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.

Analysis of nonlinear effects in a carrying rope of cableway subjected to moving load

Knawa-Hawryszków Marta1, Bryja Danuta2 1Institute of Civil Engineering, Wrocław University of Technology, Wyb. Wyspiańskiego 27, 50-370 Wrocław, Poland 2Institute of Civil Engineering, Wrocław University of Technology, Wyb. Wyspiańskiego 27, 50-370 Wrocław, Poland

email: marta.knawa@pwr.edu.pl, danuta.bryja@pwr.edu.pl

Proceedings of the 9th International Conference on Structural Dynamics, EURODYN 2014Porto, Portugal, 30 June - 2 July 2014

A. Cunha, E. Caetano, P. Ribeiro, G. Müller (eds.)ISSN: 2311-9020; ISBN: 978-972-752-165-4

3813

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 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 2. Model of an inclined carrying cable subjected to

moving in-service load.

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.

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: longitudinal displacement ui (xi, t) and transverse 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:

−−−=++ qθCqθBFKqqCqB &&&&&& ),(~),(~)(~ ttt (1)

)(),(~

),(~ N qRθθKqθK θ −− tt ,

qθKqθCqθBθSθcθJ qqq ),(~

),(~

),(~

}{}{}{ tttg −−−=++ &&&&&& , (2)

where B, C, K are mass, damping and structural stiffness block matrices, qqqθ KCBKKCB

~,

~,

~,

~,

~,

~,

~ are respective

matrices dependent on time t and pendulum angular displacements θ , )(~ tF 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

3814

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:

.)()(

~

}{)(~

)(~

)(~

}{)(~

)(~

}{)(~

)(~

N

−=

+

+

++

+

0

qRFθ

q

SK

KKK

θ

q

cC

0CCθ

q

JB

0BB

q

θ

qq

t

gt

tt

t

t

t

t&

&

&&

&&

(3)

System of equations of motion (3) remains nonlinear and coupled, however coefficients of equations are only time-dependent (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

L

N

max

max

X

X

t

t=ψ , (4)

where X = X(xi, t) is any considered response of the system in analyzed cross-section xi of a cable in i

th 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

)%1( −= ψε , (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

l

wtmax=λ . (6)

In static and dynamic analysis we use respectively: ψs – static nonlinear amplification factor, λs – static relative displace-ment and ψd – dynamic nonlinear amplification factor, λd – dynamic relative displacement. The time t in static solutions

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:

• %10p =ε , when 1,1max

maxLs

Ns

s ≤=w

w

t

tψ for every ws,

• %5p =ε , when 05,1max

maxLs

Ns

s ≤=w

w

t

tψ 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 semi-infinite 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 kgm

2, Jl = 197 kgm2.

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 semi-infinite 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

Proceedings of the 9th International Conference on Structural Dynamics, EURODYN 2014

3815

displacements, corresponding with the assumed permissible level of nonlinear effects, will be determined.

3.3.1 Cable transverse displacements

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 displace-ments 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).

(132000; 1.357)(132000; 1.384)

0.7

0.8

0.9

1.0

1.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8

104000 124000 144000 164000 184000 204000 224000

Mass of counterweight T [kg]

max

w1(

x1=

0.5

l 1)

[m]

linear statics nonlinear statics

H0

= 4

0%

Fm

in

H0

= 7

0%

Fm

in

ε do

p =

10

%

ε do

p =

5%

Figure 5. Comparison of maximal linear and nonlinear static transverse displacements of cable at the 1st mid-span.

1.5

1.7

1.9

2.1

2.3

2.5

2.7

2.9

3.1

3.3

3.5

3.7

3.9

104000 124000 144000 164000 184000 204000 224000

Mass of counterweight T [kg]

max

w2(

x2=

0.5l

2) [

m]

linear statics nonlinear statics

H0

= 4

0%

Fm

in

H0

= 7

0% F

min

(132000; 2.751)(132000; 2.636)

ε do

p =

10

%

ε do

p =

5%

Figure 6. Comparison of maximal linear and nonlinear static transverse displacements of cable at the 2nd mid-span.

0.6

0.7

0.8

0.9

1.0

1.1

1.2

1.3

1.4

104000 124000 144000 164000 184000 204000 224000

Mass of counterweight T [kg]

max

w3(

x3=

0.5l

3) [

m]

linear statics nonlinear statics

(132000; 1.060)(132000; 1.077)

H0

= 4

0%

Fm

in

H0

= 7

0%

Fm

in

ε do

p =

10

%

ε do

p =

5%

Figure 7. Comparison of maximal linear and nonlinear static transverse displacements of cable at the 3rd mid-span.

Presented results show that in the first range of counter-

weight 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 counter-weight 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.

2.1

2.3

2.5

2.7

2.9

3.1

3.3

3.5

3.7

3.9

4.1

4.3

104000 109000 114000 119000 124000 129000 134000 139000 144000Mass of counterweight T [kg]

ma

x w

2(x

2=0.

5l2)

[m]

linear dynamics nonlinear dynamics linear statics nonlinear statics

H0

= 4

0%

Fm

in

ε dop

= 1

0%

ε do

p =

5%

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

3816

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 depen-dence 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).

Static solution Dynamic solution

Mass T [kg] ψs λs

L 1) λsN 1) ψd λdL 1) λdN 1)

104000 1.14 1/128 1/113 1.23 1/125 1/102

108000* 1.10 1/133 1/120 1.27 1/131 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 1) λ = w2max/l2, L – linear solution, N – nonlinear solution ∗εdop = 10% (H0 = 1059.96 kN),** εdop = 5% (H0 = 1251.26 kN)

1.001.011.021.031.041.051.061.071.081.091.101.111.121.131.141.15

104000 108000 112000 116000 120000 124000 128000 132000Mass of counterweight T [kg]

non

linea

r am

plifi

catio

n fa

cto

r ψs

w1(x1=0,5l1) w2(x2=0,5l2) w3(x3=0,5l3)

H0 =

40

% F

min

ε dop

= 1

0%

ε dop

= 5

%

w 1(x 1 = 0.5l 1) w 2(x 2 = 0.5l 2) w 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 mid-span 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

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.

-0.50

-0.30

-0.10

0.10

0.30

0.50

0.70

0 10 20 30 40 50 60 70 80 90 100 110 120Time [s]

dyn

amic

incr

emen

t in

dis

pla

cem

ent

w2(

x2=

0.5l

2) [

m]

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).

-0.1

0.4

0.9

1.4

1.9

2.4

2.9

3.4

3.9

0 10 20 30 40 50 60 70 80 90 100 110 120Time [s]

w2(

x2=

0.5

l 2)

[m]

T = 127500 kg T = 108000 kgT = 127500 kg T = 108000 kg

nonlinear statics:nonlinear dynamics:

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%).

Proceedings of the 9th International Conference on Structural Dynamics, EURODYN 2014

3817

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

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.

(132000; -0.134)

(132000; -0.216)

-0.40

-0.35

-0.30

-0.25

-0.20

-0.15

-0.10

-0.05

0.00

104000 124000 144000 164000 184000 204000 224000

Mass of counterweight T [kg]

ma

x co

unte

rwe

ight

dis

pla

cem

ent

[m

]

linear statics nonlinear statics

H0

= 4

0%

Fm

in

H0

= 7

0%

Fm

in

ε do

p =

10%

ε do

p =

5%

Figure 12. Comparison of maximal linear and nonlinear static displacements of the counterweight.

-0.50

-0.45

-0.40

-0.35

-0.30

-0.25

-0.20

-0.15

-0.10

-0.05

0.00

104000 109000 114000 119000 124000 129000 134000 139000 144000Mass of counterweight T [kg]

max

cou

nter

wei

ght

disp

lace

men

t [m

]

linear dynamics nonlinear dynamics linear statics nonlinear statics

H0

= 4

0%

Fm

in

ε do

p = 1

0%

ε do

p = 5

%

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.

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.

Static solution Dynamic solution

Mass T [kg] ψs

L

L

w

u 1) NN

w

u 1) ψd LL

w

u 1) NN

w

u 1)

104000 1.80 0.064 0.102 2.04 0.066 0.109

108000* 1.73 0.062 0.097 2.12 0.063 0.105

127500** 1.62 0.053 0.081 1.65 0.054 0.083

132000 1.61 0.051 0.079 1.61 0.052 0.080 1) 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) 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 counter-weight 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

3818

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 cross-section 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 counter-weight 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 conside-ration 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).

REFERENCES [1] J. W. Brownjohn, Dynamics of aerial cableway system., Engineering

Structures, vol. 20, p. 826–836, 1998. [2] T. Shioneri, M. Nagai, Study on vibrational characteristics of ropeway

transport system. Proceedings of Asia-pacific Vibration Conference, Korea, 1997.

[3] R. Petrova, K. Hoffman, R. Liehl, Modelling and simulation of bi-cable ropeways under cross-wind influence, Mathematical and Computer Modelling of Dynamical Systems, vol. 13(1), p. 63–81, 2007.

[4] B. Portier, Dynamic phenomena in ropeway after a haul rope rapture. Earthquake Engineering and Structural Dynamics, vol.12, p. 433–449, 1984.

[5] H.C. Renezeder, A. Steindl, H. Troger, On the dynamics of circulating monocable aerial ropeways. In: WILEY-VCH Verlag GmbH & Co. KGaA, editors. PAMM Proceedings of Applied Mathematics and Mechanics, vol. 5, p. 123–124, Weinheim, 2005.

[6] M. Knawa, D. Bryja, Dynamics of an inclined cableway with moving cars : modelling and computation method. Proceedings of the Twelfth International Conference on Civil, Structural and Environmental Engineering Computing, Funchal, Madeira, Portugal, 1-4 September 2009, ed. by B. H. V. Topping, L. F. Costa Neves, R. C. Barros. [Stirling] : Civil-Comp Press, 2009.

[7] M. Knawa, D. Bryja, Modeling problems of steeply inclined cableway subjected to moving load., Proceedings in Applied Mathematics and Mechanics, vol. 9 (1), p. 263-264, 2009.

[8] D. Bryja, M. Knawa, Computational model of an inclined aerial ropeway and numerical method for analyzing nonlinear cable-car interaction, Computers & Structures, Civil Comp Ltd. and Elsevier Ltd., vol. 89, p. 1895-1905, 2011.

Proceedings of the 9th International Conference on Structural Dynamics, EURODYN 2014

3819