Effects of Hard Stopping of the Ski Lift
on Dynamical Strain of the Pulling
Rope
Slobodan To{i}
Professor
University of Belgrade
Faculty of Mechanical Engineering
Svetomir Simonovi}
The City of Belgrade Assembly
The ski lift has been modeled as per of the Theory of Small Oscillations of
a material system with finite number of degrees of freedom of movement,
as shown at the Fig. 1. The material system of the ski lift is allowed to six
degrees of freedom of movement, that take into account the elastic
properties of the pulling ropes and the lower foothold, as well as the
mobility of rotating and translatory masses at the upper foothold. Upon
determination of the main forms of oscillation and corresponding modal
columns, on the basis of initial parameters, the laws of ski lift movement
were set. Dynamical forces at lower parts of the up-going and downgoing rope tracks were determined by means of the said laws of
movement. The ratios of dynamical rope strains amount to 3.96 and 2.26
respectively, implying that considerable dimensions of the pulling ropes
must be set if their breakage at hard stops is to be avoided.
Keywords: Ski lift, pulling rope, static force, dynamic force, laws of
movement.
1. INTRODUCTION
Ski lift is a machine where elastic elements (ropes)
play an important role, as their task is to transfer
movement to relatively large masses originating from
the skiers’ presence and from the weight of the ropes
themselves. Consequently, the ski lift is extremely prone
to oscillations that are translated along the full length of
the rope line and affect the elements of the ski lift
structure in a very complex way. Any change in the
uniformity of ski lift movement is, therefore, a strong
impulse for appearance of oscillations. This paper is a
research into the effects of hard stopping which can
occur, for example, as a result of power supply failure.
The research was made on the basis of the Theory of
Small Oscillations and accompanied by a numerical
example.
2.
ELASTIC—INERTIAL MODEL OF THE SKI LIFT
The Fig. 1 shows an elastic inertial material system
of the ski lift, that is the system comprising elastic and
inertial elements. Moment of inertia of the lower pulley
is neglected, because his movement around vertical axe
is dumped by AC drive and gear unit. Longitudinal
oscillations of the up-going and down-going ropes are
neglected because users of the ski lift are leaned on the
Received: September 2001, revised: February 2002, accepted:
March 2002.
Correspondence to:
Slobodan To{i}, Faculty of Mechanical Engineering,
27. marta 80, 11000 Belgrade, Yugoslavia
© Faculty of Mechanical Engineering, Belgrade. All rights reserved.
ground and represent longitudinal rigidity during the
movement.
Where :
Lower foothold
m= 450 kg
Lower foothold gear mass
m1= 500kg
lower foothold height
l= 2.5 m
Lower foothold moment of inertia
J0= 1000 kg m2
Rope rigidity
C= 30 kN/m
Reduced mass of the down-going rope m3= 5300 kg
Reduces mass of the up-going rope
m4= 11300 kg
Moment of inertia of the upper foothold pulley
J2= 234 kg
m2
Lower foothold pulley diameter
r= 2.5 m
Upper foothold gear mass
m2= 400kg
Tensile rope rigidity
C1= 45 kN/m
Counter weight mass
m5= 10000 kg
m3 represents the sum of masses of the pulling rope of
the down-going track and the empty towing devices
concentrated at the mid-span.
m5 represents the masses of the pulling rope of the upgoing track, towing devices and the skiers concentrated
at the mid-span.
FME Transactions (2002) 30, 11-14
11
 ∂E p

 ∂ϕ

 = 0 ⇒ Cϕϕ st − C f st1 l − C f st3l −
0
− m1 gl sin ϕ st − 1 mgl sin ϕ st = 0
2
 ∂E p 

 = 0 ⇒ − C f st2 r − C f st4 r = 0
∂
θ

0
Fig. 1 Dynamic model of the ski lift
3.
DETERMINATION OF COEFFICIENTS OF
RIGIDITY AND INERTIA OF THE MATERIAL
SYSTEM SHOWN AT THE FIGURE 1
Potential energy of the fictive spiral spring of the
lower foothold
E p1 = 1 C ϕ (ϕ st + ϕ) 2 − 1 C ϕ ϕ 2st .
2
2
Potential spring energy of the down-going track
E p 2 = 1 C ( f st1 + x3 − l ϕ) 2 − 1 C ( f st1 ) 2 +
2
2
1
2
+ C ( f st + x2 − x3 − r θ) 2 − 1 C ( f st2 ) 2
2
2
Potential spring energy of the up-going track
E p3
= 1 C ( f st3 + x 4 − lϕ) 2 − 1 C ( f st3 ) 2 +
2
2
+ 1 C ( f st4 + x 2 − x 4 + rθ) 2 − 1 C ( f st4 ) 2
2
2
Potential spring energy of the tension rope
E p 4 = 1 C1 (ϕ 5st + x 5 − x 2 ) 2 − 1 C ( f st5 ) 2 .
2
2
Potential weight energy of the lower foothold
E p 5 = m1gl cos(ϕ + ϕst ) − m1gl cos ϕst
+ 1 mgl cos(ϕ + ϕst ) − 1 mgl cos ϕst
2
2
Potential weight energy of the down-going track
E p 6 = m3 g (h3 + x3 sin β) − m3 gh3 .
Potential weight energy of the up-going track
E p 7 = m 4 g (h4 + x 4 sin β) − m 4 gh4 .
Potential weight energy of the tension weight
E p 8 = m 5 g ( ht − x t ) − m 5 ght .
Total potential energy
8
Ep =
E pi
∑
i =1
.
As all generalized forces are equal to zero in the
point of stable balance the following six conditions are
obtained:
12 ▪ Vol. 30, No 1, 2002
 ∂E p

 ∂x2

 = 0 ⇒ C f st2 + C f st4 − C1 f st5 = 0 ,
0
 ∂E p

 ∂x3

 = 0 ⇒ C f st1 − C f st2 + m3 g sin β = 0 ,
0
 ∂E p 

 = 0 ⇒ C f st3 − C f st4 + m4 g sin β = 0 ,
 ∂x4  0
 ∂E p 
5


 ∂x  = 0 ⇒ C1 f st − m5 g = 0 .
 5 0
On the basis of said six conditions the expression for the
total potential energy is deduced to homogenous square
form:
{[
)]
(
E p = 1 Cϕ + 2Cl 2 − m + m1 gl ϕ 2 + 2Cr 2θ 2 +
2
2
+ (2C + C1 ) x22 + 2C x32 + 2C x42 + C1 x52 −
− 2Clϕ x3 − 2Clϕ x4 + 2Crθ x3 − 2Crθ x4 −
− 2C x2 x3 − 2C x2 x4 − 2C1 x2 x5
}
If the generalized coordinates are numerically marked as
follows:
q1 = x 5 ; q 2 = x 4 ; q 3 = x 3
q 4 = x 2 ; q5 = θ ; q6 = ϕ
On the basis of the expression for potential energy the
rigidity matrix [c ] is obtained, whose minors meet the
Sylvester criterion.
Kinetic energy of the material system of the ski lift
The kinetic energy system reads
[(
)
E k = 1 J 0 + m1l 2 ϕ 2 + J 2 θ 2 +
2
+ m 2 x 22 + m3 x 32 + m 4 x 42 + m5 x 52
]
and from this expression the matrix of inertia (a)
coefficients is obtained according to already introduced
numerical expression of generalized coordinates.
4. DETERMINATION OF THE FREQUENCY OF MAIN
FORMS OF OSCILLATION AND
CORRESPONDING MODAL COLUMNS
We start from the matrix
[ A] = [c]−1[a ]
After elimination of m-1 form of oscillation m-th form
of oscillation is obtained by means of iteration formula:
FME Transactions
[ A]m {µ jm +1 } = Λ m +1{µ jm +1 } ,
{A} = [µ]−1{q(0)} ,
where Λ r = 1 / ω2r is the reciprocal value of the square
of angular speed of frequency of the r-th main
oscillation form, to which corresponds the modal
column µ jr - r-th main form of system oscillation.
{ }
The matrix
It can be further shown that
(vm ) = const.(µ m )[a] .
For m defined modal columns the Gauss elimination
procedure gives:
( m −1)q + ... + v ( m −1)q = 0 .
vmm
m
ms
s
The matrix [S ]m is diagonal except in the m-th series
which looks as follows:
− 0.506697
1.045217
− 0.720186
− 0.125912
0.353827
0.15⋅10−3
0.157885 − 0.631⋅10−4
− 0.10307 − 0.376⋅10−4
− 0.52517 − 0.809⋅10−4
− 0.11525 0.368⋅10−2
0.0848
0.104⋅10−4
− 0.29⋅10−3 − 0.56⋅10−7
− 0.128⋅10−8 0.208⋅10−13 

0.485⋅10−5 − 0.697⋅10−11 
− 0.104⋅10−4 − 0.148⋅10−10 

0.308⋅10−6
0.631⋅10−13 
− 0.117⋅10−2 − 0.662⋅10−13 
− 0.270⋅10−8 0.409⋅10−7 
[ ]
Where in the matrix Adij A jj = A j
j = 1,....,6 and
Akj = 0 for k ≠ j .
The matrix of coefficients with sines is a zero
matrix. Taking into account the law of movement, the
material system of the ski lift model in the form of
matrix reads for this set of initial parameters
{q} = [K ] {cos ωt} .
.
v ( m −1)
mm
6. CONCLUSION
ω1 = 1,206 s −1; ω2 = 1,932 s −1; ω3 = 2,825 s −1;
ω4 = 16,63 s −1; ω5 = 41,423 s −1; ω6 = 197,192 s −1.
5. DETERMINATION OF LAWS OF MOVEMENT OF
THE SKI LIFT MATERIAL SYSTEM
When all model columns and frequencies of main
forms of oscillations are defined the laws of movement
of the observed material system can be determined. For
this determination initial parameters are used, which, for
the hard stop case (j=3 m/s2) read:
The static force at the lower end of the up-going
track is:
stat =
Fodl
m5 g
− m4 g sin β = 13441 N .
2
The static force at the lower end of the down-going
track is:
stat =
Fdol
m5 g
− m3 g sin β = 32360 N .
2
Dynamic force at the lower end of the up-going track is:
din = C x − lϕ ≈ Cx = Cq .
∆Fodl
( 4
)
4
2
Dynamic force at the lower end of the down-going track
is:
m4 j 11300 ⋅ 3
=
= 1,13 m ,
C
30000
din = C ( x − lϕ) ≈ Cx = Cq .
∆Fdol
3
3
3
x3 (0 ) = − x4 (0 ) = −1,13 m ,
From the matrix [K ] it can be deduced that:
x3 (0 ) 1,13
=
= 0,452 rad .
r
2,5
In tabular presentation, the initial parameters read:
θ(0 ) =
q 2 ≈ 0,187 cos 1,206 t + 1,04 cos 1,932 t − 0,103 cos 2,825 t
⇒ q 2 max ≈ 1,33 m
Table 1.
i
qi (0 )
1
0
2
1,13
3
-1,13
4
0
5
0,452
6
0
q i (0 )
0
0
0
0
0
0
Laws of movement, presented as matrix, look as
follows:
{q} = [µ] {A cos ωt} + [µ] {B sin ωt}
The matrices of the column of constants {A} and {B}
are determined as follows:
FME Transactions
[K ] = [µ][Adij ] =
( m −1)
vms
In such way modal matrix [µ] and angular speeds are
determined.
x 4 (0 ) =
The matrix of coefficients with cosines is :
 0.349

 0.188
 0.115

 0.237
 0.0145
 0.14⋅10−3

[ A]m = [ A]m −1[S ]m .
v ( m −+11)
0 0 0 ... − mm
... −
( m −1)
vmm
{Bω} = [µ]−1{q (0 )} = {0} ⇒ {B} = 0 .
q3 ≈ 0,115 cos 1,206 t − 0,72 cos 1,932 t 0,525 cos 2,825 t
⇒ q3 max ≈ 1,36 m
It follows that:
din
∆Fodl
max = Cq 2 max = 39900 N
din
∆Fdol
max = Cq 3 max = 40800 N
so that the dynamic coefficients at the lower ends of upgoing and down-going tracks amount to:
Vol. 30, No1, 2002
▪ 13
stat =
K odl
din =
K odl
stat + ∆ stat
Fodl
Fodl max
stat
Fodl
stat
stat
Fdol
+ ∆Fdol
max
stat
Fdol
= 3,96 ,
= 2,26 .
Accordingly, the pulling ropes must have large
dimensions in order to sustain the accidental stoppages.
REFERENCES
[1] Radosavljević Lj., Male oscilacije materijalnog
sistema sa konačnim brojem stepeni slobode,
Treće izdanje, Mašinski fakultet u Beogradu, 1986.
[2] Дукелъский A., Подвесые канатные дороги и
кабельны краны, Машиностроение, Москва,
1966.
[3] Simonović S., Analiza uticajnih parametara na
dinamičko ponašanje ski lifta, magistarski rad,
Mašinski fakultet u Beogradu, 1995
14 ▪ Vol. 30, No 1, 2002
UTICAJ NAGLOG ZAUSTAVQAWA SKI LIFTA NA
DINAMI^KO OPTERE]EWE NOSE]E-VU^NOG
U@ETA
S. To{i}, S. Simonovi}
Ski lift je modeliran teorijom malih
oscilacija materijalnog sistema sa kona~nim
brojem stepeni slobode kretawa kako je to
prokazano na slici 1. Materijalnom sistemu ski
lifta je dozvoqeno {est stepeni slobode
kretawa koji uzimaju u obzir elasti~na svojstva
nose}e-vu~nih u`adi i doweg oslonca, kao i
pokretqivosti obrtnih i translatornih masa
na gorwem osloncu. Posle odre|ivawa glavnih
oblika oscilovawa i odgovaraju}ih modalnih
kolona, na osnovu po~etnih uslova postavqeni
su zakoni kretawa ski lifta. Primenom zakona
kretawa odre|ene su dinami~ke sile u u`adima
na dowim delovima grana uspona i spusta.
Koeficijenti
dinami~kih
optere}awa
respektivno iznose 3,96 i 2,26 {to pokazuje da
ako se moraju odrediti velike dimenzije nose}evu~nih u`adi ski lifta ako se `eli izbe}i
wuhovo kidawe pri naglom zaustavqawu.
FME Transactions