload pendulations. However the filtering process imposes delays, as much as 2.5 seconds,
between the operator input and the actual filtered input to the crane. Further, the variable filter
characteristics produce variable responses for the same operator input. Also, the linear nature
of the filter limits its effectiveness to low crane speeds. Parker et al. [12, 13] modified this
control strategy using a roll-off coefficient linearized with respect to the natural frequency
employed in the notch filter design. Experimental verification showed an 18 dB reduction
in the payload pendulations at the end of the prescribed maneuver. Simulation results [14]
showed that, while the response of this filter is more consistent at different cable lengths
and demonstrates a slight improvement in the pendulation reduction over the original filter, it
imposes more time delays and larger changes of amplitude on the operator input. Alternatively,
Lewis et al. [14] modified the same control strategy using a roll-off coefficient linearized with
respect to the forcing (slew and luff input velocities) in the notch filter. Simulation results did
not show a significant difference in the performance of this filter as compared to that of Parker
et al. [12, 13].
The strategies described above are of limited use at sea where base excitations are always
present. To address this need, Patel et al. [15], McCormick and Witz [16], and Witz [17]
reduced a model of the in-plane motion of a ship-mounted crane to a Mathieu equation.
Based on the stability diagrams of this equation, operability conditions of the crane-ship were
derived. While this approach maximizes safety, it does not address the need to operate shipmounted cranes in high sea states, namely sea state three and above. So other control strategies
accounting for base excitations were developed.
Chin et al. [18] proposed a nonlinear feedback control scheme to suppress the parametric
instabilities in payload motions due to wave-induced base excitations. They demonstrated
analytically that, introducing a control effort in the form of a harmonic change in the cable
length at the same frequency as the base excitations, can suppress the parametric instability
and result in a smooth response.
Iwasaki et al. [19] and Imazeki et al. [20] designed an active mass-damper system to suppress payload pendulations. The system was installed on the sling of a barge-mounted boom
crane. A planar linear model of the crane was used to design a linear feedback controller.
The control effort is the acceleration applied to a 35-ton damping mass riding on the sling. A
132KW induction motor is used to drive the damping mass. Test results showed that the sling
motion was reduced to 1/2 to 1/3 of the uncontrolled motions at the test frequency.
Balachandran and co-workers [21, 22] used two-dimensional and three-dimensional [23]
nonlinear models of ship-mounted boom cranes to design a nonlinear vibration absorber, a
mechanical filter, to control the motions of the pivot point around which the payload oscillates.
The design modifies the boom crane configuration to suspend the payload from a pivot plane,
which in turn is suspended under the boom tip. The absorber has both a passive mode and an
active mode employing feedback control. Computer simulations showed that the absorber can
shift bifurcation points arising from the nonlinear dynamics of the cable-payload assembly
and suppress subcritical bifurcations. They also showed that the feedback component of the
filter can attenuate the transient and steady-state payload motions. However, in some filter
designs, suppression of the subcritical bifurcation produced a large resonance-like response
around half the natural frequency.
Using a planar model of a ship-mounted crane, Henry et al. [24] developed a delayed
feedback controller. Computer simulations and experimental results showed an effective suppression of payload pendulations due to in-plane, roll and heave, excitations. Masoud et al.
258 E. M. Abdel-Rahman and A. H. Nayfeh
[25] extended this approach to the three-dimensional case. In computer simulations and experiments, the controller successfully suppressed payload pendulations due to both in-plane
and out-of-plane base excitations.
Kral and co-workers [26, 27] used a two-dimensional nonlinear model of a ship-mounted
crane to study the influence of cable hoisting on payload pendulations. The results show
nonlinear behavior, namely chaos at cable lengths exceeding 28 m and multiple responses
for the same cable length, depending on the lowering or lifting history of the motion.
In this paper, we propose an approach using the reeling and unreeling of the hoisting
cable to reduce payload pendulations under base excitations due to sea waves. The proposed technique is to be used in conjunction with other control strategies under development
within our research group [24, 25] to produce a robust control capable of suppressing payload
pendulations in high sea states.
2. Problem Formulation
The payload is modeled as a point mass, the cable is modeled as an inextensible link, and the
assembly, a spherical pendulum, is attached to the boom tip. The assembly is subjected to an
excitation applied at the boom tip. A two-dimensional model of payload pendulations in the
plane of excitation and a three-dimensional model accounting for out-of-plane excitations and
instabilities are presented.
2.1. T WO -D IMENSIONAL M ODEL
In this model, the excitations and the payload motions are restricted to one plane. The motion
of the payload can then be described using a single degree of freedom, pendulations in the
plane. They are governed by
θ̈ +
g
sin θ + 2µθ̇ = −ẍa ,
l
(1)
where θ is the angle between the equilibrium position of the cable-payload assembly and the
cable at time t, l is the cable length, and µ is the coefficient of linear damping. The in-plane
acceleration of the boom tip is given by
ẍa = −Fx cos(t),
(2)
where Fx and are the excitation amplitude, per unit of the mass moment of inertia of the
payload around the boom tip, and frequency.
Expanding the nonlinear term in the equation of motion around the equilibrium configuration and dropping the higher-order terms, we obtain
1 3
2
(3)
ü + ω u − u + 2µu̇ = Fx cos(t),
6
where ω is the natural frequency of the cable-payload assembly and is a bookkeeping
parameter. We assume small but finite motions of the cable-payload assembly around the
equilibrium position. We also assume that µ and Fx are small and of the same order of
magnitude.
Pendulation Reduction in Boom Cranes 259
The payload motion reaches its extreme limits in the neighborhood of the natural frequency
ω of the cable-payload assembly. To examine the response of the payload to forcing in this
region, we set close to ω and express the nearness of the two frequencies by
= ω + σ.
(4)
To examine the feasibility of a control algorithm employing the reeling and unreeling of the
cable to detune ω away from , and thus reduce the pendulations of the payload, we write the
natural frequency of the payload as
g
,
(5)
ω=
lo + δt
where lo is the original length of the cable and δ is the speed of cable reeling/unreeling.
We use the method of multiple scales [28–30] to determine a first-order uniform approximation to the solution of Equation (3) in the form
u ≈ u1 (φ, τ ) + 2 u2 (φ, τ ),
(6)
where φ = ωt is a fast scale, characterizing motions close to the natural frequency ω, and
τ = t is a slow scale, characterizing the modulations of the amplitude and phase. Substituting
Equation (6) into Equation (3), using Equation (4), and equating coefficients of like powers of
, we obtain the following hierarchy of problems:
Order :
∂ 2 u1
+ u1 = 0.
∂φ 2
(7)
Order 2 :
ω2
∂u1 ∂ω 1 2 3
∂ 2 u1
∂ 2 u2
2
+
− ω u1 + 2µu̇1 = Fx cos(t).
+
ω
u
+
2
ω
2
2
∂φ
∂φ∂τ
∂φ ∂τ
6
(8)
We solve these problems sequentially.
The solution of the first-order problem can be expressed as
u1 = a cos(φ + β),
(9)
where a is the amplitude and β is the phase of the in-plane payload pendulations. Substituting
Equation (9) into Equation (8) and eliminating the secular terms, we obtain the modulation
equations
Fx
ω
a − µa −
sin β,
2ω
2ω
ω
Fx
cos β.
β = −σ − a 2 −
16
2 ωa
a = −
(10)
(11)
260 E. M. Abdel-Rahman and A. H. Nayfeh
Figure 1. Coordinate systems for the crane.
2.2. T HREE -D IMENSIONAL M ODEL
In this version of the model, we allow both in-plane and out-of-plane motions and excitations.
As shown in Figure 1, two coordinate systems: an inertial system (xyz) and a body-fixed
system (ξ ζ η) attached at the boom tip (xa , ya , za ). The Lagrangian of the cable-payload
assembly can be written as
L = T −V
1 = m (ẋa + ξ̇ )2 + (ẏa + ζ̇ )2 + ż2 − mgz,
2
(12)
where (ξ, ζ, η) is the position of the payload in the body-fixed coordinate system. The inextensiblity of the cable can be expressed as
(13)
z = za − (lo + lc )2 − ξ 2 − ζ 2 ,
where lc = δ t. Equation (13) can be used to eliminate z from the Lagrangian. Thus the
motion of the payload is described using the two degrees of freedom ξ and ζ . Then the Euler–
Lagrange equations yield the equations of motion in the ξ and ζ directions.
We order the excitations as follows: ẍa and ÿa ≤ O(ξ 3 , ζ 3 ). The masses are nondimensionalized with respect to the payload mass m, and the lengths are nondimensionalized with
respect to the original length lo of the
cable. Linear damping is added to the equations of
motion and ordered as µ ≤ O ξ 2 , ζ 2 . Expanding around the equilibrium configuration and
dropping the quartic and higher-order terms, we obtain
2
ω 2
2
2
2
2
(ξ + ζ ) + (ξ̇ + ζ̇ ) + (ξ ξ̈ + ζ ζ̈ ) = −ẍa ,
(14)
ξ̈ + ω ξ + 2 µξ̇ + ξ
2
2
ω 2
2
2
2
2
(ξ + ζ ) + (ξ̇ + ζ̇ ) + (ξ ξ̈ + ζ ζ̈ ) = −ÿa .
(15)
ζ̈ + ω ζ + 2 µζ̇ + ζ
2
Pendulation Reduction in Boom Cranes 261
We assume that both ẍa and ÿa are harmonic.
To examine the response of the payload to forcing near the resonance frequency, we set close to ω and express the nearness of the two frequencies by
= ω + 2 σ.
(16)
We seek a first-order uniform approximation to the solution of Equations (14) and (15) as
[28–30]
ξ ≈ u11 (T0 , T2 ) + 3 u13 (T0 , T2 ),
(17)
ζ ≈ u21 (T0 , T2 ) + 3 u23 (T0 , T2 ),
(18)
where To = t and T2 = 2 t. Substituting equations (17) and (18) into equations (14) and (15),
using equation (16), and equating coefficients of like powers of , we obtain the following
hierarchy of problems:
Order :
D02 u11 + ω2 u11 = 0,
(19)
D02 u21 + ω2 u21 = 0.
(20)
Order 3 :
D02 u13 + ω2 u13 = −2D0 D1 u11 − u11 (u11 D02 u11 + u21 D02 u21 )
− u11 [(D0 u11 )2 + (D0 u21 )2 ]
1
− ω2 u11 (u211 + u221 ) − 2µD0 u11 + Fx cos(t),
2
(21)
D02 u23 + ω2 u23 = −2D0 D1 u21 − u21 (u11 D02 u11 + u21 D02 u21 )
− u21 [(D0 u11 )2 + (D0 u21 )2 ]
1
− ω2 u21 (u211 + u221 ) − 2µD0 u21 + Fy cos(t),
2
(22)
where Dn ≡ ∂/∂Tn . We solve these equations sequentially.
The solution of the first-order problem, Equations (19) and (20), can be expressed as
u11 = A1 (T2 ) eiωT0 + cc,
(23)
u21 = A2 (T2 ) eiωT0 + cc,
(24)
where cc is the complex conjugate of the preceding terms. Substituting Equations (23) and
(24) into Equations (21) and (22) and eliminating the terms that produce secular terms, we
obtain the modulation equations
Fx iσ T2
ie
− µA1 −
2ω
Fy iσ T2
ie
− µA2 −
A2 = −
2ω
A1 = −
ω
A1 −
2ω
ω
A2 −
2ω
1
iωĀ1 (A21 + 3A22 ) +
4
1
iωĀ2 (3A21 + A22 ) +
4
1
iωA1 A2 Ā2 ,
2
1
iωA1 Ā1 A2 ,
2
(25)
(26)
262 E. M. Abdel-Rahman and A. H. Nayfeh
where the primes indicate the derivate with respect to T2 .
Next, we express the modulation equations in Cartesian form using the transformation
Ak =
1
[pk (T2 ) − iqk (T2 )] eiσ T2 ,
2
k = 1, 2.
(27)
Substituting Equations (27) into the modulation equations and separating real and imaginary
terms, we obtain
p1 = −σ q1 −
ω
1
p1 − µp1 − ωq1 (p12 + q12 + p22 + q22 )
2ω
16
3
− ωp2 (p1 q2 − p2 q1 ),
8
ω
1
q1 = σp1 −
q1 − µq1 + ωp1 (p12 + q12 + p22 + q22 )
2ω
16
Fx
3
,
− ωq2 (p1 q2 − p2 q1 ) +
8
ω
ω
1
p2 − µp2 − ωq2 (p12 + q12 + p22 + q22 )
p2 = −σ q2 −
2ω
16
3
+ ωp1 (p1 q2 − p2 q1 ),
8
ω
1
q2 − µq2 + ωp2 (p12 + q12 + p22 + q22 )
q2 = σp2 −
2ω
16
Fy
3
.
+ ωq1 (p1 q2 − p2 q1 ) +
8
ω
(28)
(29)
(30)
(31)
The amplitudesof the in-plane and out-of-plane motions can be expressed in terms of the pi
and qi as aξ = p12 + q12 and aζ = p22 + q22 , respectively.
3. Stationary Solutions and Their Stability
3.1. T WO -D IMENSIONAL M ODEL
We set the time derivatives in Equations (10) and (11) equal to zero. Eliminating β from the
resulting algebraic system, we obtain
a6 +
32σ 4 256 2
64
a + 2 (µ + σ 2 )a 2 − 4 Fx2 = 0
ω
ω
ω
(32)
which is a cubic equation in a 2 . To solve for a, we set the excitation frequency at =
0.601 rad/s, corresponding to the natural frequency for a nominal cable length of lo = 89 ft,
the excitation amplitude at Fx = 0.1, and the linear damping coefficient at µ = 0.01. Equation (32) is then solved for different values of σ and the resulting frequency-response curve is
shown in Figure 2. Superimposed on the curve are the results of long-time integration of the
exact equation of motion, Equation (1), for different cable lengths. Examining the eigenvalues
Pendulation Reduction in Boom Cranes 263
Figure 2. The stationary frequency-response curve obtained for = 0.601 rad/s, Fx = 0.1, and µ = 0.01.
The solid lines denote stable equilibrium solutions and the dashed line denotes unstable equilibrium solutions.
Triangles denote equilibrium solutions obtained from long-time integration of the exact equation of motion.
of the Jacobian matrix of Equations (10) and (11), we find that the equilibrium solutions are
unstable when
4ω2 a(µ2 + σ 2 ) + σ ω3a 3 +
3 4 5
ω a < 0.
64
(33)
We ran a series of simulations by integrating the modulation equations (10) and (11) and
obtained nonstationary responses corresponding to reeling (negative δ) and unreeling (positive
δ) of the cable subsequent to a near-resonance excitation for an initial cable length of lo =
87 ft. As the cable is unreeled, a decreases as the cable length is increased, thereby moving
down along the frequency-response curve. At low unreeling speeds, a slides down smoothly
along the frequency-response curve. However as δ increases, a initially oscillates around the
frequency-response curve before settling on it. On the other hand, as the cable is slowly reeled
(i.e., the cable length is decreased), a increases, thereby climbing up along the frequencyresponse curve until it reaches the peak. As l decreases further, the response amplitude jumps
down to the lower branch. As the reeling speed is increased, a progressively undershoots the
upper branch and lingers without falling onto the lower branch. Figure 3 shows a typical curve
for the reeling of the cable; it exhibits the lingering phenomenon.
3.2. T HREE -D IMENSIONAL M ODEL
We set the time derivatives in Equations (28–31) equal to zero. To solve for aξ and aζ , we set
= 0.601 rad/s, Fx = 0.0036, and µ = 0.0016. We then solve the resulting algebraic system
for different values of σ . Two sets of solutions are possible: one-mode solutions involving
only in-plane motions (i.e., aζ = 0) and two-mode solutions involving both in-plane and
out-of-plane motions.
264 E. M. Abdel-Rahman and A. H. Nayfeh
Figure 3. The influence of reeling the cable when lo = 87 ft and δ = −1.5 ft/s. It exhibits the lingering
phenomenon.
Figure 4. The stationary frequency-response curve obtained for the one-mode response when = 0.601 rad/s,
Fx = 0.1, and µ = 0.01. The solid lines denote stable equilibrium solutions and the dashed lines denote unstable
equilibrium solutions.
The stationary frequency-response curve representing one-mode solutions is shown in Figure 4. The response loses stability within a narrow band, between 85 and 92 ft, around a cable
length of 89 ft, where is equal to the natural frequency of the cable-payload assembly. In
this region, only two-mode solutions are available. Any excitation of the system at a frequency
within this band will produce in-plane and out-of-plane motions of the payload.
Pendulation Reduction in Boom Cranes 265
Figure 5. The in-plane (a) and out-of-plane (b) stationary frequency-response curves obtained when
= 0.601 rad/s, Fx = 0.1, and µ = 0.01. The solid lines denote stable equilibrium solutions and the dashed
lines denote unstable equilibrium solutions.
The stationary frequency-response curves corresponding to two-mode solutions are shown
in Figure 5. Stable solutions exist only at cable lengths longer than 89 ft, where is equal to
the natural frequency of the cable-payload assembly. In this region the payload moves both
in and out of plane, exhibiting a whirling motion. Any out-of-plane disturbance triggers this
whirling motion.
We ran a series of simulations by integrating the modulation equations (28–31) and obtained nonstationary responses corresponding to reeling and unreeling of the cable starting
with a stable two-mode response (whirling motion) for an initial cable length of lo = 92 ft.
As the cable is unreeled to 110 ft, the amplitude aξ of the in-plane motion homes onto the
lower branch of the one-mode response, while the amplitude aζ of the out-of-plane motion
vanishes gradually. Also, as the cable is reeled to 70 ft, aξ homes onto the lower branch of
the one-mode response, while aζ vanishes gradually. Figure 6 shows a typical curve for the
reeling of the cable.
Figures 7–9 show the actual time histories of some of these simulations. The results indicate that, at the same cable length, the transient dynamics depend on whether the cable is
being reeled or unreeled and on the speed of the reeling. As the cable is unreeled, Figure 7,
the response amplitudes increase first before they begin to decay exponentially. On the other
hand, as the cable is reeled, Figures 8 and 9, the response amplitudes decrease, then increase,
and finally decay exponentially. The intervals and magnitudes over which these changes in the
amplitude occur depend on the reeling and unreeling speeds. As the unreeling speed increases,
the intervals of the initial fluctuations of the amplitudes decrease. Also the magnitude of the
initial spike in the response amplitude increases with increasing reeling speed. In all cases,
the stable solutions obtained from long-time integration home onto the same points on the
one-mode frequency-response curve, regardless of the reeling/unreeling speed.
266 E. M. Abdel-Rahman and A. H. Nayfeh
Figure 6. The in-plane (a) and out-of-plane (b) responses obtained by reeling the cable starting from a length of
lo = 92 ft at a speed of δ = −0.5 ft/s.
(a)
(b)
Figure 7. Time histories of the in-plane (a) and out-of-plane (b) responses obtained by unreeling the cable from a
length of lo = 93.6 to l = 110 ft at a speed of δ = 0.5 ft/s.
4. Summary
Two-dimensional and three-dimensional models of a boom crane are presented. Comparison
of the results of the two models shows that the two-dimensional model cannot predict the
stability of the stationary response and produces qualitatively misleading transient results.
The lingering phenomenon exhibited by the two-dimensional model does not exist in the
three-dimensional model.
Pendulation Reduction in Boom Cranes 267
(a)
(b)
Figure 8. Time histories of the in-plane (a) and out-of-plane (b) responses obtained by reeling the cable from a
length of lo = 93.6 ft to l = 70 ft at a speed of δ = −0.1 ft/s.
(a)
(b)
Figure 9. Time histories of the in-plane (a) and out-of-plane (b) responses obtained by reeling the cable from a
length of lo = 93.6 ft to l = 70 ft at a speed of δ = −1.0 ft/s.
The present results show that both reeling and unreeling of the cable can be used to suppress the whirling motion and in-plane pendulations of the payload. Thus, manipulation of the
cable length is an effective technique for controlling both in-plane pendulations and whirling
motions of the payload. Rather than damping out the motion, the reeling/unreeling of the cable
changes the underlying dynamics of the system, thereby rendering it easier to control. Even
though no damping was added to the system beyond the light linear damping naturally present,
this technique reduces significantly the payload pendulations. However, the speed of the cable
reeling/unreeling has to be appropriately selected to guarantee both quick and smooth motion
268 E. M. Abdel-Rahman and A. H. Nayfeh
reduction. In summary, cable-length manipulation is an effective building block in an overall
control strategy for eliminating payload pendulations.
Acknowledgment
This work was supported by the Office of Naval Research under Grant No. N00014-96-1-1123
(MURI).
References
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
17.
Bostelman, R. and Goodwin, K., Survey of Cargo Handling Research, Intelligent Systems Division, NIST,
1998.
Sakawa, Y., Shindo, Y., and Hashimoto, Y., ‘Optimal control of a rotary crane’, Journal of Optimization
Theory and Applications 35, 1981, 535–557.
Sakawa, Y. and Nakazumi, A., ‘Modeling and control of a rotary crane’, ASME Journal of Dynamic Systems,
Measurement, and Control 107, 1985, 200–206.
Sato, K. and Sakawa, Y., ‘Modelling and control of a flexible rotary crane’, International Journal of Control
48, 1988, 2085–2105.
Takeuchi, S., Fujikawa, H., and Yamada, S., ‘The application of fuzzy theory for a rotary crane control’, in
Proceedings of the International Conference on Industrial Electronics: IECON’88, Singapore, Vol. 2, IEEE,
New York, 1988, pp. 415–420.
Yamada, S., Fujikawa, H., and Matsumoto, K., ‘Suboptimal control of the roof crane by using the microcomputer’, in Proceedings of the Conference on Industrial Electronics: IECON’83, San Francisco, CA, IEEE,
New York, 1983, pp. 323–328.
Hara, K., Yamamoto, T., Kobayashi, A., and Okamoto, M., ‘Jib crane control to suppress load swing’,
International Journal of Systems Science 20, 1989, 715–731.
Nguyen, H. T., Jackson, V., and Irvine, M., ‘Nonlinear digital controller for an off-shore crane’, in Proceedings of the IEEE International Symposium on Industrial Electronics, Xian, China, Vol. 1, IEEE, New York,
1992, pp. 88–92.
Souissi, R. and Koivo, A. J., ‘Modeling and control of a rotary crane for swing-free transport of payloads’,
in Proceedings of the 1st IEEE Conference on Control Applications, Dayton, OH, Vol. 2, IEEE, New York,
1992, pp. 782–787.
Gustafsson, T., ‘Modelling and control of a rotary crane’, in Proceedings of 3rd European Control
Conference, Rome, Italy, A. Isidori, S. Bittanti, and E. Mosca (eds.), EUCA, 1995, pp. 3805–3810.
Lewis, D., Parker, G., Driessen, B., and Robinett, R., ‘Command shaping control of an operator-in-the-loop
boom crane’, in Proceedings of the American Control Conference, Philadelphia, PA, IEEE, New York, 1998,
pp. 2643–2647.
Parker, G., Groom, K., Hurtado, J., Feddema, J., Robinett, R., and Leban, F., ‘Experimental verification of
a command shaping boom crane control system’, in Proceedings of the American Control Conference, San
Diego, CA, IEEE, New York, 1999, pp. 86–90.
Parker, G. G., Groom, K., Hurtado, J., Robinett, R. D., and Leban, F., ‘Command shaping boom crane control
system with nonlinear inputs’, in Proceedings of the IEEE International Conference on Control Applications,
Kohala Coast, HI, Vol. 2, IEEE, New York, 1999, pp. 1774–1778.
Lewis, D., Parker, G. G., Driessen, B., and Robinett, R. D., ‘Comparison of command shaping controllers
for suppressing payload sway in a rotary boom crane’, in Proceedings of the IEEE International Conference
on Control Applications, Kohal Coast, HI, Vol. 1, IEEE, New York, 1999, pp. 719–724.
Patel, M., Brown, D., and Witz, J., ‘Operability analysis for a monohull crane vessel’, Transactions of the
Royal Institution of Naval Architects 129, 1987, 103–113.
McCormick, F. and Witz, J., ‘An investigation into the parametric excitation of suspended loads during crane
vessel operations’, Underwater Technology 19, 1993, 30–39.
Witz, J., ‘Parametric excitation of crane loads in moderate sea states’, Ocean Engineering 22, 1995, 411–420.
Pendulation Reduction in Boom Cranes 269
18.
19.
20.
21.
22.
23.
24.
25.
26.
27.
28.
29.
30.
Chin, C., Nayfeh, A. H., and Mook, D. T., ‘Dynamics and control of ship-mounted cranes’, in Proceedings
of the 39th Structures, Structural Dynamics, and Materials Conference, Long Beach, CA, AIAA-98-1731,
1998.
Iwasaki, I., Tanida, K., Kaji, S., and Mutaguchi, M., ‘Development of an active mass damper for stabilizing
the load suspended on a floating crane’, in DETC’97 Proceedings of the ASME Design Engineering Technical
Conference, Sacramento, CA, DETC97/VIB-3816, ASME, New York, 1997.
Imazeki, M., Mutaguchi, M., Iwasaki, I., and Tanida, K., ‘Active mass damper for stabilizing the load
suspended on floating crane’, IHI Engineering Review 31, 1998, 60–69.
Balachandran, B. and Li, Y., ‘A mechanical filter concept to suppress crane load oscillations’, in DETC’97
Proceedings of the ASME Design Engineering Technical Conferences, Sacramento, CA, DETC97/VIB-4091,
ASME, New York, 1997.
Balachandran, B., Li, Y., and Fang, C., ‘A Mechanical filter concept for control of non-linear crane-load
oscillations’, Journal of Sound and Vibration 228, 1999, 651–682.
Li, Y. and Balachandran, B., ‘Mechanical filters for control of crane-load oscillations’, in MURI on Nonlinear
Active Control of Dynamical Systems, October, 1999.
Henry, R., Masoud, Z., Nayfeh, A., and Mook, D., ‘Cargo pendulation reduction on ship-mounted cranes via
boom-luff angle actuation’, Journal of Vibration and Control 7, 2001, 1253–1264.
Masoud, Z., Nayfeh, A., Henry, R., and Mook, D., ‘Cargo pendulation reduction on ship-mounted cranes
via boom-luff and slew angles actuation’, in Proceedings of the 41th Structures, Structural Dynamics, and
Materials Conference, Atlanta, GA, AIAA-2000-1543, 2000.
Kral, R. and Kreuzer, E., ‘Dynamics of crane ships’, in Proceedings of the International Conference on
Applied Dynamics, Hanoi, Vietnam, 1995, pp. 49–55.
Kral, R., Kreuzer, E., and Wilmers, C., ‘Nonlinear oscillations of a crane ship’, Zeitschrift für Angewandte
Mathematik und Mechanik 76-S4, 1996, 5–8.
Nayfeh, A. H., Introduction to Perturbation Techniques, Wiley, New York, 1981.
Nayfeh, A. H. and Mook, D. T., Nonlinear Oscillations, Wiley, New York, 1979.
Nayfeh, A. H., Nonlinear Interactions, Wiley, New York, 2000.