Analysis of nonlinear effects in a carrying rope of cableway

Proceedings of the 9th International Conference on Structural Dynamics, EURODYN 2014
Porto, Portugal, 30 June - 2 July 2014
A. Cunha, E. Caetano, P. Ribeiro, G. Müller (eds.)
ISSN: 2311-9020; ISBN: 978-972-752-165-4
Analysis of nonlinear effects in a carrying rope of cableway subjected to moving load
Knawa-Hawryszków Marta1, Bryja Danuta2
Institute of Civil Engineering, Wrocław University of Technology, Wyb. Wyspiańskiego 27, 50-370 Wrocław, Poland
2
Institute of Civil Engineering, Wrocław University of Technology, Wyb. Wyspiańskiego 27, 50-370 Wrocław, Poland
email: [email protected], [email protected]
1
ABSTRACT: In this study a bi-cable circulating gondola ropeway system is considered. Authors base on the comprehensive
theoretical model of a multi-span carrying cable formulated by using an analytical continuous approach. Equations of motion
describing in-plane carrying cable vibrations due to moving in-service load are derived using Ritz approximate method
combined with Lagrange equations. Traveling passenger cars are modeled as physical pendulums moving at constant speed
along cable track. Two sets of equations of motion of two sub-systems: carrying cable and traveling cars are nonlinear and
coupled, and they contain coefficients which are dependent on time. In formulated equations we identified two different reasons
of geometrical nonlinear influences. One of them is nonlinearity typical for cable structures, caused by: (i) changes in cable
route configuration due to moving load, (ii) large displacements leading to Green-Lagrange deformation formula. Such
nonlinear effects are represented in matrix equation of motion of the system by nonlinear elastic cable forces. The main
objective of this paper is to evaluate how these nonlinear forces affect the static and dynamic behavior of the cable subjected to
moving cars. Numerical analysis is performed for an example of 3-span inclined cableway tensed by counterweight.
So-called “nonlinear amplification factor” is estimated by comparing nonlinear and linear cable displacements and
counterweight displacements. Then the permissible level of nonlinear effects is assumed and related to it, permissible level of
linear transverse cable displacements is determined in static solution. Presented results show that the proposed nonlinear
analysis can be useful for determining some practical guidelines for cable displacements’ limitation in considered cableway
structures.
KEY WORDS: bi-cable ropeway, multi-span cable, geometrical nonlinearity, statics, dynamics, moving load
1
INTRODUCTION
Aerial cableways are familiar transport system which
performs well especially in difficult terrain conditions in
mountain regions (ski resorts, sightseeing areas), however
they become increasingly popular in cities as an alternative
means of public urban transport. For passenger transportation
different types of ropeways are used since various technical
systems and their combinations have evolved over many
years. They can be grouped by two main criteria: the number
of ropes with different functions (mono-, bi- and multi-cable
ropeways) and the type of motion (continuous (circulating),
reversible and pulsed operation). For further classification,
e.g. type and size of cable cars, possibility of their detach at
stations (grip types), etc., are taking into account. To cover
longer distances and higher elevations using less number of
intermediate supports, bi-cable ropeways constitute a feasible
technical solution, optionally with circulating or reversible
operation system (see Figure 1).
As we can observe the great variety of considered systems, it
is difficult to formulate a universal model and common
computational procedure to investigate static and dynamic
behavior of such structures. Then scientific publications often
handle cases of certain type of load acting on individual type
of ropeway (already built and being in use), especially when
authors undertakes the research of dynamic load [1–5] which
usually concern selected problems, e.g. dynamic behavior of
carrying and hauling rope [1], [4] or carrying-hauling rope [5],
rope and carriers reaction to lateral wind [3], etc.
Figure 1. Bi-cable circulating detachable gondola ropeway.
A literature review on the analysis of cable structures
reveals that modeling of an individual cable or cable system is
challenging, because they are highly nonlinear. Moreover the
complexity of the system, in which geometry and load
conditions are continuously changeable during exploitation,
causes that comprehensive dynamic nonlinear analysis is
quite complicated. From theoretical point of view, mostly we
can find papers which deal with certain basic problems of
suspended cable dynamics, but there are not many
publications which bring up the subject of nonlinear statics
and dynamics of ropeway cable under moving in-service load.
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Proceedings of the 9th International Conference on Structural Dynamics, EURODYN 2014
In all above-mentioned studies [1–5] the cable was regarded
as a single-span structure modeled by using FEM. The
authors of this paper created mathematical model of the
carrying (track) rope idealized as multi-span continuous cable
tensed by counterweight subjected to traveling passenger
carriers modeled by pendulums, which is described in detail in
References [6–8]. The formulated model is fully nonlinear and
allows to analyze in-plane cable displacements. The main
topic of this paper is to investigate an influence of cable
geometrical nonlinearity on static and dynamic displacements
of carrying rope and tensioning device (counterweight) in
three-span bi-cable circulating gondola ropeway under
moving load. On the basis of nonlinear analysis it would be
possible to determine maximal permissible displacement in
considered ropeway subjected to in-service load, in
accordance to assumed acceptable level of nonlinear effects.
2
Distributed mass of the carrier is lumped into concentrated
masses of individual units: carriage (Mc), hanger with cabin
(Mch) and passengers (Mlj), with rotational mass moments of
inertia Jch and Jlj. Thus, the cable car is represented by
a single-degree-of-freedom pendulum which swaying
movement in the cable sag plane is defined by rotational angle
θj, as it is shown in Figure 3.
MODEL OF A CARRYING CABLE SUBJECTED TO
MOVING PASSENGER CARRIERS
Because the proposed model of a carrying cable of bi-cable
circulating ropeway loaded by moving in-service load is fully
formulated and extensively described in previous authors’
publications [6], [8], and as in this paper the studies are
mainly aimed at the analysis of calculation results, we briefly
summarize the most important assumptions.
We consider a carrying rope as a multi-span continuous
cable, anchored at one end of the track (in a top terminal) and
pre-tensed by a sliding counterweight with mass T at the other
end (in a bottom terminal). Cable route runs in one vertical
plane as it is shown in Figure 2. The suspended cable slides
without friction on intermediate and inflexible supports. Mass
of the cable per unit length, denoted by m, and longitudinal
stiffness EA of the cable are constant lengthwise all spans. As
the small strains and large cable displacements are expected,
Green-Lagrange deformation of the carrying cable is valid.
Cable material follows Hook’s law. In-service load acting on
the carrying cable of an aerial ropeway is caused by moving
passenger carriers represented by physical pendulums (see
Figure 3). It is assumed that gondolas move along cable with
constant line speed v which is related to the direction of
spatial coordinate x measured along the average track slope
defined by ϕ angle. This simplification seems to be justified
as cable sags in a dead-load static configuration are small due
to high initial cable tension. Along x axis downhill motion of
the jth carrier is described by the function: xj = vt ‒ (j‒1)d,
where d denotes constant interval between carriers.
Figure 3. Model of a passenger carrier as swaying pendulum.
Cable displacements in the ith span due to the action of
distributed loads are described by two components:
and
transverse
longitudinal
displacement
ui (xi, t)
displacement wi (xi, t), measured in x and z directions,
respectively, in reference to initial static configuration (under
dead load). The initial tensions for all cable spans can be
obtained in the general form H0i (xi) = H0 + mgsinϕ[(li +
…+ lk) – xi], where H0 = Tg + mglk+1 describes constant initial
tension of the cable with horizontal chord (ϕ = 0).
Vibrations of a cableway system are described by two sets
of equations of motion of two sub-systems: carrying cable (1)
and passenger cars modelled as pendulums (2). Equations are
derived on the basis of Ritz approximate method combined
with Lagrange equations. Governing equations are nonlinear
and coupled, and they contain coefficients which are
dependent on time and angular displacements of pendulums:
~
~
~
(1)
&& + Cq& + Kq = F(t ) − B(θ, t )q
&& − C(θ, t )q& −
Bq
~
~
K (θ, t )q − K θ (θ, t )θ − R N (q ) ,
~
~
~
&& − C q (θ, t )q& − K q (θ, t )q , (2)
{J}&θ& + {c}θ& + g{S}θ = −B q (θ, t )q
Figure 2. Model of an inclined carrying cable subjected to
moving in-service load.
3814
where B, C, K are mass, damping and structural stiffness
~ ~ ~ ~ θ ~ q ~ q ~ q are respective
block matrices, B
, C, K , K , B , C , K
matrices dependent on time t and pendulum angular
~
displacements θ , F (t ) is time-dependent excitation vector,
RN(q) is vector of nonlinear elastic forces and {J}, {c}, {S}
are diagonal matrices of pendulums’ parameters (see
References [6–8]). We identified two different reasons of
geometrical nonlinear influences: i) nonlinearity related to
cable-car interaction, which was examined in previous
research [6–8], ii) nonlinearity typical for cable structures,
caused by changes in cable route configuration due to moving
Proceedings of the 9th International Conference on Structural Dynamics, EURODYN 2014
load, and large displacements leading to Green-Lagrange
deformation formula. In further considerations, nonlinear
components of the interaction between a cable and moving
carriers will be neglected for brevity of numerical analysis
intended mainly to investigate nonlinear elastic cable forces
assembled in vector RN(q). Then, the final matrix equation of
motion of the cable-car system modelling a multi-span
ropeway subjected to moving in-service load has the
following form:
~
~
&& C + C
B + B(t ) 0  q
(t ) 0  q& 
+
 ~q
 &&  ~ q
 & +
 B (t ) {J} θ   C (t ) {c} θ 
(3)
~
~θ
~
N
K + K (t ) K (t ) q  F (t ) − R (q)
 ~q
  =
.
g{S}  θ  
0
 K (t )

System of equations of motion (3) remains nonlinear and
coupled, however coefficients of equations are only timedependent (do not depend on pendulums’ displacements). All
components of the Equation (3) are derived and defined in
detail in References [6], [8].
3
NUMERICAL
ANALYSIS OF GEOMETRICAL
NONLINEAR EFFECTS
3.1
Definitions of introduced concepts and measures
For clear and easy interpretation of results obtained in
numerical analysis, the authors introduced some concepts and
measures defined below:
• nonlinear amplification factor ψ – measure of nonlinear
effects in static or dynamic analysis, which is defined as
the relation of maximal nonlinear response (N) to
maximal linear response (L) of the structure, given by
formula
ψ =
max t X N
,
max t X L
acts as the parameter which identifies passenger carriers’
location on a cable.
Two permissible levels of nonlinear effects are considered.
They are defined on the basis of static nonlinear amplification
factors ψs related to transverse cable displacements:
• ε p = 10% , when ψ s =
• ε p = 5% , when ψ s =
max t wsN
≤ 1,1 for every ws,
max t wsL
max t wsN
max t wsL
≤ 1,05 for every ws.
Analysis of static and dynamic relative cable displacements
and a counterweight displacement due to given in-service
load, together with analysis of corresponding levels of
nonlinear effects, enables to estimate:
• permissible level of displacements λp – maximal relative
linear displacement of cable, obtained in a static problem,
which fulfills the requirement of the assumed permissible
level of nonlinear effects (εp = 10% or εp = 5%).
3.2
Example of a cableway, used in numerical analysis
Three-span suspended carrying cable with diameter dc = 54
mm, unit mass m = 16.36 kg/m and stiffness EA = 310880 kN,
tensed by a counterweight with mass T has been considered.
Minimal breaking force for the rope is Fmin = 3252 kN and
minimal mass of a counterweight taken to calculations is
T = 104000 kg. Horizontal lengths of spans are: L1 = 275 m,
L2 = 400 m, L3 = 250 m, and average inclination angle of track
is ϕ ≈ 23° (see Figure 4). In-service load is due to a semiinfinite flow of the same carriers (gondolas) with masses:
Mch = 610 kg, Mc = 252 kg, Ml = 480 kg, travelling with speed
v = 10 m/s at the distance d = 65 m. Other carrier’s parameters
are the following: Jch = 290 kgm2, Jl = 197 kgm2.
(4)
where X = X(xi, t) is any considered response of the
system in analyzed cross-section xi of a cable in ith span,
or displacement of a counterweight; the concept of this
factor is analogical to dynamic amplification factor – used
as a measure of dynamic effects,
• level of nonlinear effects ε – percentage measure of
nonlinear effects, defined by relation
ε = (ψ − 1)% ,
(5)
• relative displacement λ – relation of maximal transverse
displacement of a cable in specified span to length of this
span (measured along x axis direction), defined by general
formula
max t w
λ=
.
l
(6)
In static and dynamic analysis we use respectively: ψs – static
nonlinear amplification factor, λs – static relative displacement and ψd – dynamic nonlinear amplification factor, λd –
dynamic relative displacement. The time t in static solutions
Figure 4. Scheme of three-span carrying cable with location of
investigated cross-sections.
3.3
Results of numerical analysis
Numerical analysis presented in this paper is limited to cable
transverse displacements at selected cross-sections located in
mid-spans and specified in Figure 4. Counterweight
movement is also analyzed, it can be treated as a maximal
longitudinal cable displacement which appears at the last
cable support: u3(x3=l3). Calculations are carried for semiinfinite flow of gondolas which are fully loaded to simulate
the largest possible load acting on 3-span carrying cable. On
the basis of obtained results, the permissible level of cable
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Proceedings of the 9th International Conference on Structural Dynamics, EURODYN 2014
1.4
displacements, corresponding with the assumed permissible
level of nonlinear effects, will be determined.
Cable transverse displacements
1.8
linear statics
1.7
nonlinear statics
1.6
(132000; 1.384)
(132000; 1.357)
1.4
1.3
1.2
0.9
0.8
0.7
104000
124000
H 0 = 70% F min
1.0
ε dop = 5%
H 0 = 40% F min
1.1
ε dop = 10%
max w 1(x 1=0.5l 1) [m]
1.5
144000
164000
184000
Mass of counterweight T [kg]
204000
224000
Figure 5. Comparison of maximal linear and nonlinear static
transverse displacements of cable at the 1st mid-span.
(132000; 1.077)
(132000; 1.060)
1.1
1.0
0.7
ε dop = 10%
0.8
0.6
104000
124000
H 0 = 70% F min
ε dop = 5%
H 0 = 40% F min
0.9
144000
164000
184000
Mass of counterweight T [kg]
H 0 = 40% F min
4.1
3.9
nonlinear statics
3.7
2.9
(132000; 2.751)
(132000; 2.636)
2.7
2.5
1.5
104000
124000
H 0 = 70% F min
1.7
ε dop = 5%
H 0 = 40% F min
1.9
144000
164000
184000
204000
224000
Mass of counterweight T [kg]
Figure 6. Comparison of maximal linear and nonlinear static
transverse displacements of cable at the 2nd mid-span.
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3.3
3.1
2.9
2.7
2.5
2.3
2.1
3.5
2.3
2.1
104000
109000
ε dop = 5%
3.1
ε dop = 10%
max w 2(x 2=0.5l 2) [m]
3.3
ε dop = 10%
max w 2(x 2=0.5l 2) [m]
3.5
224000
Presented results show that in the first range of counterweight mass values (T = 104000÷132000 kg) an influence of
nonlinear effects is significant. Maximal cable displacements
decrease monotonically when the mass T increases.
Simultaneously, nonlinear effects become smaller, so linear
and nonlinear solutions fit together finally. The convergence
of solutions is clearly visible in the second range of counterweight mass values (T = 132000÷230000 kg), when initial
cable tension is much more higher than allowable 40%Fmin.
The greatest differences between nonlinear and linear
solutions are observed in the 2nd span (for the cable
displacement w2 (x2 = 0.5l2) – see Figure 6). This is consistent
with expectations as the 2nd span is the longest one,
therefore, its displacements are largest and in consequence –
nonlinear effects are higher than in other cable spans. Taking
this under consideration, the dynamic analysis has been
limited only to cross-section x2 = 0.5l2. Figure 8 illustrates
maximal values selected from time histories of linear and
nonlinear dynamic responses w2 (x2 = 0.5l2). A range of
counterweight mass is assumed as: T = 132000÷146000 kg.
As a background, the maximal static solutions are presented.
3.9
linear statics
204000
Figure 7. Comparison of maximal linear and nonlinear static
transverse displacements of cable at the 3rd mid-span.
4.3
3.7
nonlinear statics
1.2
Graphs presented in Figures 5–7 illustrate maximal static
transverse displacements of the carrying cable at the mid-span
of each cable section, in relation to a counterweight mass T
which provides cable pre-tension. The considered displacements are calculated in a wide range of counterweight mass
values (T = 104000÷230000 kg) to show a convergence of
nonlinear and linear solutions, which can be observed when
displacements are small due to very high initial cable tension
(H0 = 2256.78 kN = 70%Fmin). However, such a high initial
tension of a rope should not be applied in real structures. In all
figures, red and black lines are applied to nonlinear and linear
solutions, respectively. Vertical dashed grey lines indicate
such values of a counterweight mass for which we notice
permissible levels of nonlinear effects εp = 10% and εp = 5%.
Vertical continuous grey line denotes the counterweight mass
T = 132000 kg corresponding to the maximal initial tension of
analyzed cable, recommended by technical requirements and
from practical point of view (H0 = 1295.40 kN = 40%Fmin).
max w 3(x 3=0.5l 3) [m]
3.3.1
linear statics
1.3
114000
linear dynamics
119000 124000 129000 134000
Mass of counterweight T [kg]
nonlinear dynamics
linear statics
139000
144000
nonlinear statics
Figure 8. Comparison of maximal linear and nonlinear
dynamic displacements of cable at the 2nd mid-span.
The nonlinear effects on dynamic displacements obtained
for lower initial cable tensions are much more significant
Proceedings of the 9th International Conference on Structural Dynamics, EURODYN 2014
when compared with static solutions. Moreover, maximal
dynamic displacements do not decrease monotonically in
contrary to static solutions.
The results of quantitative analysis of nonlinear effects on
maximal static and dynamic cable transverse displacements
w2 (x2 = 0.5l2) are synthetically summarized in Table 1. Four
characteristic values of counterweight mass (T = 104000 kg,
108000 kg, 127500 kg, 132000 kg) and corresponding values
of static and dynamic nonlinear amplification factors ψs and
ψd (defined in subsection 3.1 by formula (4)) are set together.
Table 1 also contains values of relative displacements, λs and
λd, expressed by general formula (6). They are calculated for
maximal, static and dynamic, linear (L) and nonlinear (N)
displacements, in relation to a length of the 2nd span.
Additionally, a static nonlinear amplification factor dependence on counterweight mass is presented graphically in
Figure 9 for three analyzed displacements: w1 (x1 = 0.5l1),
w2 (x2 = 0.5l2) and w3 (x3 = 0.5l3).
Table 1. Comparison of linear and nonlinear maximal cable
displacement in the 2nd mid-span: w2(x2= 0.5l2).
Dynamic solution
Mass T
[kg]
104000
108000*
1.14
1.10
1/128
1/133
1/113
1/120
1.23
1.27
1/125
1/131
1/102
1/103
127500**
1.05
1/156
1/150
1.07
1/155
1/145
132000
1.04
1/160
1/155
1.06
1/159
1/150
ψs
λs
L 1)
λs
N 1)
ψd
λd
L 1)
λd
1)
ε dop = 5%
ε dop = 10%
10
20
30
40
Time [s]
50
60
70
80
90
100
110
120
-0.50
-0.30
-0.10
0.10
0.30
0.50
0.70
nonlinear solution T=108000 kg
nonlinear solution T=127500 kg
Figure 10. Time-history of dynamic increment in cable
displacement at the 2nd mid-span (nonlinear solution).
H 0 = 40% F min
0
10
20
30
40
50
Time [s]
60
70
80
90
100
110
120
-0.1
0.4
108000
112000 116000 120000 124000 128000 132000
Mass of counterweight T [kg]
w1(x1=0,5l1)
w 1(x 1 = 0.5l 1)
w2(x2=0,5l2)
w 2(x 2 = 0.5l 2)
w
w3(x3=0,5l3)
3(x 3 = 0.5l 3)
Figure 9. Comparison of nonlinear amplification factors ψs
calculated for three considered cable cross-sections.
Judging from presented results we can come to conclusion
that maximal transverse displacements and maximal nonlinear
effects in the considered 3-span cableway appear at a midspan of the longest cable section: x2 = 0.5l2. Hence, maximal
displacement w2 (x2 = 0.5l2) has been selected to determine the
permissible level of displacements λp, for which the assumed
level of permissible nonlinear effects is not exceeded.
According to subsection 3.1, two levels of permissible
nonlinear effects are considered: εp = 10% and εp = 5%. They
are related to maximal static cable displacements. As we can
0.9
w 2(x 2=0.5l 2) [m]
nonlinear amplification factor ψ s
λ = w2max/l2, L – linear solution, N – nonlinear solution
∗
εdop = 10% (H0 = 1059.96 kN),**εdop = 5% (H0 = 1251.26 kN)
1.15
1.14
1.13
1.12
1.11
1.10
1.09
1.08
1.07
1.06
1.05
1.04
1.03
1.02
1.01
1.00
104000
0
N 1)
dynamic increment in displacement w 2(x 2=0.5l 2) [m]
Static solution
see in Figure 9, the requirements for the first level εp = 10%
(ψs ≤ 1,1) are achieved when counterweight mass amounts
T = 108000 kg, which provides initial cable tension
H0 = 1059.96 kN (see also Table 1). When counterweight
mass is given as T = 127500 kg, that assures the tension
H0 = 1251,26 kN, the requirements for more restrictive level
εp = 5% (ψs ≤ 1,05) are fulfilled (see Figure 9 and Table 1). It
is important to notice that in the case: εp = 10%, the dynamic
nonlinear amplification factor of displacement w2 (x2 = 0.5l2)
is ψd = 1.27 (see Table 1). It means that in this case a quite
high level of nonlinear influences on the dynamic cable
response is permitted – nearly 30%. In authors’ opinion, such
high nonlinear influences should not be allowed because they
lead to significant increasing the amplitude of cable vibrations
(see Figure 10) that is undesirable in view of the passengers
ride comfort as well as the bearing capacity of cable which
decreases due to material fatigue. Therefore, more restrictive
limitation should be recommended: εp = 5%. Then, all static
nonlinear amplification factors meet the requirement
ψs ≤ 1.05, and maximal value of dynamic nonlinear
amplification factor is: ψd = 1.07.
1.4
1.9
2.4
2.9
3.4
3.9
nonlinear statics:
nonlinear dynamics:
T = 127500 kg
T = 127500 kg
T = 108000 kg
T = 108000 kg
Figure 11. Time-history of nonlinear static and dynamic cable
displacements at the 2nd mid-span for two initial tension
values (when εp = 5% and εp = 10%).
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Proceedings of the 9th International Conference on Structural Dynamics, EURODYN 2014
Summing up the results obtained for the analyzed cableway
we can state that:
• the permissible level of displacements, corresponding to
recommended permissible level of nonlinear effects
εp = 5%, is λp = 1/156,
• the recommended initial cable tension which assures
satisfying the permissible level of displacements for the
considered load is H0 = 1251.26 kN and it is realized by
counterweight mass T = 127500 kg.
Time-histories of nonlinear static and dynamic cable
displacement w2 (x2 = 0.5l2) obtained for recommended initial
cable tension are presented in Figure 11. For comparison the
same responses for tension H0 = 1059.96 kN (T = 108000 kg)
are presented as well. It is clearly noticeable that in the case of
lower initial cable tension, when nonlinear effects level is
εp = 10%, dynamic cable response is too great to be accepted.
3.3.2
Counterweight displacements
Dynamic solutions are obtained for the shorter range
T = 104000÷146000 kg (Figure 13) in order to reduce
computational effort.
Maximal values of counterweight displacements are much
smaller than maximal transverse cable displacements
w2 (x2 = 0.5l2), however, nonlinear effects are much greater.
Particularly, for the lowest initial cable tension provided by
counterweight mass T = 104000 kg we observe a large
counterweight nonlinear displacement (app. 0.38 m) which is
almost doubled in comparison to linear displacement. Linear
and nonlinear solutions presented in Figures 12 and 13 are
convergent, nevertheless, even when cable tension is high (it
means that counterweight mass is more than T = 132000 kg)
nonlinear effects are still clearly noticeable. An influence of
nonlinear effects on dynamic responses is more significant
than in a static case but only when counterweight mass is in
a low range: T = 104000÷114000 kg.
Table 2. Comparison of linear and nonlinear solutions of
maximal counterweight displacements.
Let’s consider now the influence of nonlinear effects on
tensioning counterweight displacements. In Figure 12 and 13
we compared maximal linear and nonlinear displacements of
the counterweight.
Static solution
-0.40
linear statics
nonlinear statics
-0.30
-0.25
(132000; -0.216)
-0.20
-0.15
0.00
104000
124000
H 0 = 70% F min
-0.05
ε dop = 5%
H 0 = 40% F min
-0.10
(132000; -0.134)
ε dop = 10%
max counterweight displacement [m]
-0.35
144000
164000
184000
204000
224000
Mass of counterweight T [kg]
Figure 12. Comparison of maximal linear and nonlinear static
displacements of the counterweight.
H 0 = 40% F min
-0.45
-0.40
-0.35
-0.30
-0.25
-0.20
-0.10
-0.05
ε dop = 5%
-0.15
ε dop = 10%
max counterweight displacement [m]
-0.50
0.00
104000 109000 114000 119000 124000 129000 134000 139000 144000
Mass of counterweight T [kg]
linear dynamics
nonlinear dynamics
linear statics
nonlinear statics
Figure 13. Comparison of maximal linear and nonlinear
dynamic displacements of the counterweight.
Static solutions are calculated for the wide range of
counterweight mass T = 104000÷230000 kg (Figure 12) to
examine the convergence of linear and nonlinear solutions.
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uN
wN
Dynamic solution
ψs
uL
wL
104000
108000*
1.80
1.73
0.064
0.062
0.102
0.097
2.04
2.12
0.066
0.063
0.109
0.105
127500**
1.62
0.053
0.081
1.65
0.054
0.083
1)
1)
ψd
uL
wL
uN
wN
Mass
T [kg]
1)
1)
132000
1.61 0.051 0.079 1.61 0.052 0.080
u = umax = max. counterweight displacement,
w = wmax = maxt w2(x2 = 0.5l2),
L – linear solution, N – nonlinear solution
∗
εdop = 10% (H0 = 1059.96 kN),**εdop = 5% (H0 = 1251.26 kN)
1)
In Table 2 we compared values of static and dynamic
nonlinear amplification factors, ψs and ψd, calculated for four
characteristic values of counterweight mass – the same as in
Table 1. These factors are considerably greater in comparison
with analogical factors of transverse cable displacements
w2 (x2 = 0,5l2) – see Table 1. Nonlinear effects on counterweight displacements occurred to be extremely high
(ψs = 1.80÷1.61 i ψd = 2.04÷1.61). Even for the recommended
initial cable tension H0 = 1251.26 kN that corresponds with
permissible level of nonlinear effects εp = 5%, the influence of
nonlinearity on counterweight displacements remains high
and exceeds ε = 60%. Such a high level we found as specific
and unavoidable for an analyzed scheme of carrying cable
tensed by a counterweight. It can be accepted regarding all
advantages provided by a tensioning counterweight (i.e.
compensation of thermal effects, negligible small axial force
increments due to in-service load).
Table 2 lists also ratios of maximal counterweight
displacement to maximal transverse cable displacements (wmax
= maxt w2(x2 = 0.5l2)), selected from static and dynamic, linear
(L) and nonlinear (N) solutions, respectively. These ratios are
introduced to identify the percentage relation between
maximal cable and counterweight displacements. In case of
the recommended level of nonlinearity εp = 5%, maximal
static and dynamic linear counterweight displacements
Proceedings of the 9th International Conference on Structural Dynamics, EURODYN 2014
constitute app. 5% of maximal linear transverse cable
displacements, respectively, and nonlinear counterweight
displacements app. 8% of cable ones. It means that
counterweight displacements are relatively small. Moreover,
they are technically acceptable because they do not exceed
25 cm in a nonlinear case (see Figure 13).
It is necessary to underline that calculated counterweight
displacements can be also treated as maximal longitudinal
cable displacements. It results from the fact that if we neglect
elongation of vertical, tensioning cable section, the
displacement of a counterweight is roughly the same as
longitudinal cable displacement measured on the last sliding
support: u3 (x3 = l3) – see Figure 4. This cable displacement is
greater than displacements measured in any other crosssection of a cable.
4
GENERAL CONCLUSIONS
Basing on an analysis of 3-span cable scheme, the possible
methodology for evaluation of permissible displacements in a
multi-span carrying cable of bi-cable ropeway with counterweight has been presented in the paper. Analogical analysis
can be performed for any other similar scheme of a cable with
different geometrical and material parameters, and different
in-service load.
On the basis of nonlinear and linear, static and dynamic
displacement analysis, the permissible level of geometrical
nonlinearity effects has been recommended, that can be used
for any similar structures. The imposed permissible level
εp = 5% assures limitation of nonlinear influences on the
dynamic cable response that follows significant decreasing of
the vibrations amplitude. It is a very important advantage
considering problem of safety and passengers ride comfort, as
well as cable material fatigue. A displacement limitation
suggested in the paper assures that nonlinear effects do not
exceed the permissible level.
We proposed to impose a displacement limitation on the
maximal relative static transverse displacement of a cable,
calculated in static analysis. It is a representative value for all
transverse and longitudinal cable displacements and
counterweight movement because they are coupled in the
analysed scheme of cableway. It is also a representative value
for both static and dynamic responses since they are due to the
same moving load. Imposing a displacement limitation on
linear static displacements is also justified. Generally, in the
first stage of structure design, a simplified static analysis is
performed with a linear approach. Only after determination of
essential parameters, in a conceptual project, more advanced
and sophisticated analysis is held with taking into consideration dynamic effects or nonlinear effects. Computational
algorithm applied in this paper and elaborated on the basis of
theory formulated by authors in References [6–8] can be used
for both types of structural analysis.
Permissible level of cable displacements, defined in this
paper, can be recognized as one of the important parameters
that allow to assess whether analyzed cableway structure can
be exploited in changed load conditions. When the structure is
subjected to different load it is necessary to calculate a new
initial cable tension. It has been demonstrated that the
suggested algorithm is an effective tool for determining the
initial cable tension recommended for a given load. It allows
to evaluate (easily and quickly) such value of counterweight
mass which guarantees that maximal linear cable
displacement (transverse) will not exceed the permissible
level λp.
ACKNOWLEDGMENTS
Calculations have been carried out using resources provided
by Wrocław Centre for Networking and Supercomputing
(http://wcss.pl).
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