Effect of Hoisting Cable Elasticity on the Period of

Oscillation of Quay-Side Container Cranes

Ziyad N. Masoud∗

Oscillation frequency of crane payloads is the main and most important factor in craneanti-sway control systems design. In the summer of 2005, a Smart Sway Control system(SSC) was developed and installed on a 65-ton quay-side container crane at Jeddah Port.During the calibration phase, it was observed that heavy payloads combined with thedynamic stretch of the hoist cables had a significant impact on the configuration of thehoisting mechanism and the pattern of oscillation. This introduced considerable changein the oscillation frequency of the payload, which resulted in a significant impact on theperformance of the anti-sway control system. Empirical formulas were used to compensatefor the change in the frequency approximation used in the controller algorithm. In thiswork, an analytic approximation of the oscillation frequency of the hoisting mechanism ofa quay-side container crane is developed, which takes into consideration the elasticity ofthe hoisting cables. A parametric study is performed to investigate the extent of the effectof the hoisting cables stretch on the system behavior for a typical range of payload massesand cable lengths.

I. Introduction

Since the introduction of Super-Post-Panamax container ships, demand on larger and faster quay-sidecontainer cranes became an essential requirement. A Super-Post-Panamax ship has a capacity of up to 12,000containers and up to 22 containers across the width of the ship. The large span of quay-side containers cranesthat handle such ships necessitates higher trolley speeds and accelerations, figure 1. Operating such cranesdemands special skills and long experience. Accidents during the operations of these cranes are usuallysevere and costly. These accidents may result is a severe damage to the ship, its payload, and sometimesto the crane itself. Consequently, more stringent motion suppression requirements are the norm rather thanthe exception.

The last four decades have seen mounting research interest in the modeling and control of cranes.1

Container cranes are traditionally modeled as a two-dimensional simple pendulum with a lumped mass atthe end of a rigid inextensible massless link. However, the hoisting mechanism of a quay-side container craneis significantly different. The actual hoisting mechanism of a quay-side container crane consists typically ofa multi-cable hoisting arrangement. The cables are hoisted using a number of hoisting cables dropping fromfour pulleys on a trolley to four pulleys on a spreader bar used to lift containers. An anti-sway controllerdesign based on the actual model of a quay-side container crane is most likely to result in a performancesuperior to those based a simple pendulum model.

Input-shaping is one of the most widely used open-loop control strategies for quay-side container cranes.Controllers using various forms of input-shaping are incorporated into cranes currently used in ports.2 Apartfrom command filtering, these techniques are used to move the trolley of the crane a preset distance alonga preset path. These controllers have also been used for inching maneuvers in tight work spaces and neartarget points. The acceleration profile of the trolley is designed to induce minimum payload oscillationduring travel and to deliver the payload at the target point free of residual oscillations.

However, input-shaping techniques are limited by the fact that they are sensitive to variations in the pa-rameter values about the nominal values, and to changes in the initial conditions and external disturbances,and that they require “highly accurate values of the system parameters” to achieve satisfactory system per-formance.3–5 Singhose et al.6 developed four input-shaping controllers. Simulations of their best controller

∗Assistant Professor, Department of Mechanical Engineering, The Hashemite University, Zarqa 13115, Jordan.

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49th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference <br> 16t7 - 10 April 2008, Schaumburg, IL

AIAA 2008-2269

Copyright © 2008 by Ziyad Masoud. Published by the American Institute of Aeronautics and Astronautics, Inc., with permission.

Figure 1. Typical quay-side container crane.

produced a reduction of 73% in transient oscillations over the time-optimal rigid-body commands. However,they reported that “transient deflection with shaping increases with hoist distance, but not as severely as theresidual oscillations”. There simulations showed that the percentage in reduction with shaping is dependenton system parameters. As a result, these controllers suffered significant degradation in crane maneuvers thatinvolved large hoisting.

Kress et al.7 showed analytically that input-shaping is equivalent to a notch filter applied to a generalinput signal and centered around the natural frequency of the payload. Based on this conclusion, theyproposed a Robust Notch Filter, a second-order notch filter, applied to the acceleration commands. Numericalsimulation and experimental verification of this strategy on an actual bidirectional crane, moving at arbitrarystep accelerations and changing cable length at a slow constant rate, showed that the strategy was able tosuppress residual payload oscillation. Parker et al.8 developed a command shaping notch filter to reducepayload oscillation on rotary cranes excited by the operator commands. They reported that in general, therewas no guarantee that applying such a filter to the operator’s speed commands would result in excitationterms having the desired frequency content, and that it only works for low speed and acceleration commands.Parker et al.9 experimentally verified their simulation results.

To improve the performance of input-shaping controllers, research shifted towards developing betterfrequency approximations. Dadone and VanLandingham10 derived a nonlinear approximation of the payloadoscillation period using the method of Multiple Scales. Using numerical simulation, they compared theresidual oscillations due to single-step input-shaping strategies based on their approximation, a simplifiedform of that approximation, and the linear approximation of the oscillation period. They obtained significantenhancement of as much as two orders of magnitude in the performance using their approximation overthe linear approximation. The enhanced performance of the nonlinear frequency approximation was mostpronounced for longer coasting distances and higher accelerations.

Agostini et al.11 used an optimal control scheme to demonstrate that for open-loop controllers, neglectingsystem parameters such as joint friction and actuator saturation leads to large residual oscillations. Lewiset al.12 neglected the lift-line and boom dynamic coupling in ship-mounted boom cranes. As a result,significant residual oscillations were observed.

While closed-loop control may be used to alleviate these problems in input-shaping techniques, it cannot be used with time-optimal control techniques because it can lead to the development of limit cycles.13

Further, the use of closed-loop control in conjunction with either approach requires a very accurate plantmodel and cannot therefore offer significant improvement over open-loop control.14

Linear control strategies are invariably tuned to counter the effects of the natural frequency of thepayload at a single cable length. As a result, they are most sensitive to changes in the hoisting cablelength. Therefore, linear control imposes restrictions on raising and lowering the payload during motion andrequires low operating speeds, and thus imposing unrealistic constraints on crane operations. Burg et al.15

reported that the neglected nonlinearities in a state-space model of a gantry crane may significantly impact

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the performance of a linear controller. Their computer simulations show that a linear controller providesacceptable performance only within a fixed operating range of small oscillation angles around the equilibriumpoint of the payload.

In an earlier work, Masoud et al.16 developed a two-dimensional model of a quay-side container cranewhich included the multi-cable hoisting mechanism of the crane. The model was used to develop a delayedposition-feedback control system for sway reduction on quay-side container cranes. This control techniquedemonstrated reduced sensitivity to the natural frequency of the payload. Based on this control strategy,a Smart Sway Control system (SSC) was developed and installed on a 65-ton quay-side container cranein Jeddah Port (Kingdom of Saudi Arabia) in the summer of 2005. During the calibration phase of theinstallation, it was observed that heavy payloads combined with the dynamic stretch of the hoist cables hada significant impact on the geometrical configuration of the hoisting mechanism, and hence, the pattern ofoscillation. This introduced considerable variation in the oscillation frequency of the payload resulting in asignificant impact on the performance of the SSC anti-sway control system. Empirical formulas had to beused to compensate for the change in the frequency approximation.

In the inextensible hoisting cables models, the mass of the payload drops out of the equations of motion,which prevents predicting its effect on the oscillation frequency. In the distributed mass approach, thehoisting cable is modeled as a distributed-mass cable, and the hook and payload, lumped as a point mass,are applied as a boundary condition to this distributed-mass system. The only model available in thiscategory is the planar model of d’Andrea-Novel et al.17,19 and d’Andrea-Novel and Boustany18 for a gantrycrane linearized around the cable’s equilibrium position. They ignored the inertia of the payload and modeledthe cable as a perfectly flexible, but inextensible body using the wave equation.

In this work, an analytic approximation of the oscillation frequency of the hoisting mechanism of a quay-side container crane, which takes into consideration the elasticity of the hoisting cables is developed. Aparametric study is performed to investigate the extent of the effect of the hoisting cables stretch on thesystem behavior for a typical range of payload masses and cable lengths. This improved approximation of theoscillation frequency should enhance the performance of control systems that suffer significant dependence onthe oscillation frequency, especially for such real-world application where many uncertainties are encountered.

II. Mathematical Modeling

A geometrically accurate model of the quay-side container crane was developed earlier by Masoud et al.16

The hoisting cables were modeled as rigid massless links. An approximation of the frequency of oscillationsof the inextensible hoisting cables model was derived as

ω2 =lo + a2R

(lo − aR)2g (1)

Their frequency approximation resulted in an outstanding controller performance. However, the mass ofthe payload used in their experiments was not large enough to produce any noticeable stretch in the hoistingcables due to the high cable stiffness to payload mass ratio, which was found to be much higher than thatof the actual cranes.

In the following analysis, a model with a similar geometry is developed, which includes the effect of thedynamic stretch of the hoisting cables on the frequency of oscillations.

A. Generalized Constraint Forces

To develop an analytic expression for the oscillation frequency, the concept of virtual work and generalizedforce of a constrained multi-body system is used to derive the equations of motion.20 For this purpose, aplanar model of two constrained rigid bodies is considered, figure 2.

The general equation of motion of a rigid body under the action of generalized forces can be expressedas

Mq = Q (2)

where M is the inertia matrix, q is the generalized coordinates vector, and Q in the generalized force vector.The virtual work of the constraint force Q between point A on body i and point B on body j is defined

asδW = −Qδdij (3)

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where dij is the distance between points A and B. The vector dij from point A to point B is

dij = rBj − rA

i (4)

The position vectors of points A and B can be written as

rAi = ri + AisA

i (5)

rBj = rj + AjsB

j (6)

where ri and rj are position vectors to reference points fixed on the rigid bodies i and j. To simplify thederivation of the equations of motion of rigid bodies, these points are usually selected to be the centers ofmasses of these bodies. sA

i and sBj are the position vectors of points A and B in the corresponding body

fixed coordinate systems. The rotational transformation matrices Ai and Aj are

Ai =

[cos(φi) − sin(φi)sin(φi) cos(φi)

](7)

Aj =

[cos(φj) − sin(φj)sin(φj) cos(φj)

](8)

Substituting Eqs. (5) and (6) into Eq. (4), the vector dij becomes

dij = rj + AjsBj − ri −AisA

i (9)

The distance dij can be written as a dot product in terms of the vector dij as

d2ij = dT

ijdij (10)

Assuming that the spatial variations and differentiations are interchangeable, the variation in dij can beobtained from Eq. (10) as

2dijδdij = 2dTijδdij (11)

The virtual change δdij is obtained from Eq. (9) as

δdij = δrj + BjsBj δφj − δri −BisA

i δφi (12)

where

Bi =

[− sin(φi) − cos(φi)cos(φi) − sin(φi)

](13)

Figure 2. Planar rigid body model.

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Bj =

[− sin(φj) − cos(φj)cos(φj) − sin(φj)

](14)

Using Eqs. (11) and (12), The virtual change in the distance dij can be written as

δdij =(dij

dij

)T

(δrj + BjsBj δφj − δri −BisB

i δφi) (15)

Substituting Eq. (15) into Eq. (3), the virtual work of the constraint forces becomes

δW = − Q

dijdT

ij(δrj + BjsBj δφj − δri −BisB

i δφi) (16)

By definition, the generalized forces acting on both bodies are simply the coefficients of variations in thegeneralized coordinates that define the position and orientation of each body. This leads to a generalizedforce vector on the body i as

Qi =Q

dij

[dij

dTijBisB

i

](17)

B. Full Nonlinear Model

In this model, the hoisting mechanism of the crane is modeled by four cables, each running from the centerof a pulley on the trolley to the center of the corresponding pulley on the spreader bar. The spreader barand the container are lumped into one rigid body. Because of the planar nature of the transfer maneuversperformed by the crane, a two dimensional model is used, figure 3. To model their elasticity, the hoistingcables are modeled as two linear springs with an unstretched length of lo.

Following the analysis in the previous section, and to determine the generalized force acting on thecontainer due to the hoisting cable AC, the container will be treated as body i and the trolley will be treatedas body j. The [x, y, θ]T will be used as a generalized coordinates vector. The inertial position vectors ofthe two bodies are

ri = [x, y]T (18)

rj = [u, h]T (19)

where h is the height of the trolley tracks. The position vectors of points C and A in their correspondingbody fixed coordinate systems are

sCi = [−w/2, R]T (20)

sAj = [−d/2, 0]T (21)

The rotational angles φi of the payload is represented by the angle θ while the angle φj of the trolley isset to zero.

Substituting the above position vectors into Eq. (9), we get

dij =[u− d

2− x +

w

2cos(θ) + sin(θ)R, h− y +

w

2sin(θ)− cos(θ)R

]T

(22)

The linear spring force in the cable AC is

Q1 = k(l1 − lo) (23)

Hence, the generalized force vector acting on the container due to the cable AC is obtained by substitutingEqs. (18) - (23) into Eq. (17) as

Q1 =k(l1 − lo)

l1

u−d

2− x +

w

2cos(θ) + sin(θ)R

h− y +w

2sin(θ)− cos(θ)R

(u− d

2− x +

w

2cos(θ) + sin(θ)R)(

w

2sin(θ)− cos(θ)R)

−(h− y +w

2sin(θ)− cos(θ)R)(

w

2cos(θ) + sin(θ)R)

(24)

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Figure 3. Planar rigid body model.

Similarly, the generalized force vector acting on the container due to the cable BD can be derived as

Q2 =k(l2 − lo)

l2

u+d

2− x− w

2cos(θ) + sin(θ)R

h− y − w

2sin(θ)− cos(θ)R

(u +d

2− x− w

2cos(θ) + sin(θ)R)(−w

2sin(θ)− cos(θ)R)

+(h− y − w

2sin(θ)− cos(θ)R)(

w

2cos(θ)− sin(θ)R)

(25)

An additional generalized force vector acting on the container is the vector that includes the force dueto gravity, which can be written as

Qg = m

0−g

0

(26)

The final nonlinear equations of motion of the container become

Mq = Q1 + Q2 + Qg (27)

where the inertia matrix M is

M =

m 0 00 m 00 0 J

The nonlinear equations of motion of the container Eq. (27) will be used throughout the followingnumerical simulations of the behavior of the crane payload oscillations.

C. Modified Model

Extracting an expression for the oscillation frequencies of the container from Eq. (27) is a complex nontrivialprocess. A simpler approach is to separate the inertia and the stiffness terms of the equations of motion. Onceinertia and stiffness matrices are obtained, frequency determination becomes trivial. To do so, a simplifiedlinear model with an alternative set of generalized coordinates is used to derive an analytic approximation

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of the oscillation frequency of the container. The new set of generalized coordinates used in the proceedinganalysis consists of the stretch δ in an imaginary center line l connecting the midpoint of the trolley E to themidpoint on the top of the spreader bar F , the oscillation angle of this imaginary line φ, and the rotationangle of the container θ.

To project the system representation onto the new set of generalized coordinates, geometrical coordinatetransformations is used as follows. First, the closing constraint equations of the loop AEFC are

l1 sin(φ1) = l sin(φ)− w

2cos(θ) +

d

2(28)

l1 cos(φ1) = l cos(φ) +w

2sin(θ) (29)

and the closing constraint equations of the loop EBDF are

l2 sin(φ2) = l sin(φ) +w

2cos(θ)− d

2(30)

l2 cos(φ2) = l cos(φ)− w

2sin(θ) (31)

To eliminate the variables φ1 and φ2, Eqs. (28) and (29) are squared and added together, and Eqs. (30)and (31) are also squared and added together. The following equations are obtained

l21 =(

l sin(φ)− w

2cos(θ) +

d

2

)2

+(l cos(φ) +

w

2sin(θ)

)2

(32)

l22 =(

l sin(φ) +w

2cos(θ)− d

2

)2

+(l cos(φ)− w

2sin(θ)

)2

(33)

The length of the hoisting cables and the imaginary center line can be written in terms of the unstretchlengths of the cables and the cables stretch as

l1 = lo + δ1

l2 = lo + δ2 (34)l = lo + δ

To simplify the equations in the following analysis, the following parameter is used, which is defined as

a =d− w

w(35)

Substituting Eqs. (34) into Eqs. (32) and (33), and assuming small oscillation angles φ and θ, a linearapproximation for the cables stretch δ1 and δ2 can be obtained as

δ1 =w2a2

8lo+

w

2(aφ + θ) + δ (36)

δ2 =w2a2

8lo− w

2(aφ + θ) + δ (37)

The equations of motion of the new model are derived using Lagrange’s equations. To determine thekinetic and potential energy of the container, the position vector to the center of mass of the container isused as

x = u + l sin(φ) + R sin(θ) (38)y = h− l cos(φ)−R cos(θ) (39)

The kinetic energy of the container is

T =12m(x2 + y2) +

12Jθ2 (40)

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Substituting Eqs. (36) - (39) into Eq. (40), the kinetic energy in terms of the new generalized coordinatesbecomes

T =12m

[(δ sin(φ) +

(lo + δ cos(φ)

)φ + R cos(θ)θ

)2

+(δ cos(φ)− (

lo + δ sin(φ))φ−R sin(θ)θ

)2]

+12Jθ2 (41)

The potential energy includes both gravitational potential energy and elastic potential energy in the twocables, which can be written as

V = mgy +12k

(δ21 + δ2

2

)(42)

Substituting Eqs. (36) - (39) into Eq. (42), the potential energy in terms of the new generalized coordinatesbecomes

V = mg(h− (lo + δ) cos(φ)−R cos(θ)

)

+12k[(w2a2

8lo+

w

2(aφ + θ) + δ

)2

+(w2a2

8lo− w

2(aφ + θ) + δ

)2](43)

Substituting the kinetic energy Eq. (41) and the potential energy Eq. (43) equations into Lagrange’sequations results in a nonlinear system of equations of motion. A linearized version of the resulting equationsof motion can be written as

m 0 00 ml2o mRlo

0 mRlo mR2 + J

δ

φ

θ

+

2k 0 00 mglo + 1

2kw2a2 12kw2a

0 12kw2a mgR + 1

2kw2

δ

φ

θ

=

mg − 14kw2a2/lo

00

(44)

The nonzero term in the right-hand-side vector can be eliminating by considering the oscillation ofδ around its equilibrium position. The frequencies of oscillations can now be determined by solving thefollowing eigenvalue problem

| − λM + K| = 0 (45)

The resulting frequency associated with the δ generalized coordinate is obtained as

λδ =2k

m(46)

which is in agreement with a mass-spring system in free vibration. Since payload lateral positioning is themost important factor in quay-side crane operations, the more important frequency is the lateral oscillationfrequency of the container associated with the oscillation angle φ, which is obtained as

λφ =1

4mJl2o

{2loR(lo + R)gm2 +

(2logJ + (lo − aR)2w2k

)m + kw2a2J

−[(

2loR(lo + R)g)2

m4 + 4loR((lo + R)(lo − aR)2kw2 − 2lo(lo −R)gJ

)gm3

+(4lo

(logJ2 − (l2o − 2a2R2 + 2aRlo + a2Rlo)kw2J

)g + (lo − aR)4k2w4

)m2

+ 2(2loJg + (lo − aR)2kw2

)ka2w2Jm + k2a4w4J2

] 12}

(47)

Equation (47) shows significant dependence on the mass of the payload and the stiffness of the hoistingcables. These two parameters were not factors in the determination of the oscillation frequency of theinextensible model Eq. (1).

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III. Numerical Validation

To obtain the frequencies of the payload oscillations of the nonlinear model Eq. (27), the trolley isgiven a short square acceleration input. Using long time integration, FFT plots of the payload response aregenerated. Typical crane parameters were used for the simulations (d = 2.85 m, w = 1.83 m, and R = 3.5 m).The stiffness of the hoisting cables was approximated by k = 106 N/m. Figure 4 shows sample simulations ata cable length of 25 m and a combined payload/spreader bar mass of 30 tons and 80 tons. Figure 4(a) showsa clear peek at 0.102 Hz which is the frequency associated with the oscillation frequency of the φ degree offreedom, which is the most important for sway controller’s design. The other peek at approximately 1.14 Hzcoincides with the frequency associated with the δ1 and δ2 degrees of freedom. Similarly, figure 4(b) showsa peek at 0.1 Hz associated with the φ degree of freedom. Several other simulations are performed to obtainoscillation frequencies for different loading conditions and different hoisting cables length. The data obtainedis used to validate the accuracy of the new frequency approximation Eq. (47).

On the other hand, the frequency associated with the extension degrees of freedom show considerabledeviation from the approximation obtained in Eq. (46). This can be attributed to the fact that the derivedapproximation agrees with a spring-mass model in axial motion. Therefore, the sway angles in the simula-tions result in a significant deviation from the approximated behavior for the extension degree-of-freedom.However, the frequency approximation of the payload sway was not affected by the linearization of the cranemodel. This is due to the fact that the slow dynamics of the payload sway justifies excluding the nonlinearterms in the model equations, which on the other hand, is not valid for the case of the extension degrees-of-freedom due to their fast dynamics when compared to the order of magnitude of hoisting cables extensions.And since the extension frequency is generally not a factor in the sway controllers’ design, this frequencyapproximation can be neglected.

Figure 5 shows the period of oscillation for a combined container/spreader bar load of 50 tons. Cablelengths varying from 15 m to 35 m are used to compare the approximate frequency calculated using Eq. (47)and the inextensible frequency approximation Eq. (1) with the simulated frequency obtained by long timeintegration of the full nonlinear equations of motion Eq. (27). The results in the figure demonstrate excellentagreement between the elastic frequency approximation Eq. (47) and the simulated frequency. The figurealso shows the significant deviation of the inextensible model frequency from the true simulated frequency.

The effect of the payload mass on the oscillation frequency is demonstrated in figure 6. The figure showsthat the payload mass has a moderate effect on the oscillation frequency demonstrated by the small averageslope of the simulated and elastic periods of oscillation. However, the effect of the cable elasticity has thehigher significance demonstrated by the large shift between the simulated and the approximated periods ofthe flexible cables model, and the approximated period of the inextensible cables model.

0 0.2 0.4 0.6 0.8 1 1.20

50

100

150

200

250

300

350

400

Frequency (Hz)

(a) 30 ton payload.

0 0.2 0.4 0.6 0.8 1 1.20

100

200

300

400

500

600

700

Frequency (Hz)

(b) 80 ton payload.

Figure 4. An FFT plot of the container oscillations with a cable length of 25 m

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15 20 25 30 355

6

7

8

9

10

11

12

Cable length [m]

Peri

od [

s]Simulated responseElastic cable approximationInextensible cable approximation

Figure 5. Oscillation period for a combine payload of 50 ton.

IV. Concluding Remarks

The frequency of lateral payload oscillations is the most important factor in most of the anti-sway controlsystems design. The accuracy of the frequency approximation has a major impact on the performanceand efficiency of these control systems. Errors in the frequency approximation can result in a significantdegradation of the performance of feedback control systems, and can lead to complete system failure in thecase of open-loop control systems such as input-shaping and optimal control systems.

It is shown in this paper that for large quay-side container cranes, the frequency of the payload oscillationsis significantly different from the frequency approximations derived from inextensible cable models. Theanalytic form of the frequency of oscillation derived in this work shows significant dependence on the stiffnessof the hoisting cables and the payload mass.

The dependance of this frequency approximation on the mass of the crane payload will not pose a problemin real application and will not require additional hardware for control systems since this information isgenerally available and accessible from the PLC on quay-side container cranes.

An improved approximation of the oscillation frequency would enhance the performance of control systemsthat suffer significant dependence on the oscillation frequency especially for such real-world application where

20 30 40 50 60 70 808

8.5

9

9.5

10

10.5

11

11.5

12

Payload mass [ton]

Peri

od [

s]

Simulated responseElastic cable approximationInextensible cable approximation

Figure 6. Effect of payload mass on the oscillation period for a cable length of 30 m.

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many uncertainties are encountered. We believe that this new frequency approximation will contribute to asignificant improvement in the performance of many existing anti-sway control systems.

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