Jtam-A4.dvi


JOURNAL OF THEORETICAL

AND APPLIED MECHANICS

50, 4, pp. 953-966, Warsaw 2012
50th Anniversary of JTAM

THE INFLUENCE OF A SHOCK ABSORBER ON DYNAMICS OF

AN OFFSHORE PEDESTAL CRANE

Jerzy Krukowski

National Oilwell Poland, Gdańsk, Poland

e-mail: jerzy.krukowski@nov.com

Andrzej Maczyński, Marek Szczotka

University of Bielsko-Biała, Faculty of Management and Computer Science, Bielsko-Biała, Poland

e-mail: amaczynski@ath.bielsko.pl; mszczotka@ath.bielsko.pl

Offshore pedestal cranes are devices installed on offshore platforms or vessels. A characteri-
stic feature of any floating object is the significant movement caused by sea waving. These
movements cause that the offshore cranes are exposed to dynamic loads reasonably higher
than structures of similar operational parameters but operated on land. Therefore, they are
equipped with special systems for overload reduction. One of them is the shock absorber.
The paper presents amathematical model of an offshore pedestal cranewith a shock absor-
ber. Results of numerical simulations are presented to assess the effectiveness of the shock
absorber in conditions when large dynamic overloads occur.

Key words: modelling, offshore crane, shock absorber

1. Introduction

Despite general economic slowdown in theworld, we can still notice fast development of offshore
technology connected with the exploration of underwater natural resources and its transport.
Offshore devices are the ones assembled away from the shore line, usually on oil rigs and vessels.
One of the major group is formed by offshore cranes.
One of the main features differentiating offshore devices from land–based devices is the

significant movement of the base caused by sea waves. This phenomenon causes that they are
exposed to much bigger dynamic overloads than their onshore counterparts. These overloads
have significant influence on the permissible load capacity of the offshore crane. The aim of
engineers is to design an offshore crane capable to work at possibly high rate of waving. As a
result, a lot of emphasis is put on correct definition of dynamicoverload appearing in this system.
Hence, engineers, designers and analysts are interested in gaining access to the mathematical
model of a crane which would allow one to perform fast calculations of crane dynamics in
various conditions. Such simulationswould be particularly useful at the early stage of the device
development process.
We can findmany publications connected with themodelling of the offshore crane dynamics

in the literature. The authors develop models of different extent of details, 2D or 3Dwith rigid
elements or those taking into account flexibility of chosen submodels. Operation of a winch in
order to limit overload of a system or vertical movements of a load caused by the sea waving
were consideredbyOsiński andWojciech (1994, 1998). Similar issueswere examined in thepaper
Pedrazzi and Barbieri (1998) concerning the cranes serving the purpose of launching Remotely
OperatedVehicles. The stability of the load depthbymeans of anA-framewinchwas the subject
of papers: Fałat (2004), Adamiec-Wójcik et al. (2009). A 2Dmodel of the system: supply vessel-
load-crane-vessel, developed for the base and load dynamic analysis was presented by Osiński



954 J. Krukowski et al.

et al. (2004). The nonlinear dynamical response to a regular waving of a crane mounted on the
vessel was examined by Ellermann et al. (2002, 2003) and Ellermann andKreuzer (2003). Both
numerical simulations and experimental research were carried out. The possibility of parametric
vibration of the crane under the influence of weak waving was analyzed by Witz (1995). The
subject of consideration presented by Urbaś et al. (2010) was a gantry crane, installed on the
oil platform. Amechanical anti–pendulum systemwas presented byBalachandran et al. (1999),
Li and Balachandran (2001). Reduction of load swinging via proper steering of the crane slew
and boom hoist was considered by Masoud (2000) and Nayfeh and Masoud (2004). A different
concept of stabilizing the load position of an offshore cranewas the subject ofMaczyński (2005).
Maczyński and Wojciech (2009) proved that stabilization of the load also minimizes the taut-
slack phenomenon in a rope.Twodifferent algorithms of control thatminimize loadwavingwere
discussed by Schaub (2008). One of them was based on measurement data, the other one on
generated calculations obtained from the numericalmodel of the system.The analyses presented
byCozijn et al. (2008) concentrated on the permissiblework conditions for offshore installations
including the offshore crane of S7000 type.
Currently designed offshore cranes are equipped with various systems that allow dynamic

overloads to be reduced.One of the essential system is the shock absorber.However, the authors
have not encountered any model in the literature which would include the existence of such a
system.

2. Characteristics of a shock absorber

In the offshore crane design, two kinds of shock absorber systems are most commonly used.
The first one is mounted on the boom structure of the crane. The minimalization of dynamic
overload is obtained by leading the hoist rope through an additional sliding sheave connected
with a hydraulic system. The other one, constituting the system of hydraulic accumulators, is
mounted in the hook block. The subject of analysis presented in this work is the first type of
the shock absorber because of its effectiveness, simple and compact construction.
Figure 1 shows in principle the hydraulic part of the shock absorber. It is the system con-

sisting of an accumulator filled with gas and a hydraulic cylinder. When a force S applied to
the piston is big enough, the piston is pulled out and oil is streaming from the cylinder to the
accumulator. The end piston stroke length ∆2 must normally be shorter than the maximum
possible piston stroke length ∆max minus a safety piston stroke length ∆safe, to make sure
there is no risk for the piston to reach the bottom at normal operation. The safety piston stroke
length ∆safe shall not be less than about 50mm.

Fig. 1. Shock absorber



The influence of a shock absorber... 955

3. Model of the offshore crane

The subject of the paper is the offshore pedestal crane equipped with the system reducing
dynamic overload, situated on the boom, see Fig. 2. The main assumptions adopted at the
design stage and the most important connections used during derivation of the equation of
motion will be given below. The modelling of the shock absorber was particularly emphasised.
For the description of the system, joint coordinates and homogenous transformations were used
based on the Denavit-Hartenberg notation.
In the model, the following assumptions were taken into consideration:

• The base of the crane (the platform of a vessel) and the supply vessel are rigid bodies
with 6 degrees of freedom.Themovement is caused bywaving defined by pseudoharmonic
functions.

• The pedestal, the A-frame and the boom are modelled by means of the Rigid Finite
Element Method using a modified approach (MRFEM), (Wittbrodt et al., 2006).

• The king frame, including the slewing part, is treated as a rigid structure with one degree
of freedomwith respect to the pedestal – the slew angle.

• Thebasic element of the shock absorber is the hydraulic cylinder, whichwasmodelled as a
pointmass (additionally including themass of themoving sheave) connected to the boom
by means of a spring damping system.

• The hoist and the luffing ropes are modelled as massless elements with equivalent longi-
tudinal flexibility. The damping was taken into account.

• The load is treated as amaterial point.

• Thedrive functionof thehoistwinch canbeassumed in twoways: as akinematic excitation
or force excitation by a given moment.

• The luffing winch drive and the slew of the crane was adopted as a kinematic excitation.

Fig. 2. Scheme of an offshore pedestal crane

The equations of motions were derived from the Lagrange equations of the second kind

d

dt

∂E

∂q̇k
−
∂E

∂qk
+
∂V

∂qk
+
∂D

∂q̇k
= Qk for k =1, . . . ,n (3.1)



956 J. Krukowski et al.

where q, q̇ are the vectors of genaralized coordinates and their velocities, E, V – kinetic
and potencial energy, D – function of energy dissipation, Qk – nonpotential generalized force
corresponding to the k-th generalized coordinate, n – number of generalized coordinates.

3.1. Modelling of the crane and cargo base motion

It is assumed that the crane basemotion and thusmovement of the system {A}with respect
to the system {0} is known and described by functions

y
(A)
i =

n
(A)
i∑

j=1

A
(A)
i,j sin(ω

(A)
i,j t+ϕ

(A)
i,j ) i =1, . . . ,6 (3.2)

where A
(A)
i,j , ω

(A)
i,j , ϕ

(A)
i,j are respectively the amplitude, phase, frequency and forcing phase angle,

n
(A)
i – number of harmonic series.
Motion of the supply vessel, i.e system {D}, will be described in the same way.
In further considerations, the coordinate system {0}will be identifiedwith the inertial coor-

dinate system {·}, and the following notation will be used for the homogeneous transformation
matrix from the coordinate system {p} to the coordinate system {0}

0
pT=T

(p) (3.3)

where p is the number of members in the kinematic chain.
The homogeneous transformationmatrix 0AT, taking into accountmotion of the system {A}

in {·}, depends on time. It is to be noticed that if r̃ = [x̃, ỹ, z̃,1]T is a vector describing the
coordinates of dm mass (point) in the local system {·} connected to any part of the system.
The coordinates of such amass in the system {·}may be described as

r= 0AT(t)T(q)r
′ =Tr′ (3.4)

where T(q)=A
{·}′
T(q1, . . . ,qn) is transformationmatrix of coordinates fromthe local coordinate

system {·}′ into the {A} system, dependent on the generalized coordinates (q1, . . . ,qn) of the
body T= 0AT(t)T(q).

3.2. Crane pedestal

The crane pedestal was digitized by means of MRFEM. The number of rigid elements on
which the pedestal was divided, equals n1 +1. The first rigid element (rfe(1,0)) is added to
the vessel body. The generalized coordinates describing location of the second and other rigid
elements modelling the pedestal with respect to its predecessors, may be presented as vectors

q̃(1,i) = [ϕ(1,i)x ,ϕ
(1,i)
y ,ϕ

(1,i)
z ]

T = [q̃(1,i)x , q̃
(1,i)
y , q̃

(1,i)
z ]

T (3.5)

where ϕ
(1,i)
x , ϕ

(1,i)
y , ϕ

(1,i)
z are the rotation angles presented in Fig. 3.

The vector of generalized coordinates of the “rfe” yields

q
(1,1) = q̃(1,1) = [q

(1,1)
1 ,q

(1,1)
2 ,q

(1,1)
3 ]

T

q
(1,i) =

[
q
(1,i−1)T, q̃(1,i)

T
]T
= [q

(1,i)
1 ,q

(1,i)
2 , . . . ,q

(1,i)
3i ]

T i =2, . . . ,n1
(3.6)

Inaccordance to theaboveconsideration, duringderivationof the equationsofmotion,kinetic
and potencial energy of the rfe(1,0) have been omitted. On the basis of equations presented in



The influence of a shock absorber... 957

Fig. 3. Pedestal discretized bymeans of MRFEM

previousworks, for example inWittbrodt et al. (2006), the kinetic energy of the bodydiscretized
by theMRFEM can be calculated as

E1 =
n1∑

i=1

E(1,i) E(1,i) =
1
2
tr
{
Ṫ
(1,i)
H
(1,i)
Ṫ
(1,i)T
}

(3.7)

where H(1,i) is the inertia matrix of the rigid element (1, i) defined in its own coordinate sys-
tem, T(1,i) – transformationmatrix from the coordinate system of rfe{1, i} into the coordinate
system {·} for i =1, . . . ,n1 and

T(1,i) =T(1,i−1)T̃(1,i) = 0ATT̃
(1,0)T̃(1,1) · · ·T̃(1,i−1)T̃(1,i)

T̃(1,i) – transformationmatrix fromthe coordinate systemof rfe{1, i} into systemof rfe{1, i−1}.
For the Lagrange equations, the concept of Lagrange operators is often introduced

εi(E)=
d

dt

∂E

∂q̇i
−
∂E

∂qi
(3.8)

Such operators for other rfe(1, i), i =1, . . . ,n1, can be written in the vector form as

εq(1,i)(E(1,i))=A
(1,i)q̈(1,i)+e(1,i) (3.9)

where

A
(1,i) =(a

(1,i)
k,j
)k,j=1,...,3n1 = tr

{
T
(1,i)
k
H
(1,i)
T
(1,i)T

j

}

e
(1,i) =(e

(1,i)
k
)k=1,...,3n1 =

3n1∑

j=1

3n1∑

l=1

tr
{
T
(1,i)
k
H
(1,i)
T
(1,i)
j,l

}
q̇
(1,i)
j q̇

(1,i)
l

+tr
{
T
(1,i)
k
H
(1,i)
[
0
AT̈T

(1,i)
+20AṪṪ

(1,i)]T}

T
(1,i)
=
i∏

j=0

T̃(1,j) T
(1,i)
k
=
∂T(1,i)

∂q
(1,i)
k

T
(1,i)
j,l
=

∂

∂q
(1,i)
j

(
∂T(1,i)

∂q
(1,i)
l

)



958 J. Krukowski et al.

The potential energy due to gravity forces acting on rigid elements of the pedestal can be
described

V
g
(1,i)
= m(1,i)gθ3T

(1,i)
r̃
(1,i)
C i =1,2, . . . ,n1 (3.10)

where m(1,i) is the mass of the rfe(1, i), g – gravity acceleration, θ3 = [0,0,1,0] r̃
(1,i)
C – vector

of the element mass centre (1, i) expressed in its own coordinate system.
The corresponding derivatives, which are elements of the Lagrange equations, are

∂V
g
(1,i)

∂q(1,i)
=G(1,i) (3.11)

where

G
(1,i) =(g

(1,i)
k
)k=1,...,3n1 g

(1,i)
k
= m(1,i)gθ3T

(1,i)
k
r̃
(1,i)
C

InMRFEM, the successive rfe are connected with each other bymeans of massless, spring-
damping elements (sde). Potential energy of the elastic deformation rfe(1, i) is defined as

V s(1,i) =
1
2

(
c
(1)
i,x[ϕ

(1,i)
x ]

2+ c
(1)
i,y [ϕ

(1,i)
y ]

2+ c
(1)
i,z [ϕ

(1,i)
z ]

2
)
=
1
2

3∑

j=1

c
(1)
i,j [q̃

(1,i)
j ]

2 (3.12)

where c(1)i,x, c
(1)
i,y , c

(1)
i,z are the adequate coeficients of the rotational stiffness of rfe(1, i).

Equation (3.12) can be presented in the form

V s(1,i) =
1
2
q̃(1,i)

T

C(1,i)q̃(1,i) (3.13)

where

C
(1,i) = diag[c

(1)
i,x,c

(1)
i,y ,c

(1)
i,z ] = diag[c

(1)
i,1 ,c

(1)
i,2 ,c

(1)
i,3 ]

The required derivatives of the potential energy of elastic deformation have a simple form

∂V s
(1,i)

∂q̃(1,i)
=C(1,i)q̃(1,i) (3.14)

It may be additionally assumed that in rfe(1, i) dissipation of energy appears

D(1,i) =
1
2

(
b
(1)
i,x[ϕ̇

(1,i)
x ]

2+ b
(1)
i,y [ϕ̇

(1,i)
y ]

2+ b
(1)
1,z[ϕ̇

(1,i)
z ]

2
)
=
1
2

3∑

j=1

b
(1)
1,j[
˙̃q
(1,i)
j ]

2 (3.15)

where b(1)i,x, b
(1)
i,y , b

(1)
i,z are respective damping coefficients of rfe(1, i).

Equation (3.15) may be also written as

D(1,i) =
1
2
˙̃q
(1,i)T

B(1,i) ˙̃q
(1,i)

(3.16)

where

B
(1,i) = diag[b

(1)
i,x,b

(1)
i,y ,b

(1)
i,z ] = diag[b

(1)
i,1 ,b

(1)
i,2 ,b

(1)
i,3 ]

and the adequate derivatives can be obtained from

∂D(1,i)

∂ ˙̃q
(1,i)
=B(1,i) ˙̃q

(1,i)
(3.17)



The influence of a shock absorber... 959

3.3. King frame – slewing part

Let us define the following vector of the generalized coordinates for the slewing part

q
(2) =

[
q
(1,nk)

T
,ϕ(2)z

]T
= [q

(2)
1 ,q

(2)
2 , . . . ,q

(2)
n2
]T (3.18)

where ϕz is the angle of rotation of the slewing part with respect to the pedestal.
The kinetic energy of the slewing part can be described as

E2 =
1
2
tr
{
Ṫ
(2)
H
(2)
Ṫ
(2)T
}

(3.19)

where H(2) is the inertial matrix of the slewing part.
The Lagrange operators for the slewing part are formulated in the form

εq(2)(E2)=A
(2)
q̈
(2)+e(2) (3.20)

where

A
(2) =(a

(2)
i,l
)i,l=1,...,n2 = tr

{
T
(2)
i H

(2)
T
(2)
l

T}

e
(2) =(e

(2)
i )i=1,...,n2 = tr

{
T
(2)
i H

(2)
[
0
AT̈T

(2)
+20AṪ

n2∑

j=1

T
(2)
j q̇
(2)
j +

n2∑

l=1

n2∑

j=l

δl,jT
(2)
l,j
q̇
(2)
l
q̇
(2)
j

]T}

δl,j =

{
1 for l = j
2 for l 6= j

T
(2)
= T̃(1,0)T̃(1,1) · · ·T̃(1,n1)

(1,n1)
2T

and
(1,n1)
2 T is the transformationmatrix form the coordinate system of slewing part {2} to the

last “rfe” coordinate system of pedestal {1,n1}.
The potential energy of gravity forces of the slewing body equals

V
g
2 = m

(2)gθ3T
(2)r̃
(2)
C

(3.21)

where m(2) is the mass of the slewing part, r̃
(2)
C – position vector of the mass center of the

slewing part, expressed in system {2}.
The necessary derivatives are defined bellow

∂V
g
2

∂q(2)
=G(2) (3.22)

where

G
(2) =(g

(2)
k
)k=1,...,n2 g

(2)
k
= m(2)gθ3T

(2)
k
r̃
(2)
C

3.4. A-frame and boom

TheA-frame ismodelledbymeansof theMRFEMincompliancewithonlybendingflexibility
in the perpendicular direction to the plane of the A-frame. Additionally as for the pedestal,
rfe(3,0) is added to the slewing part, and as a result, it does not have its own generalized
coordinates, Fig. 4. The following vectors of generalized coordinates for each “rfe” of the A-
frame are defined:
— one-element vectors of the flexible coordinates

q̃
(3,1) = [ϕ(3,1)y ] . . . q̃

(3,n3) = [ϕ(3,n3)y ] (3.23)



960 J. Krukowski et al.

Fig. 4. Simplified model of flexible A-frame and boom

— coordinate vectors describing the position of the rigid element with respect to the base
coordinate system

q
(3,i) =

[
q
(2)T, q̃(3,1)

T
, . . . , q̃(3,i)

T
]T
= [q

(3,i)
1 , . . . ,q

(3,i)
n2+i
]T i =1,2, . . . ,n3 (3.24)

In contradistinction to the pedestal and A-frame, in the case of boom, it was assumed that
there is a rotational connection between rotating part {2} and rfe(4,0), Fig. 4 – angle ψ. We
can define the following vectors of generalized coordinates for the boom:
— vectors of the flexible coordinates

q̃
(4,0) = [ψ] = [ϕ(4,0)y ] . . . q̃

(4,i) = [ϕ(4,i)x ,ϕ
(4,i)
y ,ϕ

(4,i)
z ]

T i =1,2, . . . ,n4 (3.25)

— coordinate vectors describing the position of the rigid element with respect to the base
coordinate system

q(4,i) =
[
q(2)
T
, q̃(4,0)

T
, . . . , q̃(4,i)

T
]T
= [q

(4,i)
1 , . . . ,q

(4,i)
3i+n2+1

]T i =0,1, . . . ,n4 (3.26)

The necessary elements of the Lagrange equations related to the A-frame and boom subsys-
tems were calculated in the same way as presented in Section 3.2.

3.5. The model of shock absorber

The model of the shock absorber is presented in Fig. 5. Its basic element is sheave (3)
possessing the mass msA mounted to the boom bymeans of a paraller spring-damping system.
Relative motion of sheave (3) is possible only along the longitudinal axis of the boom. The
mass msA is enlarged due to movable parts of the hydraulic cylinder.
In the presented model, the characteristic shown in Fig. 6a was assumed for the elastic

element. It represents the characteristic cSA = cSA(x). The curve given in Fig. 6a can be
described as follows

y =





S + c(x−∆) for x ­ a∆

kx+αx2eβx for 0¬ x < a∆

−S + c(x−∆) for x ¬−a∆

kx−αx2e−βx for −a∆ < x < 0

(3.27)



The influence of a shock absorber... 961

Fig. 5. Model of the shock absorber

Fig. 6. Characteristic of: (a) elastic element cSA = cSA(x), (b) shock absorber

By selecting appropriate values of α and β, one obtains smooth transition curves at the
point x = a∆ (and x =−a∆). Then the following conditions must be fulfilled

ka∆+α(a∆)2eβa∆ = S + c(a∆−∆)= k∆+ c(a−1)∆

k+2αa∆eβa∆+αa2∆2βeβa∆ = c
(3.28)

After some transformations, parameters α and β are obtained

β =
2−a

a∆(a−1)
α =

√
(a−1)(c−k)
a2∆eβa∆

(3.29)

The shock absorber is designed in such a way that it works only under the tensioning load.
Up to the value of force S1, the stiffness is a very high value. Within the limit of forces S1
up to S2, the stiffness decreases (shock absorber working range), and beyond the force S2, its
stiffness increases significantly. The characteristic of the elastic element from Fig. 6a must be
appropriately scaled to the form shown in Fig. 6b. The shock absorber working parameters are
defined by the following variables:

S1,S2 – minimum/maximum force within which the shock absorber is active,
∆1,∆2 – displacement of the shock absorber sheave corresponding to the force S1

and S2,
a – parameter specifing where the point of curvilinear part of the characteristic is

becoming rectilinear, a > 1,
α,β – parameters defining the shape of the characteristic described by equation

(3.29).

Form Fig. 6b, it is easy to read that the constants c and k are described bymeans of

c =
S1

∆1
k =

S2−S1
∆2−∆1

(3.30)



962 J. Krukowski et al.

and the values of x and y are determined as

y = y−S x = x−∆ (3.31)

where x, y, ∆ and S are shown in Fig. 6b.

3.6. Hoisting and luffing ropes

The potential energy of elastic deformation and function of energy dissipation for the hoist
and luffing rope can be described by the following equations

Vl =
1
2
δc(l)∆2l Dl =

1
2
δb(l)∆̇2l (3.32)

where

δ =

{
0 for ∆l ¬ 0

1 for ∆l > 0

and ∆l is the elongation of the hoist or luffing rope, c
(l), b(l) – stiffness and damping coefficients

of the ropes, respectively.
Because of the possibility of significant changes in the active length of the hoist rope during

craneoperation, the stiffness coefficient of thehoist rope isdeterminedbymeansof the expression

c(l) =
E6F6

L6,0−α(6)r(6)
(3.33)

where L6,0 is the initial length of the hoist rope, E6, F6 – Young’s modulus and cross section
of the wire rope core, respectively, α(6) – rotation angle of the hoist winch drum, r(6) – radius
of the hoist which drum.
The stiffness coefficient c(l) of the luffing rope is considered as a constant value. A method

of determination of the necessary derivatives of equations (3.32) was described by Maczyński
(2008), which was applied in this work as well.

3.7. Load

The load was modelled as a material point. The weight of the hook block was added to the
weight of the load. The vector of generalized coordinates is defined as

q(L) = [x(L),y(L),z(L)]T = [q
(L)
1 ,q

(L)
2 ,q

(L)
3 ]
T (3.34)

The kinetic and potential energy of the load are described bymeans of

EL =
1
2
m(L)
(
ẋ(L)

2

+ ẏ(L)
2

+ ż(L)
2
)

V
g
L = m

(L)gz(L) (3.35)

where m(L) is the mass of the load. On this basis it is possible to write

εq(L) =A
(L)q̈(L)

∂V
g
L

∂q(L)
= [0,0,m(L)g]T (3.36)

where A(L) = diag[m(L),m(L),m(L)].
The computer program allowed simulation of the following cases:
• the load is in the air (water) – does not remain in contact with the deck of the supply
vessel,

• the load remains stationary on board of the supply vessel; its coordinates are defined by
motion of the supply vessel,

• the load is permanently connected to the supply vessel (e.g. frozen).



The influence of a shock absorber... 963

3.8. Drive systems

The slewing, hoisting and luffing drive systems aremodelled as kinematic inputs. Therefore,
the following function is known

φd = φd(t) (3.37)

where φd is respectively the slewing angle, hosting or luffing winch rotation angle.
From the perspective of planned applications of themodel presented, the hostingmachinery

is one of themost significant drive systems.Therefore, the secondmethod of itsmodelling, using
force excitation, is carried out. Based on the analysis of literature (e.g., Osiński et al., 2004) as
well as the gained experience from crane operators and designers, the hoist winch characteristic
was assumed as shown in Fig. 7.

Fig. 7. Hoist winch characteristic

3.9. Agregation of the equations of motion

Substituting to Lagrange equations (3.1), the equations of motion of the crane can be writ-
ten as

Aq̈=F (3.38)

where A is the inertiamatrix, q – vector of generalized coordinates, F – right-hand side vector;
its elements are designated as the partial derivatives of the kinetic energy, potential forces of
gravity and flexibility, function of energy dissipation, etc.
Equations (3.38) were solved by a computer program using the Runge-Kutta method of

the fourth order with fixed step integration. Before the integration of (3.38), initial conditions
were calculated by solving the above equations assuming q̈ = q̇ = 0. The resulting system of
nonlinear algebraic equations was solved using Newton’s method.

4. Numerical calculations

Examplary simulation results obtained from the developed computer program are presented in
this Section. Two load cases are considered: LC-1: hoisting of the load from the stationary deck,
and LC-2: hoisting from the deck whose movement is described by the function

zP =
3
4
sin
(2π
6
t
)
m (4.1)

A load of 18000kg (including the wire rope and hook block mass) is lifted from the supply
boatdeck.Assuming that thewire rope is loose at thebeginning of the cycle (by a length of 1m),



964 J. Krukowski et al.

some dynamic overload can be expected. The hoisting speed is assumed 0.4m/s for quadruple
operation,with the drumrotation characteristics consistentwithFig. 7. The shock absorberwas
defined by the following parameters: S1 = 97500N, S2 = 1.4S1, ∆1 = 0.02m, ∆2 = 0.52m,
a =1.1.
Plots presented in Figs. 8 and 9 present time courses of the main hoist wire tension force,

luffing wire force and z-coordinate of the load during lifting operation. Two crane models:
working with and without the shock absorber are compared. The whole crane structure was
assumed as rigid.

Fig. 8. LC-1 load case results: (a) hoist rope force, (b) luffing rope force, (c) z coordinate of the load

Fig. 9. LC-2 load case results: (a) hoist rope force, (b) luffing rope force, (c) z coordinate of the load

The conditions assumed in the presented examples are rather theoretical – the winch accele-
ration during the first phase (when the rope is loose)makes a high dynamic peak load when the
wire is suddenly pretensioned.However, this scenario is simulated in order to showhow effective
the shock absorber could be. In some cases, without such a system, makes the life time of it
much shorter.

Fig. 10. LC-2 load case, comparison of the results with rigid and flexible crane boom: (a) hoist rope
force, (b) luffing rope force

The plots shown in Fig. 10 were obtained for the load case LC-2, considering the flexible
boom.The results for the rigid crane boomwere comparedwith those obtainedwith the flexible
structure. Discretization of the crane boomwas performed using n4 =7 rigid finite elements.



The influence of a shock absorber... 965

5. Summary

The results of simulations performed using the crane model having a shock absorber installed
confirm a significant decrease of the dynamic overload experienced by the structural systems.
Application of the shock absorber subsystem in real constructions would allow the crane to
work in much more difficult conditions. Without such systems, the same crane has to be de-
rated, which would make it in a higher sea state less efficient handling tool, causing that the
whole vessel or platform can not perform planned lifts until the weather conditions improve.
Consequently, the load chart of a crane equipped with a system decreasing dynamic loads will
bemuch different from the same construction without such a device. Therefore, proper working
of shock absorbers is very important in the offshore cranes.
The flexibility of the boom has a small influence on the obtained results. Some slight diffe-

rences are observed in the time history of the luffing rope force. It therefore appears that, for
preliminary calculations or for test purposes, the flexibility of the boom can be omitted. On
the basis of a model with a few degrees of freedom, an engineer obtains a quick software tool,
supporting him during the design process. The calculation model presented enable us to deter-
mine the crane overload in variousworking conditions. Thatmakes it possible to predict limiting
weather conditions for a given crane design and specific operation scenarios. Implementation of
themodel in a standalone desktop application makes it attractive for various conceptual ad-hoc
analyses.

References

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A-frame – a type of an offshore crane,The Archive of Mechanical Engineering, 56, 1, 37-59

2. Balachandran B., Li Y.Y., Fang C.C., 1999, A mechanical filter concept for control of non-
linear crane-load oscillations, Journal of Sound and Vibrations, 228, 651-682

3. Cozijn J.L.. van derWal R.J., DunlopC., 2008,Model testing and complex numerical simu-
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5. Ellermann K., Kreuzer E., Markiewicz M., 2002, Nonlinear dynamics of floating cranes,
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7. Fałat P., 2004, Dynamic analysis of a sea crane of an A-frame type, PhD Thesis, Bielsko-Biała
[in Polish]

8. LiY.Y., BalachandranB., 2001,Analytical study of a systemwith amechanical filter, Journal
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Research is financed from the project N N502 464934

Wpływ układu anty-przeciążeniowego liny na dynamikę kolumnowego żurawia morskiego

Streszczenie

Żurawie montowane na statkach i platformach należą do grupy najczęściej wykorzystywanych urzą-
dzeń związanych z techniką morską. W procesie projektowania struktury i doboru mechanizmów napę-
dowych tych maszyn szczególną uwagę poświęca się znaczącym ruchom unoszenia spowodowanym fa-
lowaniem morza. Ponieważ prace przeładunkowe muszą być wykonywane w warunkach intensywnego
falowania, żurawie te są poddane znacznie większym siłom dynamicznym, niż odpowiadające im żurawie
pracujące w warunkach lądowych. Są one wyposażane w różne systemy anty-przeciążeniowe, redukujące
gwałtowne siły dynamiczne pojawiające się w trakcie podnoszenia ładunku. Jednym z takich układów
jest urządzenie działające na zasadzie podobnej do amortyzatora.Wpracy zaprezentowanomodel mate-
matycznyumożliwiający przeprowadzanie analiz dynamicznych żurawi offshorewyposażonychw systemy
anty-przeciążeniowe. Wykonano przykładowe symulacje numeryczne obrazujące efektywność działania
układuw trakcie przykładowej operacji podnoszenia ładunkuwwarunkach powodujących znaczące prze-
ciążenia liny nośnej.

Manuscript received January 1, 2011; accepted for print December 7, 2011