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Systems
ISSN: 1387-3954 (Print) 1744-5051 (Online) Journal homepage: https://www.tandfonline.com/loi/nmcm20
Parameter sensitivity analysis of non-dimensional
models of quayside container cranes
N.Đ. Zrnić , S.M. Bošnjak & K. Hoffmann
To cite this article: N.Đ. Zrnić , S.M. Bošnjak & K. Hoffmann (2010) Parameter sensitivity analysis of non-dimensional models of quayside container cranes, Mathematical and Computer Modelling of Dynamical Systems, 16:2, 145-160, DOI: 10.1080/13873951003676534
To link to this article: https://doi.org/10.1080/13873951003676534
Published online: 26 Mar 2010.
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Parameter sensitivity analysis of non-dimensional models
of quayside container cranes
N.Đ. Zrnića*, S.M. Bošnjaka
and K. Hoffmannb
a
Department of Material Handling, Constructions and Logistics, Faculty of Mechanical Engineering, University of Belgrade, Belgrade, Serbia;bInstitute for Engineering Design and Technical Logistics,
Vienna University of Technology, Vienna, Austria (Received 2 August 2009; final version received 8 January 2010)
This paper gives an analysis of dynamic behaviour of the waterside boom, previously identified as the most important structural part of the mega quayside container cranes (QCC) under moving concentrated mass. The non-dimensional mathematical model used in this paper presents a conceptual substitution of the real system of the mega container crane boom and provides understanding and prediction of its dynamic behaviour under the action of a moving trolley. The paper discusses the procedure for setting up the nondimensional mathematical model of the crane boom as a necessary condition for an estimation of structural and trolley drive parameters, such as the effects of stiffness, mass, spatial position of structural segments as well as trolley velocity and acceleration, on the dynamic values of deflection under the moving mass. The results obtained by the simulation of the trolley motion alongside the QCC boom during container transfer from quay-to-ship are implemented in the parameter sensitivity analysis in order to obtain mutual influences of the dynamic parameters.
Keywords: quayside container crane; structural dynamics; moving mass; non-dimensional modelling; parameter sensitivity analysis
1. Introduction
The moving load problem is one of the fundamental problems in structural dynamics. In contrast to other dynamic loads this load varies not only in magnitude but also in position. The significance of this problem is manifested in numerous applications in the field of transporta-tion. The detailed review of previous researches, including a comprehensive references list, can be found in a dedicated excellent monograph written by Fryba [1] on the subject of moving load problems, where most of the previously used analytical methods are described, as well as
in the following review papers recommendable for further reading [2–6]
Applications of moving load problem have been presented in mechanical engineering studies for the past 30 years. Its solution requires appropriate modelling of the structure and a trolley. Typical structure under moving mass in mechanical engineering are overhead cranes, gantry cranes, unloading bridges, slewing tower cranes, cableways, guideways, shipunloa-ders or quayside container crane (QCC). Beam structures are commonly considered as three-dimensional bodies with one dimension significantly larger than the other two. In general, a length to width (height) ratio of 5–10 of the structural members is sufficient. That means that mechanical structures subjected to moving loads such as e.g. large span cranes, are often well
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*Corresponding author. Email: [email protected]
ISSN 1387-3954 print/ISSN 1744-5051 online © 2010 Taylor & Francis
DOI: 10.1080/13873951003676534 http://www.informaworld.com
qualified for being modelled as beam structures. Commonly applied is the fundamental beam theory according to Euler-Bernoulli and Saint Venant, which does not include the warping (Vlasov) torsion because cross sections of large cranes relevant for the moving load analysis usually have closed thin-walled profiles, such as bridge cranes, gantry cranes, container cranes etc. Some of the structural assumptions in the analysis of the moving load problem are: initially straight beam, undeformable beam cross-section, linear elastic material and constant material properties over the beam cross section, small structural deformations, shear strains according to Saint Venant and negligible damping effects [4]. The Bernoulli-Euler beam theory is valid for the lower modes analysis of the small vibration of slender beams at relatively low velocities that is the case for supporting structures of cranes, where merely the lower modes of natural vibrations are of practical relevance for the analysis of dynamic behaviour and where the maximum currently existing velocities, for e.g. mega QCC, are not exceeding 6 m/s which is small compared to the velocities of trains. For low velocities the bending effect is the dominant factor on structural vibrations.
The basic approaches in trolley modelling are: the moving force model; the moving mass
model and the trolley suspension model– existing in some special structures of unloading
bridges. The simplest dynamic trolley models are the moving force models. The conse-quences of neglecting the structure-vehicle interaction in these models may sometimes be minor. In most moving force models the magnitudes of the contact forces are constant in time. A constant force magnitude implies that the inertia forces of the trolley are much smaller than the dead weight of the structure. Thus the structure is affected dynamically through the moving character of the trolley only. All common features of all moving force models are that the forces are known in advance. Therefore structure-trolley interaction cannot be considered. On the other hand the moving force models are very simple to use and yield reasonable structural results in some cases. The moving mass model is an interactive model. The moving mass model as well as the moving force model, is the simplification of the suspension model, but it includes transverse inertia effects between the beam and the mass. The interaction force between the moving mass and the structure during the time the mass travels along the structure takes into account the contribution from the inertia of the moving mass, the centrifugal force, the Coriolis force and the time-varying velocity-depen-dent forces. These inertia effects are mainly caused by structural deformations (structure-trolley interaction) and structural irregularities. Factors that contribute to creating (structure-trolley inertia effects include: high trolley velocity, flexible structure, large vehicle mass, small structural mass, stiff trolley suspension system and large structural irregularities. Finally, the trolley speed is assumed to be known in advance and thus not dependent on structural deformations. For moving mass models the entire trolley mass is in direct contact with the structure. In general, the dynamic structure-trolley interaction predicted by such models is very strong. The trolley suspension model is representing the physical reality of the system more closely (moving oscillator problem [6]), but it is of little interest in cranes dynamics because, as a rule, the frame of the crane trolley is rigid.
The application of moving load problem in crane dynamics has obtained special attention in engineering research in the last few years, but unfortunately little literature on the subject is available. The paper [7] is according to the authors’ best knowledge the first attempt to increase the understanding of the dynamics of cranes due to the moving load. Some other papers dealing with a similar kind of problem, such as [8–11], should also be mentioned.
2. Non-dimensional mathematical model of the large container crane boom
The past 40 years have shown an increasing interest in the research of modelling and control of cranes, but only a few of them have treated QCC dynamics. These models can be
distinguished by a different complexity of modelling and by the nature of the neglected parameters. The most common modelling approaches are the lumped-mass and distributed mass approach, as well as the combination of the first two approaches. A relatively recent review on cranes dynamics, modelling and control is given in [12], but without considering the problems of the moving load influence on the dynamic response of cranes.
Modern mega QCCs have already tripled in outreach and load capacity compared to the first QCC built 50 years ago (1959). This is not easily accomplished given the cantilevered nature of QCC. A cantilever is structurally inefficient because almost all of the structural strength and weight is needed to support its own weight. The stiffness of the structure affects the deflection magnitude and the vibration frequency. By increasing the stiffness of the crane structure, the deflection will decrease and the vibration frequency will increase. In practice it is very complicated and expensive to do experimental research on a real size mega QCC. This makes investigation using mathematical models necessary, especially during the design stage. The simpler models of mega QCC enable easier mathematical analysis and give better insight into the design and the possibilities of different control algorithms. On the other hand, more complex models are necessary to get a closer approximation of the physical reality e.g. the flexibility of the crane structure will certainly affect the behaviour of the controller. Yet it is impossible to include all real life effects in a mathematical model of a large container crane [13]. However, numerical examination of a model that is not a prototype of some real system is of little interest except for some general conclusions which can be applied to other, related configurations [11]. The choice of an adequate model of the container crane should be determined by the particular problem under consideration and must take into account the eigenfrequencies of the container crane structure as a whole in order to enable suitable dynamic analysis of single structural parts e.g. the crane boom, [11,13].
That is why it was necessary to primarily consider the flexibility of the container crane upper structure. That was done in [13] after analysing the dynamic behaviour of the whole structure of the container crane i.e. after defining natural modes of the QCC structure by the Finite Element Method (FEM). Because a FEM study uses numerical models that have a greater degree of resolution and refinement than experimental models, results obtained from a finite element analysis may be more accurate than those of any experimental model. Hence, FEM is a valuable tool for evaluating the structural dynamic characteristics of machines and structures, and can be used to estimate the natural frequencies and mode shapes for equip-ment and supporting structures [14]. The equip-mentioned facts apply for all types of cranes. The whole crane structure is modelled in [13] by using FEM and by applying beam elements, Figure 1 [15]. In [13] the excitation of the QCC structure in service due to the motion of the trolley is considered most important from the aspect of dynamic analysis, while the water-side boom, due to its large dimension and flexibility, is identified as the most representative structural part for the dynamic structural analysis. It is also observed that the vibrations of the boom on the water side leg in vertical direction perpendicular to trolley direction are practically independent from other structural parts, and this vibration is recognized as one in the range of the first three vibrations with lowest frequencies most important for dynamic analysis [13].
The modelling process, consisting of several intermediate stages, is presented in detail in
[13] and the adopted model of the QCC boom is shown in Figure 2. Equivalent stiffnesses c1
and c2represent respectively the reduced stiffnesses of the inner forestay and the outer
forestay including the stiffness of the upper structure with mast while the lumped masses M1
and M2comprise the masses from the forestays weight. The boundary conditions in point A
connection with other parts of the upper structure, Figure 2. The following parameters for the real construction of mega QCC are used during analysis [13]:
L¼ 65:8 m; L1¼ 30 m; E ¼ 2:1 1011N=m2; I ¼ 0:2484389 m4;
A¼ 0:1925 m2; m ¼ 1848:46 kg=m; M ¼ 175; 000 kg; M1 ¼ 3690 kg;
M2 ¼ 21; 170 kg; c1¼ 111:7462 105N=m; c2¼ 31:8675 105N=m:
Figure 2. Model of the QCC boom acted upon by the concentrated moving mass M. Figure 1. FEM model of the large STS container crane.
The problem of the moving load is treated in the analysis presented in [13] as a moving mass problem i.e. the inertia of the trolley mass has not been neglected. Differential equations of motion are obtained by Lagrange’s equations using the Assumed Modes Method as an alternative to the Rayleigh-Ritz method and by neglecting the dissipation function (damping) because cranes are typically lightly damped, which means that any transient motion takes a long time to dampen out (e.g. according to the data collected in review paper [12] damping of cranes is estimated from 0.1 to 0.5% of their critical damping, up to an higher estimation of 1% for vertical motion). This method is employed to approximate the structural response in terms of the finite number of admissible functions that satisfy the geometric boundary conditions of the mathematical model shown in Figure 2. The selection and estimation of admissible functions is done by using variational approach. The moving mass mathematical model in itself includes the influence of the moving mass inertia, the influence of the Coriolis centripetal force, and the influence of the moving mass deceleration (braking).
The main goal of this paper is to present the non-dimensional mathematical model for predicting the boom dynamics of a mega QCC. The non-dimensional mathematical model used in this paper presents a conceptual substitution for the real system of the mega QCC and provides a general understanding of the dynamic behaviour of a container crane boom under the action of a moving trolley. So, the obtained results can be applied for analysing dynamic behaviour of a series of similar constructions of QCC. The model for set up non-dimensional equations of motion is shown in Figure 2.
The co-ordinate system shown in Figure 2 is assumed to be fixed in the inertial frame with
the~i unit vector parallel to the undeformed beam and the~j unit vector pointing downward in
the direction of the gravitational field g. The position of the mass at any instant is induced by
x¼ s in the i direction. The parameter s is a known and prescribed function of time. The
quantity y is defined as the transverse deflection of an arbitrary point located at x along the beam. The position vector of a general point p on the deformed beam is given by [15]:
~p ¼ x~iþ y~j: (1)
The velocity at the point is:
~vp ¼ _y~j; (2)
where _y ¼dy
dt:
For simplicity in subsequent computations, the following dimensionless (normalized) quantities similar as in [16], are introduced:
τ ¼ t ffiffiffiffiffiffiffiffiffi EI mL4 r ; L1¼ L1 L ; L ¼ L L; y ¼ y L; ¼ x L; s ¼ s L; c1 ¼ c1L3 EI ; c2¼ c2L3 EI M ¼ M mL; M1¼ M1 mL; M2¼ M2 mL; v ¼ v ffiffiffiffiffiffiffiffiffi mL2 EI r ; g ¼gmL3 EI ; a ¼ amL3 EI : (3)
By using the assumed mode method, the dimensionless deflectiony can be expressed as
y ¼ y
L ¼
X5
i¼1
qiðτÞfiðÞ; (4)
where qiðτÞ are generalized coordinates which are unknown functions of time and fiare
dimensionless (normalized) assumed spatial functions that satisfy the prescribed geometric boundary conditions at the two ends of the beam. Those functions are adopted from a set of admissible functions. Admissible functions are employed because eigenfunctions of the
system shown in Figure 2 cannot be determined in practice due to the non-standard boundary conditions and added lumped masses. Admissible functions are any set of functions that satisfy the geometric boundary conditions of the eigenvalue problem and are differentiable ‘n’ times. Finally, after several iterations the following five (n ¼ 1; 5) admissible functions
(these functions should not be‘blindly’ selected) are assumed as [13, 15]:
f1ðÞ ¼ ; f2ðÞ ¼ sin π; f3ðÞ ¼ sin 2π; f4ðÞ ¼ sin 3π; f5ðÞ ¼ sin 4π: (5)
The equation of motion can be formulated using the Lagrangian approach either by considering the presence of the moving mass in terms of the contact force [13], or by including the mass to be part of the system, with the external force acting on the system given by gravitational force alone [15]. For the second approach, including the mass to be part of
the system shown in Figure 2, and by adopting x ¼ s and s ¼ s(t) having in mind that
_s ¼ v;€s ¼ a, the kinetic energy of the moving mass TMis:
TM ¼ 1 2M v 2þ dyðs; tÞ dt 2 " # ; (6) where dyðs; tÞ dt ¼ @ yðs; tÞ @s vþ @ yðs; tÞ @t ; (7)
and by plugging the expression (7) into the expression yðs; tÞ ¼X
5 i¼1 fiðsÞqiðtÞ yields dyðs; tÞ dt ¼ X5 i¼1 f0iðsÞvqiðtÞ þ fiðsÞ_qiðtÞ
. The expression for the kinetic energy of the moving mass is finally
TM ¼ 1 2Mv 2þ1 2 X5 i¼1 X5 j¼1 f0iðsÞvqiþ fiðsÞ_qi ðf0jðsÞvqjþ fjðsÞ_qj:Þ (8)
The kinetic energy of the beam and concentrated masses M1and M2is:
TB;M ¼ 1 2 ðL 0 m @yðx; tÞ @t 2 dyþ1 2M1 @yðL1; tÞ @t 2 þ1 2M2 @yðL; tÞ @t 2 ¼1 2 ðL 0 m X 5 i¼1 fiðxÞ_qiðtÞ " # X5 j¼1 fjðxÞ_qjðtÞ " # dx þ1 2M1 X5 i¼1 fiðL1Þ_qiðtÞ " # X5 j¼1 fjðL1Þ_qjðtÞ " # þ1 2M2 X5 i¼1 fiðLÞ_qiðtÞ " # X5 j¼1 fjðLÞ_qjðtÞ " # : (9)
By using the well-known notation for the kinetic energy T ¼ 1
2 Xn i¼1 Xn j¼1 mij_qiðtÞ_qjðtÞ; the
TB;M ¼1 2 X5 i¼1 X5 j¼1 _qiðtÞ_qjðtÞ ðL 0 mfiðxÞfjðxÞdx þ M1fiðL1ÞfjðL1Þ þ M2fiðLÞfjðLÞ 2 4 3 5: (10) The bending elastic strain of the beam including potential energy of the spring elements c1and c2is: VB:M ¼12 ðL 0 EI X 5 i¼1 @2f iðxÞ @x2 qiðtÞ " # X5 j¼1 @2f jðxÞ @x2 qjðtÞ " # dx þ1 2c1 X5 i¼1 fiðL1ÞqiðtÞ " # X5 j¼1 fjðL1ÞqjðtÞ " # þ1 2c2 X5 i¼1 fiðLÞqiðtÞ " # X5 j¼1 fjðLÞqjðtÞ " # : (11)
By using the well-known notation for the potential energy V ¼1
2
Pn
i¼1
Pn
j¼1cijqiðtÞqjðtÞ; the
expression (11) for five admissible functions (n¼ 5) can be written as:
VB;M ¼1 2 X5 i¼1 X5 j¼1 qiðtÞ qjðtÞ ðL 0 EI@ 2f iðxÞ @x2 @2f jðxÞ @x2 dxþ c1fiðL1ÞfjðL1Þ þ c2fiðLÞfjðLÞ 2 4 3 5: (12) The potential energy of the moving mass is:
VM ¼ Mgyðs; tÞ ¼ Mg
X5
i¼1
fiðsÞqi: (13)
The total kinetic energy of the system is T ¼ TB;Mþ TM, while the total potential energy
of the system is V ¼ Vmþ VM. By plugging the expressions for kinetic and potential energy
into the Lagrangian of the system defined as L ¼ T – V and by using the well-known
Lagrange’s equations for the considered system d
dtð dL @ _qjÞ
dL
@qj¼ 0 [17] and dimensionless
quantities given in Equation (3), finally yields the non-dimensional differential equation of motion in the matrix form:
M ½ f g þ B€q ½ f g þ _q ½ qCf g ¼ Mg Φ; (14) where ½ ¼M ½ mijþ M ½ Hij, ½ ¼ 2 B Mv A½ ij, ½ ¼C ½ cijþ Mv2½ K ijþ Ma A½ ij, ½ ¼Φ fðsÞ ½ i, while m ½ ij ¼ ðL 0 fiðÞfjðÞd þ M1fiðL1ÞfjðL1Þ þ M2fiðLÞfjðLÞ; c ½ ij ¼ ðL 0 ð@2fiðxÞ @x2 Þð @2f jðxÞ @x2 Þd þ c1fiðL1ÞfjðL1Þ þc2fiðLÞfjðLÞ; ½ Hij¼ fiðsÞfjðsÞ; A ½ ij ¼ fiðsÞf0jðsÞ and K½ ij¼ fiðsÞf00jðsÞ:
It is obvious that the non-dimensional matrix equation of motion (14) can only be solved numerically. This system of differential equations is solved by using the Runge-Kutta
method of the V order (Method Runge-Kutta-Fehlberg, RK45), and by using in-house software written in C++. The system of differential equations is non-stiff, so the implemen-tation of the fifth order Runge-Kutta method is strictly sustainable.
During the transhipment of containers from quay-to-ship (ship loading) in real-life
circumstances it happens that the moving mass (trolley with pay load– container) reaches
its maximum velocity in the vicinity of the hinged-end A, Figure 2. It is therefore assumed that the moving load is starting to move from the left end (hinge A) of the beam. Also is assumed the maximum path of the trolley i.e. loading of the endmost container ship cell. For that reason the simulation of the moving load motion is done by assuming that the motion
consists of two parts: uniform motion during the time tuand transient motion (constant
deceleration– braking) during the time tb, i.e. t¼ tuþ tb. Hence, the displacement of the
trolley with pay load x(t) from the left end of the beam, Figure 2, becomes xðtÞ ¼ vtuþ12at2b.
The dimensionless deflection under the moving load (y ¼ ð; τÞ), depending on the number
of admissible functions is shown in Figure 3. It can be seen that fast convergence of the solution for adopted five admissible functions is obtained, while excellent agreement is
found for n¼ 4 and n ¼ 5. This fast convergence confirms the adequate number of adopted
admissible functions. External validation of the model is done by experts and experienced professional engineers (expert scrutiny) dealing with problems of large container cranes, as a way of model validation suggested in [18]. The structure of the mathematical model is consistent with the computer program, without problems in algorithm structure; thereby the internal verification of the mathematical model is also done as described in [19].
3. Parameter sensitivity analysis of the dynamic system of the QCC boom
For obtaining qualitative estimation of the structural parameters and dependencies of the non-dimensional boom deflection on the dimensionless structural and trolley drive para-meters, the parameter sensitivity analysis method is used (sensitivity analysis of dynamic systems is elaborated in [20]). Such a technique, although currently somewhat neglected by
7.0×10–3 6.0×10–3 5.0×10–3 4.0×10–3 3.0×10–3 2.0×10–3 1.0×10–3 0.0 0.0 0.2 0.4 ξ n=5 n=4 n=3 n=2 0.6 0.8 1.0 y ( ξ , τ )
Figure 3. Dependence of the deflectiony under the moving mass M on the number of admissible functions.
many in the system modelling and simulation fields, still has considerable relevance, particularly in structural dynamic problems. The obtained sensitivity functions can be used to establish the dependence of each part of a response time-history to each of the model parameters. Parameter sensitivity analysis techniques also provide useful methods for some types of validation problems [18]. This paper discusses the effects of changing structural and trolley drive parameters on the values of deflections of the QCC boom. In this case the parameter sensitivity analysis is used to validate the simulation model in relation to the model obtained by FEM, as the substitution for the real system of a mega QCC. It is important to mention that the variation of some dynamic parameters is done in the scope of real values (defined by an engineering-expert assessment and imposed by a rational approach to the design of structure) for the modern constructions of a mega QCC shown in Figure 1. This means that this investigation is particularly important from an engineer’s viewpoint, especially in the early stage of design in order to analyse all variants of structural solutions. The scope of the real values of considered dynamic parameters is expanded to the theoretical ones in order to obtain the fitted sensitivity functions. This was necessary to validate the dynamic model.
Hereafter a parametric study is proposed which describes QCC boom behaviour (dimen-sionless deflection) in terms of dimen(dimen-sionless parameters, such as the forestays stiffness
variation c12 {54.94; 111.7462-designed value; 164.8; 219.8} 10
5
N/m, c22 {17.42;
31.8675-designed value; 54.94; 91.57} 105N/m, Figures 4–7, the ratio between moving
mass and mass of the boom M2 {206.8; 175-designed value; 121.6; 85.14} t, Figures 8 and 9,
the geometry of the inner forestay position along the boom L12 {23.03; 30-designed value;
36.19; 42.77} m, Figures 10 and 11, the trolley velocity v2 {4; 6; 8; 10} m/s, Figure 12, and
the trolley deceleration a2 {0.4; 0.8; 1.2; 1.6} m/s2, Figure 13.
The dimensionless deflection under the moving load y increases with its movement
towards the free end of boom; the smaller the inner forestary is, the faster it increases, Figure 4. This trend is maintained until the load is not found below the point of attachment of mentioned forestay ( ¼ 0.456). From this position up to the position defined by coordinate ¼ 0.83, the dimensionless deflection increases, while maintaining approximately con-stant differences of observed deflections for minimal and maximum analysed stiffness of
7.0×10–3 6.0×10–3 5.0×10–3 4.0×10–3 3.0×10–3 2.0×10–3 1.0×10–3 0.0 0.0 0.2 0.4 ξ 0.6 0.8 1.0 y ( ξ , τ ) C1=30 C1=61.02 C1=90 C1=120
Figure 4. Deflectiony under the moving mass versus inner forestay stiffness c1(designed value
the inner forestay. During further movement of the moving load, the speed of the dimen-sionless increment of deflection follows the inner forestay stiffness: the higher it is, the higher the speed.
The change in normalized stiffness of the inner forestay fromc1;A ¼ 0:49c1;D¼ 30 up to
c1;B¼ 1:97c1;D¼ 120, Figure 5, causes the change of maximum value of dimensionless
deflection fromymax;A¼ 1:09ymax;D up toymax;B ¼ 0:94ymax;D, whereymax;D is maximum
dimensionless deflection for the designed stiffness of inner forestay. It is observed that for
c1 > 5c1;D 300 functionymaxis, from an engineering standpoint, practically unaffected by
the change of inner forestay stiffness. Moreover, based on the parameter dependence
function the model validation can be done for the boundary cases whenc1 ! 0 (no support,
maximum deflection) andc1! 1 (rigid support, minimum deflection).
9.0×10–3 A A D B D B 6.8 × 10–3 6.6 × 10–3 6.4 × 10–3 6.4 × 10–3 6.0 × 10–3 5.8 × 10–3 30 40 60 80 100 120 8.0×10–3 7.0×10–3 6.0×10–3 5.0×10–3 0 100 200 300 400 500 600 700 800 900 1000 y ( L, τ )max C1
Figure 5. Parameter dependence function: deflectionymaxversus inner forestay stiffnessc1(designed
valuec1;D¼ 61; 02). 1.0×10–2 8.0×10–3 6.0×10–3 4.0×10–3 2.0×10–3 0.0 0.0 0.2 0.4 0.6 0.8 1.0 y ( ξ , τ ) ξ C2=10 C2=30 C2=50 C2=17.402
Figure 6. Deflectiony under the moving mass versus outer forestay stiffness c2(designed value
Dimensionless deflection under the moving loady, increases with its movement towards the free end of boom; the smaller the outer forestay stiffness, the faster the increase, Figure 6.
The change in normalized stiffness of the outer forestay fromc2;A ¼ 0:58c2;D¼ 10 up to
c2;B¼ 2:87c2;D¼ 50, Figure 7, causes the change of maximum value of dimensionless
deflection fromymax;A¼ 1:39ymax;Dup toymax;B¼ 0:4ymax;D.
The values of dimensionless deflection under the moving load y, increase with its
movement towards the free end of boom; the larger the mass of the considered moving load, the higher the values and the quicker their growth, Figure 8.
The change in normalized mass of the moving load from MA¼ 0:49 MD¼ 0:7 up to
MB¼ 1:18 MD¼ 1:7, Figure 9, causes the change of maximum value of dimensionless
deflection fromymax;A¼ 0:67ymax;Dup toymax;B¼ 1:63ymax;D.
0 10 20 30 40 50 60 2.5×10–2 A A D B B 1.0×10–2 9.0×10–3 8.0×10–3 7.0×10–3 6.0×10–3 5.0×10–3 4.0×10–3 3.0×10–3 2.0×10–3 10 20 30 40 50 2.0×10–2 1.5×10–2 1.0×10–2 5.0×10–3 0.0 y ( L, τ )max C2
Figure 7. Parameter dependence function: deflectionymaxversus outer forestay stiffnessc2(designed
valuec2;D¼ 17; 402). 8.0×10–3 7.0×10–3 6.0×10–3 5.0×10–3 4.0×10–3 3.0×10–3 2.0×10–3 1.0×10–3 –1.0×10–3 0.0 0.2 0.4 0.6 M=1.7 M=1.439 M=1.0 M=0.7 0.8 1.0 0.0 ξ y ( ξ , τ )
For maximum considered distance of the inner forestay point of attachment in relation to
hinged-end A, L1;max¼ 0:65, dimensionless deflection under moving load has the fastest
increase and gives the maximum value until the trolley reaches the position defined by
coordinate ¼ 0.375, Figure 10. The curves corresponding to L1;max¼ 0:65 and
L1;min¼ 0:35 intersect for ¼ 0.375. After that, the curve which corresponds to L1;max
intersects the curves corresponding to L1;D¼ 0:456 and L1¼ 0:55 for ¼ 0.46 and
¼ 0.56, respectively, and for further motion of trolley give the lowest values of y. The
curve which corresponds to L1;minafter its intersection with the curve corresponding to L1;max
has the fastest increase and gives the maximum values of deflection.
8.0×10–3 7.0×10–3 6.0×10–3 5.0×10–3 4.0×10–3 3.0×10–3 0.6 0.7 0.8 1.0 1.2 A D B 1.4 1.6 1.7 1.8 y ( L, τ )max M
Figure 9. Deflectionymaxversus mass M (designed value MD¼ 1:439) in the scope of real values.
7.0×10–3 6.0×10–3 5.0×10–3 4.0×10–3 3.0×10–3 2.0×10–3 1.0×10–3 –1.0×10–3 0.0 y ( ξ , τ ) 0.0 0.2 0.4 ξ 0.6 0.8 1.0 L1=0.35 L1=0.55 L1=0.65 L1=0.456
Figure 10. Deflectiony under the moving mass versus inner forestay geometry L1(designed value
The change of location of the inner forestay in relation to hinged-end A from
L1;A¼ 0:77L1;D¼ 0:35 up to L1;B¼ 1:43L1;D¼ 0:65, Figure 11, causes the change of
maximum value of dimensionless deflection from ymax;A¼ 1:13ymax;D up to
ymax;B¼ 0:78ymax;D. The model validation can also be done for two theoretical cases,
without inner forestay L1 ! 0 (maximum deflection) andL1! 1, i.e. both forestays are
attached at the endpoint of the boom giving increased stiffness (minimum deflection). It should be mentioned that the current maximum velocities of the mega QCC do not exceed 6 m/s, but higher values of velocity are used to show that there is no practical significance of the velocity increase on the values of boom deflection, Figure 12. For
9.0×10–3 8.0×10–3 7.0×10–3 6.0×10–3 5.0×10–3 4.0×10–3 3.0×10–3 2.0×10–3 1.0×10–3 0.1 0.2 0.3 0.4 0.5 A A D B 0.6 0.7 0.8 0.9 0.0 1.0 7.0 × 10–3 6.5 × 10–3 6.0 × 10–3 5.5 × 10–3 5.0 × 10–3 4.5 × 10–3 0.35 0.40 0.45 0.50 0.55 0.60 0.65 y ( L, τ )max L1
Figure 11. Parameter dependence function: dimensionless deflection ymax versus inner forestay
geometry L1(designed value L1;D¼ 0:456).
7.0×10–3 6.0×10–3 5.0×10–3 4.0×10–3 3.0×10–3 2.0×10–3 1.0×10–3 0.0 0.0 0.2 0.4 ξ 0.6 0.8 1.0 v=0.074 v=0.05 v=0.099 v=0.124 y ( ξ , τ )
example, the increase of the currently maximum velocity of 6 m/s to the possible future value of 10 m/s will increase the absolute deflection for 0.53% which is irrelevant for an engineering analysis.
It is obvious that the boom deflection is practically unaffected by the value of decelera-tion, Figure 13.
4. Conclusions
The crane is not only a part of the terminal system but is also a system in its own right, and the optimum design requires balance. The deterministic computer simulation of the non-dimen-sional mathematical model of the QCC boom, as the most important structural part, is a kind of numerical experiment used instead of the extremely costly experiments on a real size crane or scale-model.
The results obtained by simulating the trolley motion alongside the STS container boom from quay-to-ship are used for parameter sensitivity analysis in order to obtain the estimation of the structural parameters and the dependencies of the non-dimensional boom deflection on the dimensionless structural and trolley drive parameters such as stiffnesses, masses, geometric configuration, velocity and deceleration. At the same time, the parameter sensi-tivity analysis is used as a way of model validation.
The main conclusions obtained by parameter sensitivity analysis which are important for dynamic structural analysis of QCC are:
The variation of the values of structural parameters (stiffness, geometry and mass) defined by an engineering-expert assessment and imposed by a rational approach to the design of structure has significant impact on the dynamic values of deflections. This conclusion can be used for making a new approach in the design (particularly in the early stage of design) of a modern mega QCC with flexible supporting structure without strict deflection limits. 0.0 0.2 0.4 ξ 0.6 0.8 1.0 7.0×10–3 6.0×10–3 5.0×10–3 4.0×10–3 3.0×10–3 2.0×10–3 1.0×10–3 0.0 y ( ξ , τ ) a=0.004037 a=0.00875 a=0.012 a=0.016
Before adopting the final design solution of the QCC it is necessary to analyse in detail the values for stiffnesses and geometry of forestays, as well as the location of the point of attachment of the inner forestay to the boom.
The influence of the parameters of trolley drive (including the anticipated values in the future), such as velocity and deceleration, are not significant from the aspect of dynamic values of deflections. So, there are no mechanical restrictions for the future increase of trolley velocity, but the current practice has shown that even the velocities higher than 5 m/s are too fast for the crane operator, and most manufacturers have rejected the idea of further increase of trolley velocity.
The results obtained in this work may be used as a basic point for setting new and improved control algorithms by considering the structural flexibility of the boom, instead of the traditional and common approach where the crane structure is assumed to be rigid.
The modelling procedure used in this paper can be applied in solving structural dynamics problems for some other related large constructions, such as slewing tower cranes and ship unloaders for bulk materials.
Acknowledgments
A part of this work is a contribution to the Ministry of Science and Technological Development of Serbia funded project TR 14052.
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