Static and dynamic analysis of the cableway

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T H E A R C H I V E O F M E C H A N I C A L E N G I N E E R I N G

VOL. LV 2008 Number 4

DOI: 10.24425/ame.2008.131632

Key words: cableway, wire ropes, catenary, rope vibrations

_JANUSZ_ _KOWAL_ ∗ _,_ _JACEK_ _SNAMINA_ ∗∗ _,_ _ANDRZEJ_ _PODSIADŁO_ ∗∗∗ _,_

_JAROSŁAW_ _KONIECZNY_ ∗∗∗∗

STATIC AND DYNAMIC ANALYSIS OF THE CABLEWAY

The paper is concerned with an analysis of behaviour of the cableway. On
the basis of design data and results of adequate experiments, a physical model of
cableway was formulated. The static of cableway was developed assuming a full
nonlinear model based on elastic catenary curve. The tension of the rope and the
reactive forces between the rope and the supports were calculated. Assuming various
loadings of the rope, the relation between the tension in bottom and upper stations
and the length of the rope was determined. The model describing the motion of the
system is linear. Finite elements were used to formulate the model. Two methods of
accelerating the system were investigated.

1. Introduction

Due to complex dynamics of cableway and random character of loads
acting in the system, the description of cableway motion is difficult. Thus,
the proper methods of experimental and numerical investigation of the system
should be worked out. To improve the system’s dynamics, the mathematical
model emulating the static and dynamic properties of cableway installations
is very helpful [1, 2].

∗ _Department_ _of_ _Process_ _Control,_ _Faculty_ _of_ _Mechanical_ _Engineering_ _and_
_Robotics,_ _University_ _of_ _Science_ _and_ _Technology;_ _Av._ _Mickiewicza_ _30,_ _30-059_ _Krakow;_
_janusz.kowal@agh.edu.pl_

∗∗ _Department_ _of_ _Process_ _Control,_ _Faculty_ _of_ _Mechanical_ _Engineering_ _and_
_Robotics,_ _University_ _of_ _Science_ _and_ _Technology;_ _Av._ _Mickiewicza_ _30,_ _30-059_ _Krakow;_
_jacek.snamina@agh.edu.pl_

∗∗∗ _Department_ _of,_ _Process_ _Control,_ _Faculty_ _of_ _Mechanical_ _Engineering_ _and_
_Robotics,_ _University_ _,of_ _Science_ _and_ _Technology;_ _Av._ _Mickiewicza_ _30_ _30-059_ _Krakow;_
_andrzej.podsiadlo@agh.edu.pl_

∗∗∗∗ _Department_ _of_ _Process_ _Control,_ _Faculty_ _of_ _Mechanical_ _Engineering_ _and_
_Robotics,_ _University_ _of_ _Science_ _and_ _Technology;_ _Av._ _Mickiewicza_ _30,_ _30-059_ _Krakow;_
_jaroslaw.konieczny@agh.edu.pl_

358 JANUSZ KOWAL, JACEK SNAMINA ANDRZEJ PODSIADŁO, JAROSŁAW KONIECZNY

The results of experiments and numerical calculation presented in this
paper concerns the cableway “Strugi Szczawiny”, running from Korbielów
to Hala Miziowa (the Pilsko Mountain, Beskid Żywiecki). There are two stations in this cableway installation: the upper station incorporating the driving
system and the bottom station with the rope tensioning system. There are 22
supports between the stations. The rope connecting the two stations acts as a
truck and a haul rope. 219 double chairs are mounted to the rope by conical
grips. The maximal loading is 16000 kg.

The main purpose of the present study is to develop a mathematical model
to be used for synthesis of a vibration control system in real life conditions.

2. Experiments

Acceleration and braking of the cableway causes local changes of rope
tension. They are the reason of vibrations. Chairs vibrations are sensed by
passengers, making them feel uncomfortable. Vibrations are more intensive
in longer installations, where rope elasticity is of key importance.

The aim of the work was identification of cableway and description of
static and dynamic behaviour of the cableway installation. An accurate model
would allow for designing such a construction, in which the negative phenomena are reduced. Extensive tests were performed, involving the measurements
of a chair’s acceleration in three directions:
x - longitudinal direction (coinciding with the rope axis and the direction
of travel),
y - transverse direction (horizontal and perpendicular to the direction of
the travel),
z - direction normal to the rope axis.
Measurements were taken with an adhesive-mounted three-axial piezoelectric accelerometer. DAQ is a signal acquisition module for conducting
high-accuracy measurements from integrated electronics piezoelectric sensors (IEPE). 24-bit resolution ADCs with 102 dB of dynamic range deliver
and incorporate IEPE signal conditioning for the accelerometers. The input
channels simultaneously digitize input signals up to 50 kS/s (kilo sample per
second). A computer was used for an introductory analysis and acquisition
of measurement signals. The measurements of acceleration were taken every
few minutes, at the same time the cableway velocity signals were recorded.
The measurement data include three ranges:
I - accelerating up to the nominal speed,
II - travel at the steady speed (the chair passes the support),
III - emergency braking between two subsequent supports.

STATIC AND DYNAMIC ANALYSIS OF THE CABLEWAY 359

_2_

_1.5_

_1_

_0.5_

_0_

_-0.5_

_0.6_
_0.4_
_0.2_
_0_

_-0.2_
_-0.4_
_-0.6_

_0.3_
_0.2_
_0.1_
_0_

_-0.1_
_-0.2_
_-0.3_

_1.5_
_1_
_0.5_
_0_

_-0.5_
_-1_

_-1.5_

I II III

_65_ _70_ _75_ _80_ _85_ _90_ _95_
_Time [s]_

Fig. 1. Rope velocity and chair acceleration components in three directions

Selected results of measurement are shown in Fig. 1. The first graph
presents velocity of the rope and the three others present components of
acceleration associated with chair vibrations.

During emergency braking, all braking systems were applied. After braking, the rope and chairs oscillate with frequency of about 0.5 Hz. The largest
accelerations are registered in the x-and z-direction. The accelerations in the
transverse direction are negligibly small.

3. Static and dynamic analysis of cableway

The rope in the cableway is subjected to forces distributed along its
length and forces applied at the end points of the overall structure. The load
distributed along the rope results from the rope’s mass and the mass of
passenger chairs attached to the rope. The rope balance system is placed at
the end point. In the installation considered in this study, the tensioning force
is produced by the weight of the mass placed at the bottom station. At the
upper end, the driving moment (or braking force) is applied to the driving
wheel.

Due to a strongly nonlinear behaviour of the rope, the analysis of static
and dynamic becomes a complex task, and the motion of a cableway during
braking and accelerating is particularly difficult to capture.

360 JANUSZ KOWAL, JACEK SNAMINA ANDRZEJ PODSIADŁO, JAROSŁAW KONIECZNY

3.1. Static analysis

The equilibrium state of cableway can be described using various methods

[4, 3]. In this calculation, a nonlinear model based on a catenary was chosen.
The equilibrium equations of a catenary passing through the points located
on supports should be solved in order to obtain the length of the rope in each
of two tracks and the vertical component of tension.

An assumption is made that the mass of each chair is uniformly distributed
along the rope. Thus, the linear density of the rope is adequately increased.

In order to describe the static equilibrium of rope in each span, the
following parametric equations of catenary were used:

- _smg_ µ
_N_ 0

��

(1)

- ( _s_ - _sm_ ) _g_ µ
_N_ 0

+ asinh

�2
+ ( _s_ - _sm_ ) 2

_x_ = _N_ 0

_g_ µ

asinh

�2
+ _sm_ 2

��0 _N_ 0
_g_ µ

_y_ =

_g_ µ

�� _N_ 0
_g_ µ

where: ( _x_, _y_ ) - coordinates in local Cartesian system, _N_ 0 - horizontal component of rope tension, µ - linear density, _g_ - gravitational acceleration, _s_ arc-coordinate, _sm_ - catenary parameter. Equations (1) are supplemented by
the equation describing the rope tension:

_N_ =

_N_ 0 2 + (( _s_ − _sm_ ) _g_ µ)2 (2)

In the above equations, the arc coordinate is equal to zero at the point
where the rope is supported ( _x_ = 0, _y_ = 0). The coordinates of the second
support point and the tension at the bottom end of the span are known.
The length of the rope and the rope tension at the upper end of the span
can be determined. The calculation starts from the bottom span, where the
tension at the end point is equal to a half of the weight of the tensioning
mass. Then, the adjacent span is taken into account in calculations. The rope
tension determined for the previous span is used in calculations concerning
the next span.

The tension at the bottom end of the rope is equal to _60_ kN. Taking into
account the mass of passenger chairs, the unit mass was assumed to be _17.4_
kg/m (chairs are empty) or _28.1_ kg/m (full load - chairs with passengers).
The local coordinate system is introduced in each span. Selected results of
calculations are presented in Fig. 2. The graphs illustrate the forces at the
upper and bottom points of the rope as functions of the length of rope in two
cases: with full load (Fig. 2a) and without the load of passengers (Fig. 2b).

STATIC AND DYNAMIC ANALYSIS OF THE CABLEWAY 361

× 10 5

1.8

1.6

1.4

1.2

1.0

0.8

0.6

0.4

0.2

0.0

~~lower force~~
upper force

1546 1547 1548 1549 1550

length [m]

~~lower force~~
upper force

× 10 5

1.8

1.6

1.4

1.2

1.0

0.8

0.6

0.4

0.2

0.0

1546 1548 1550 1552 1554

length [m]

Fig. 2. Rope tension in bottom and upper stations versus the length of rope: a) full load, b) chairs

are empty

It is apparent that the driving friction wheel has to be blocked when one
rope is loaded by passengers, and the second one is loaded by empty chairs
only. Using the characteristics presented in Fig. 2, we can easily calculate
the static displacement of the mass in the tensioning system due to nonsymmetric loading of ropes.

3.2. Study of dynamic behaviour

The dynamic behaviour of the cableway can be described using the system
of ordinary and partial differential equations of motion [5]. They are difficult
to solve analytically. For this reason, we introduce a simplified, finite-element
model of the cableway. Finite-element method is frequently used in solving
the static and dynamic problems of rope structures [6].

Similarly as in the static calculations, we assume that the mass of each
chair is uniformly distributed along the rope. The bottom and upper wheels,
as well as the mass in tensioning system, are assumed to be rigid bodies.
The coordinate _x_ 1 describing the displacement of the tensioning mass and
the angles ϕ1, ϕ2 describing the revolutions of wheels are joined the nodal
displacements _q_ 1,...., _q_ 40 of finite elements that describe the motion of the
rope. Finally, the following vector of generalized coordinate is defined [ _x_ 1,
_r_ ϕ1, _q_ 1, ..., _q_ 20, _r_ ϕ2, _q_ 21, ..., _q_ 40] _T_ . We assume that at _t_ = 0 the nodes
are located at the point placed on supports (cableway towers). Thus, each
span is modelled by only one finite element. The coordinates are shown
schematically in Fig. 3.

362 JANUSZ KOWAL, JACEK SNAMINA ANDRZEJ PODSIADŁO, JAROSŁAW KONIECZNY

Fig. 3. Scheme of coordinates used in calculations

To create the FEM model of a rope, linear interpolation functions are
applied to each line element. The tangent displacement _ue_ inside the element
is interpolated using the corresponding nodal displacements by:

_ue_ = ψ ¯ _T_ _q_ ¯ _e_ (3)

where: _q_ ¯ _e_ = [ _qe_ 1, _qe_ 2] _T_ is the nodal displacements vector for element and

ψ¯ = [ψ1, ψ2] _T_ is the vector of interpolation functions.

The normal displacement _we_ inside the element can be interpolated using nodal displacement and quasi-static equations. Thus, the analysis of the
cableway motion can be reduced to solving a system of ordinary differential
equations.

The calculations were performed for the following parameters: length of
each rope _l_ = 1548 m, stretching stiffness _EA_ = 1.9 × 10 8 N, unit mass of
the rope with full chairs 28.1 kg/m, unit mass of the rope with empty chairs
17.4 kg/m.
It is apparent that the motion is a superposition of translation and vibration motion. The natural frequencies and modes are the basis for dynamic
analysis of the systems. The spectrum analysis shows that the amplitudes
of vibrations significantly increase for the frequencies close to the system’s
natural frequencies. Using the FE model of the cableway, one can calculate
the natural frequencies and mode shapes. The results are shown in Tables 1
and Fig 4.

Table 1.
Non-zero natural frequencies

Mode
Number

Natural frequency

Number [Hz]

1 0.40

2 0.93

3 1.24

4 1.87

5 2.12

6 2.83

1.0

0.5

0.0

-0.5

-1.0

STATIC AND DYNAMIC ANALYSIS OF THE CABLEWAY 363

1.0

0.5

0.0

-0.5

-1.0

0 10 20 30 40

coordinate number

0 10 20 30 40

coordinate number

Fig. 4. The first and the second mode shapes of vibration

Taking into account the results of the modal analysis, the following conclusions can be formulated:

- Because of the large rope length, the first non-zero frequency is very low.
Neighbouring natural frequencies are separated, in general, only by about
0.4 Hz. Thus, the cableway is very sensitive to any loads and disturbances.

- The displacement of tensioning mass takes the maximum value for the
modes in which the displacements in each rope have the same directions.
When the full loaded cableway is started, displacements and accelerations
strongly depend on the driving moment control. Two cases of the control were
considered. In the first case, the dynamic component of driving moment is
constant up to the moment when the velocity reaches the desired value. In
the second case of the control, the angular acceleration of the driving wheel
is constant until the velocity achieves the desired value.

Selected results of calculations of the cableway behaviour under a constant driving moment are presented in Figs. 5-7. The constant driving moment
is applied to the driving wheel for 11.5 s. Fig.5 shows the dynamic component
of driving force (the dynamic component of driving moment divided by the
radius of driving wheel) as a function of time.

15000

10000

5000

0.0 2.5 5.0 7.5 10.0 12.5
time [s]

Fig. 5. Dynamic component of driving force

364 JANUSZ KOWAL, JACEK SNAMINA ANDRZEJ PODSIADŁO, JAROSŁAW KONIECZNY

Arc displacement of the point placed on the driving wheel and its tangent
velocity are presented in Fig. 6. As shown, the increase of velocity is not a
smooth function. The motion has an oscillating component.

12.5

10.0

7.5

5.0

2.5

0.0

0.0 2.5 5.0 7.5 10.0 12.5
time [s]

2.0

1.5

1.0

0.5

0.0

0.0 2.5 5.0 7.5 10.0 12.5
time [s]

Fig. 6. Arc displacement and tangent velocity of the point placed on driving friction wheel

Fig.7 presents the displacement and the velocity of the tensioning mass. As
shown, the main frequency of vibrations is equal to the first natural frequency
(Tab. 1). The motion of tensioning mass is very important for operation of
the cableway because it is directly associated with proper tension of the rope.

0.04

0.02

0.00

-0.02

-0.04

0.0 2.5 5.0 7.5 10.0 12.5
time [s]

0.10

0.05

0.00

-0.05

-0.10

0.0 2.5 5.0 7.5 10.0 12.5
time [s]

Fig. 7. Displacement and velocity of tensioning mass

STATIC AND DYNAMIC ANALYSIS OF THE CABLEWAY 365

In the following calculations it was assumed that the cableway accelerates
from 0 to 1.5 m/s in 11.5 s and the acceleration of driving wheel is constant.

In this case, the motion of cableway is different than that in the case with
constant dynamic moment. Some selected results, arranged in the same way
as in the previous case, are presented in Figs. 8-10.

15000

10000

5000

-5000

-10000

-15000

0.0 2.5 5.0 7.5 10.0 12.5

time [s]

Fig. 8. Dynamic component of driving force

As shown in Fig. 8, the driving force has an oscillating component in the
period when acceleration of the driving wheel is constant, and later when
the driving wheel rotates continuously.

12.5

10.0

7.5

5.0

2.5

0.0

0.0 2.5 5.0 7.5 10.0 12.5
time [s]

2.0

1.5

1.0

0.5

0.0

0.0 2.5 5.0 7.5 10.0 12.5
time [s]

Fig. 9. Arc displacement and tangent velocity of driving wheel

366 JANUSZ KOWAL, JACEK SNAMINA ANDRZEJ PODSIADŁO, JAROSŁAW KONIECZNY

0.04

0.02

0.00

-0.02

-0.04

0.0 2.5 5.0 7.5 10.0 12.5
time [s]

0.10

0.05

0.00

-0.05

-0.10

0.0 2.5 5.0 7.5 10.0 12.5
time [s]

Fig. 10. Displacement and velocity of tensioning mass

The values of displacement and velocity of the tensioning mass are greater
than those in the case with constant moment. During acceleration of the
cableway, the displacement of tensioning mass takes positive values. The
positive value of displacement is associated with the decrease of rope tension
(see Fig. 3) and a simultaneous decrease of forces between the rope and the
support rollers placed on the supports.

In order to prevent this effect and improve the system’s dynamics, a proper
damper can be mounted in the tensioning system. The damper does not act
on the mass when it stretches the rope. The damper prevents the motion of
the mass when it releases the rope. The results of numerical simulation are
presented in Fig. 11. As in previous calculations, it was assumed that the
angular acceleration of motion of the driving wheel is constant. The damping
coefficient equals 1.5×10 5 Ns/m. It was found out by a process of trial and
error.

The displacements presented in Fig. 11 are more advantageous than those
presented in Fig. 10. During acceleration of the cableway with additional
damper, the displacement of the tensioning mass is negative. This means
that the rope is stretched better than in the case without the damper.

STATIC AND DYNAMIC ANALYSIS OF THE CABLEWAY 367

0.02

0.01

0.00

-0.01

-0.02

0.0 2.5 5.0 7.5 10.0 12.5
time [s]

0.10

0.05

0.00

-0.05

-0.10

0.0 2.5 5.0 7.5 10.0 12.5
time [s]

Fig. 11. Displacement and velocity of tensioning mass with additional damper

4. Conclusion

A cableway is a very complex system. It is important to have an accurate,
general model which makes it possible to predict effects of possible dynamic
behaviours of the cableway in various situations. In this paper, a model using
finite-element method is presented. In spite of simplifying assumptions, the
model is useful for calculations of characteristics of the overall system e.g.
accelerating and braking of the cableway, and the behaviour of the tensioning
system. The model should be extended, if necessary, to describe the motion
of some subsystems. Based on the results of the performed calculations, we
can draw the following conclusion:

- As expected, the cableway is very sensitive to any loads and disturbances.
The first non-zero natural frequency is very low. The neighbouring natural
frequencies are separated, in general, only by about 0.4 Hz.

- During accelerating, the motion of the cableway depends on the driving
moment control. Two cases of the control were considered. In the case
when the angular acceleration of the motion wheel is constant, disadvantageous phenomena occur in tensioning system, which is associated with
large displacement of the tensioning mass.

- In order to prevent the disadvantageous phenomena, a proper damper can
be mounted in the tensioning system. The damper does not act on the
mass when it stretches the rope. The proposed solution can be realized
by applying a viscous damper with directional flow valve.

368 JANUSZ KOWAL, JACEK SNAMINA ANDRZEJ PODSIADŁO, JAROSŁAW KONIECZNY

This research work is supported by the state Committee for Scientific
Research as a part of the research program No. KBN 4 T07A 005 30.

Manuscript received by Editorial Board, June 06, 2008;

final version, October 27, 2008.

REFERENCES

[1] Bruno D., Leonardi A., 2001. Nonlinear structural models in cableway transport systems Sim
ulation Practice and Theory, 7 (1999), pp. 207-218.

[2] Brownjohn J. M. W., 1998. Dynamics of an aerial cableway system Engineering Structures,

Vol. 20 No.9, pp. 826-836.

[3] Irvine, H. M., 1981. Cable structure. Cambridge, Massachusetts: MIT Press.

[4] Shan Q., Zhai W. M., 1998. A macroelement method for catenary mode analysis Computer

and Structures, pp. 767-772.

[5] Simpson A., 1972. Three-dimensional vibration of travelling elastic cables Journal of Sound

and Vibration. 20. pp. 177-189.

[6] Sung-Pil Chang, Jung-Il Park, Kyoung Chan Lee, 2008. Nonlinear dynamics analysis of spatial
ly suspended elastic catenary cable with finite element method. Journal of Civil Engineering.
Vol. 12, No. 2, pp. 121-128.

Analiza statyczna i dynamiczna kolejki linowej

S t r e s z c z e n i e

W pracy przedstawiono ogólną analizę statyczną i dynamiczną kolejki linowej. Na podstawie danych konstrukcyjnych oraz wyników odpowiednio dobranych pomiarów zbudowano model
obliczeniowy konstrukcji. W obliczeniach statycznych wykorzystano model nieliniowy, linię statycznej równowagi liny opisano krzywą łańcuchową. Wyznaczono siłę naciągu oraz siły oddziaływania liny na podpory. Do opisu dynamiki zbudowano model liniowy wykorzystujący metodę elementów skończonych. W obliczeniach dynamiki układu rozpatrzono dwa sposoby rozruchu kolejki.
Wyniki obliczeń zostały zilustrowane na wykresach.

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