Wind-induced swinging of ropeway gondola can be reduced using dynamic absorbers. To maximize the performance of conventional dynamic absorbers, their location should be as high as possible. However, absorbers can not be installed at high positions due to interference issues with structures such as towers and stations. To address this problem, a new type of dynamic absorber that moves vertically is proposed. This absorber is composed of a mass supported by a spring. The mass moves in the radius direction (up and down) and it induces Coriolis force in the circumference direction to prevent the swing of gondola. If the natural frequency of the absorber is tuned to twice that of the gondola, the absorber moves spontaneously with a large amplitude due the resonance. This absorber is more effective when it is located at lower positions. The experiment with a small model and an actual gondola for 10 passengers were carried out and the results agreed well with the theoretical predictions

Matsuhisa, Hiroshi

Yasuda, Masashi

Vietnam Academy of Science and Technology: Journals Online

Vietnam Journal of Mechanics, VAST, Vol. 30, No. 4 (2008), pp. 291 – 298
Special Issue of the 30th Anniversary
DYNAMIC ABSORBER FOR ROPEWAY GONDOLA
USING CORIOLIS FORCE
Hiroshi Matsuhisa1 and Masashi Yasuda2
1Kyoto University, Japan
2Tokkyokiki Corp, Japan
Abstract. Wind-induced swinging of ropeway gondola can be reduced using dynamic
absorbers. To maximize the performance of conventional dynamic absorbers, their lo-
cation should be as high as possible. However, absorbers can not be installed at high
positions due to interference issues with structures such as towers and stations. To ad-
dress this problem, a new type of dynamic absorber that moves vertically is proposed.
This absorber is composed of a mass supported by a spring. The mass moves in the ra-
dius direction (up and down) and it induces Coriolis force in the circumference direction
to prevent the swing of gondola. If the natural frequency of the absorber is tuned to
twice that of the gondola, the absorber moves spontaneously with a large amplitude due
the resonance. This absorber is more effective when it is located at lower positions. The
experiment with a small model and an actual gondola for 10 passengers were carried out
and the results agreed well with the theoretical predictions.
1. INTRODUCTION
Wind loading often induces swinging of cable suspension carriers such as gondolas
and chair lifts. In general, the operation of gondola is suspended for the wind speeds greater
than 15m/s. There are two means for reducing swing of cable suspension carrier, dynamic
absorbers and gyrostabilizers. Because the structure of the gyrostabilizers is complicated
and requires a power supply, it is not suitable for cable suspension carriers. Alternatively,
passive dynamic absorbers do not have such problems. Since their effectiveness is generally
proportional to the square of the vibration amplitude, many researchers have tried to
install dynamic absorbers at the bottom of carriers. In these setups, the absorbers worked
very poorly, and it was believed that dynamic absorbers were not effective for reducing the
swing of pendulum-type structures. However, Matsuhisa et al. [1] established a new theory
regarding the location of dynamic absorber on pendulum structures in 1993. This theory
revealed that their effectiveness is proportional to the square of the distance between the
absorber and the center of oscillation. In the case of gondola, the center of oscillation is
approximately the mid-point between the center of gravity and the bottom of the gondola.
This means that absorbers located at the bottom of gondola do not work well and the
absorber should be located as high as possible. Even if the absorber is above the fulcrum
it works very well. This location theory is applicable for all kind of absorbers, which move
in the circumference direction as shown in Figure 1. Since the first installation of the
292 Hiroshi Matsuhisa, and Masashi Yasuda
dynamic absorber on the ropeway chair lifts in 1995 (Japan), dynamic absorbers have
been installed on about 20 ropeways in Japan [2]. However, there are many obstacles such
as towers and stations that prevent installation of absorbers at a high position above the
roof of gondola, then the actual absorbers are installed just above the roof without enough
distance and they can not have good performance. In this paper, a new absorber that uses
gondola
Coriolies absorberConventional absorber
mass
spring motion
motion
Fig. 1. Conventional and Coriolies absorbers on a gondola.
Coriolies force and moves in the radial direction of the pendulum-type structure as shown
in Figure 1 is introduced. This absorber can be located at the low position of gondola.
Because the Coriolies force is proportional to the mass, the angular velocity of swing, and
the velocity of the absorber’s mass in the radial direction, the effectiveness of absorber
increases rapidly with the amplitude of swing. This phenomenon is opposite to the concept
of pumping on a swing in a park, i.e., the swing angle increases very little when it is small,
but it increases rapidly when it is large. Theoretical analysis and experiments with a small
model were carried out and it was shown that the Coriolies absorber worked well. Then
the Coriolies absorbers for an actual gondola was produced and its effect was investigated.
2. THEORETICAL ANALYSIS
2.1. Equations of motion
As shown in Figure 2, a ropeway gondola can be modeled as a simple pendulum
(the center of oscillation and the center of gravity are assumed to be the same). The
model parameters are defined as the following: m1 is the mass, l1 is the length between
the gravity center G and the fulcrum (center of swing) O, c1 is the damping coefficient,
θ is the rotational angel of the pendulum, u is the displacement of the absorber, l is
the distance between the fulcrum and the absorber in the static condition, and g is the
acceleration of gravity; m2, k and c are mass, spring constant and damping coefficient of
the absorber, respectively. Assuming that the cable does not move, the locations of the
gondola and dynamic absorber masses are
Dynamic absorber for ropeway gondola using coriolis force 293
Fig. 2. Schematic diagram of pendulum with a Coriolis dynamic absorber
x1 = l1 sin θ, (1)
y1 = l1 cos θ, (2)
x2 = (l− u) sin θ, (3)
y2 = (l− u) cosθ. (4)
The kinetic energy T and the potential energy V are
T =
1
2
m1l
2
1θ˙
2 +
1
2
m2
{
(l− u)2θ˙2 + u˙2
}
, (5)
V = m1gl1(1− cos θ) +m2g(l− u)(1− cos θ) +
1
2
ku2. (6)
The energy dissipation function is
F =
1
2
c1θ˙
2 +
1
2
cu˙2 . (7)
The Lagrange’s equations of motion are obtained from Eqs. (5), (6), and (7) as
m1l
2
1θ¨ +m2(l− u)
2θ¨ + c1θ˙ − 2m2(l− u)u˙θ˙ + (m1gl1 +m2gl−m2gu) sinθ = Pl1e
jωt, (8)
m2u¨+ cu˙+ ku +m2(l− u)θ˙
2
−m2g(1− cos θ) = 0, (9)
where, P is the external force assumed to acts on the gravity center G.
The fourth term of Equation (8) is the moment caused by the Coriolis force 2m2u˙θ˙
that acts as a damping force. The fourth term of Equation (9) is centrifugal force that is
proportional to θ˙2 and this force drives m2. Therefore, the Coriolis force is approximately
proportional to θ˙3. The fifth term m2g(l−u) sin θ in Equation (8) is the moment given by
the gravity force, and this moment is also non-linear due to variation of the arm length
(l−u). Because the fourth and fifth terms in Equation (9) are the driving force on m2, the
displacement u has a resonant peak at the damped natural frequency that is two times
that of the frequency of θ. Considering the non-linearity of the damping force, the Coriolis
absorber works poorly for small swing angles, but remarkably well for large angles. This
294 Hiroshi Matsuhisa, and Masashi Yasuda
phenomenon can be experienced when one pumps a swing in a park by moving his body
up and down.
2.2. Frequency response and optimum tuning
The distance of the absorber massm2 from the fulcrum O is l−u, and u is a variable.
However, it is assumed that u is small enough compared with l and the distance is l. The
equivalent arm length of the combined pendulum of the main mass m1 and the absorber
mass m2 is
le =
m1l
2
1
+m2l
2
m1l1 +m2l
, (10)
The natural frequencies of the gondola and absorber are
ω1 =
√
g
le
, (11)
ωa =
√
k
m2
. (12)
The following non-dimensional parameters are introduced.
µ =
m2
m1
, γ =
l
le
, f =
ωa
ω1
, h =
ω
ω1
, ς1 =
c1
2m1l2eω1
, ς =
c
2m2ω1
, Θ =
P
m1g
0
10
20
30
40
50
0.8 0.9 1 1.1 1.2
frequency ratio h
θ(
de
g) 03.0=z
08.0=z
2.0=z
absorberWithout
f=1.95
0
10
20
30
40
50
0.8 0.9 1 1.1 1.2
frequency ratio h
θ(
de
g) 03.0=z
2.0=z
08.0=z
absorberWithout
f=2.05
0
10
20
30
40
50
0.8 0.9 1 1.1 1.2
frequency ratio h
θ(
de
g) 03.0=z
2.0=z
08.0=z
absorberWithout
f=2.05
Fig. 3. Frequency response of the pendulum with Coriolis absorber
Because this system is nonlinear, its response to sinusoidal excitation is numerically
calculated. The calculated frequency responses with parameters µ = 0.1, γ = 1, Θst =
0.015, ζ1 = 0.005, f=1.95, 2.001, and 2.05 and ζ= 0.03, 0.08 and 0.2 and are shown in
Figure 3. The peak response is significantly reduced by the absorber and its magnitude
depends on f and ζ. In general, the optimum values fopt is 1.99 and ζopt is between 0.06
and 0.1. When the damping is zero, fopt must be 2.0. However, fopt is decreased by a
small amount by the damping. When the exciting force is large, ζopt becomes large. The
effectiveness of the absorber depends on its mass and location. Figure 4 and 5 shows the
frequency responses for various mass ratios µ and location parameters γ, respectively,
where f =2.0, ζ=0.08, Θst=0.015, µ=0.1, and γ=1.0. The effectiveness of the absorber
is proportional to the mass ratio. The location of the absorber also has a large influence
on its effectiveness. Because motion of the absorber is caused by the centrifugal force, the
absorber must be attached to a place far from the center of swing.
Dynamic absorber for ropeway gondola using coriolis force 295
0
10
20
30
40
50
0.8 0.9 1 1.1 1.2
frequency ratio h
θ
(d
eg
) 05.0=m
1.0=m
2.0=m
absorberWithout
0
10
20
30
40
50
0.8 0.9 1 1.1 1.2
frequency ratio h
θ
(d
eg
)
)03.0,99.1(0 === zg f
)06.0,2(5.0 === zg f
)08.0,2(1 === zg f
absorberWithout
Fig. 4. Effect of mass ratio on frequency Fig. 5. Effect of location on frequency
response response
2.3. Free response
The free response of a gondola with an initial angle of θ = 30◦ is shown in Figure 6 for
f =2.0, ζ=0.1, µ=0.1, ζ1 = 0.005 and γ=1.0. This figure shows that the absorber oscillates
with twice the frequency of the pendulum. In the case of a linear system, the damping ratio
is independent on the amplitude of oscillation. However, because the Coriolies absorber is
non-linear, the damping ratio depends on the magnitude of the oscillation. Figure 7 shows
the damping ratio for each period of oscillation for three different initial angles: 30, 35
and 40 degrees. This figure shows that when the oscillation is small, the damping ratio is
negligible, but it increases with the magnitude of the oscillation.
-40
-20
0
20
40
θ
(d
eg
)
absorberWithout
absorberWith
Absorber response
-2
0
2
0 10 20 30 40
time(s)
u
(m
) 0
0.005
0.01
0.015
0 5 10 15 20 25
(deg)
n
A
deg30)0( =q
deg35)0( =q
deg40)0( =q
n
eq
z
Fig. 6. Free response of pendulum with Fig. 7. Damping ratio calculated for
Coriolies absorber each period of free vibration
2.4. Random response
Random wind force is imposed on the carrier using Davenport’s [3] formulations. The
wind-induced vibration of a gondola with Coriolies absorbers for wind velocities of 15, 20,
and 25 m/s are shown in Figure 8. In these calculations, values for parameters are selected
to model a middle-size gondola (12 passengers) asm1=790 kg (without passengers), l1=3.8
m, ζ1=0.01, f =2.0, ζ=0.1, µ=0.1, and γ=1.0. The results show that the absorber works
296 Hiroshi Matsuhisa, and Masashi Yasuda
very well. Typically, the operation of gondola without absorbers is suspended for the wind
velocity greater than 15 m/s. However, for a gondola with the Coriolies absorber, operation
is feasible up to wind speeds of 20 m/s.
Without absorber
-50
0
50
θ
(d
eg
)
With absorber
-50
0
50
0 100 200 300
time(s)
θ
(d
eg
)
Without absorber
-50
0
50
θ
(d
eg
)
With absorber
-50
0
50
0 100 200 300
time(s)
θ
(d
eg
)
With absorber
-50
0
50
0 100 200 300
time(s)
θ
(d
eg
)
Without absorber
-50
0
50
θ
(d
e
g
)
Fig. 8. Wind induced vibration of a gondola for three wind velocities
3. EXPERIMENT WITH A SMALL MODEL
Figure 9 shows an experimental apparatus composed of a pendulum and a passive
Coriolies absorber. The absorber is placed in parallel with the pendulum and its mass is
suspended by two coil springs (one is above the mass and the other is below). The damping
is provided by the electromagnetic force that is induced by a permanent magnet attached
to the absorber mass and aluminum bar on the pendulum. The swing angle is measured
using a potentiometer at the fulcrum. Values for the system parameters are: m1=0.93 kg,
l1=0.67 m, ζ1=0.002, m2=0.2 kg, l=0.65 m, µ=0.24, f=2.0, and ζ=0.045.
spring of absorber
angle sensor
mass of absorber
mass of pendulum
support stand
aluminum bar
magnet
Fig. 9. Pendulum with a Coriolies absorber
Figure 10 shows the free response of the experimental system. This response agrees
well with the theoretical prediction. Figure 11 shows the system’s frequency response. In
this experiment, instead of the external force P , the fulcrum is moved horizontally with
harmonic displacements. The resonant peak value is significantly reduced.
Dynamic absorber for ropeway gondola using coriolis force 297
0
10
20
30
40
0.85 0.95 1.05 1.15
frequency ratio h
θ
(d
eg
)
absorberWithout
absorberWith
-40
-20
0
20
40
θ
(d
eg
)
absorberWithout
absorberWith
Absorber response
-0.2
0
0.2
0 5 10 15 20
time(s)
u
(m
)
Fig. 10. Experimental results of free response Fig. 11. Experimental results of
frequency response
4. EXPERIMENT WITH AN ACTUAL MODEL
Figure 12 shows an gondola with two Coriolies absorbers. The mass of absorber is
suspended by two coil springs and the damping is given by electromagnetic induced force
with permanent magnet on the moving mass and aluminum guide. The mass of gondola is
Fig. 12. Pictures of gondola with absorbers
-30
-20
-10
0
10
20
30
0 20 40 60 80 100
Time (s)
D
e
gr
e
e
(
‹
j
Without damper
With damper
Fig. 13. Free responses of gondola with and without absorber
298 Hiroshi Matsuhisa, and Masashi Yasuda
650 kg, the natural frequency is 0.262 Hz and damping ratio is 0.0035. The mass of absorber
is 145 kg, the moving mass is 75 kg, the natural frequency is 0.531 Hz and damping ratio
is 0.055. Figure 13 shows the free response. In the first several periods, the absorber works
well, however after that the amplitude of the swing of gondola becomes small and the
absorber does not work well. This is resulted by the non-linearity of the Coriolies force.
Therefore the Coriolies absorber can reduce the large amplitude as 20 degrees to middle
amplitude as 10 degrees.
5. CONCLUDING REMARKS
In this paper, a passive absorber that uses Coriolies force to reduce vibration is
introduced. This absorber moves in the radial direction of a swinging pendulum. There-
fore, it has many differences with conventional absorbers that move in the circumference
direction. For example, the relationship between the absorbers’ effectiveness and their lo-
cation is opposite. In the case of conventional absorber on a gondola, the location should
be as high as possible, but for the Coriolies absorber, it should be as low as possible.
The damping ratio of the conventional absorber is constant, but that of the Coriolies
absorber increases rapidly with the magnitude of vibration. Considering the absorber’s
characteristics, it is applicable to many systems such as gondola, boats, and cranes.
REFERENCES
[1] H. Matsuhisa, R. Gu, Y. Wang, O. Nishihara, and S. Sato, Vibration control of a ropeway
carrier by passive dynamic vibration absorbers, Jpn. Soc. Mech. Eng. International Journal
(C), 38 4 (1995) 657-662 .
[2] H. Matsuhisa and M. Yasuda, Location effect of dynamic absorbers on rolling structures,
Asia-Pacific Vibration Conference (2003), pp.439-444.
[3] A.G.Davenport,The Spectrum of Horizontal Gustiness Near the Ground in High
Winds,Quarterly Journal of the Royal Meteorological Society 87 (1961) 194-211.
Received May 14, 2009
BỘ HẤP THỤ ĐỘNG LỰC SỬ DỤNG LỰC CORIOLIS
LẮP CHO CABIN CÁP TREO
Sự lúc rung do gió của cabin cáp treo có thể được giảm bớt nhờ việc sử dụng bộ hấp thụ
động lực. Để tối ưu hóa hiệu quả, bộ hấp thụ động lực thông thường cần có vị trí càng cao càng
tốt. Tuy nhiên, bộ hấp thụ không thể đặt ở vị trí cao do sự cản trở của các kết cấu khác như tháp
treo cáp hoặc các trạm dừng. Để giải quyết vấn đề này, một dạng bộ hấp thụ động lực mới, chuyển
dịch theo phương thẳng đứng, đã được đề xuất. Bộ hấp thụ này bao gồm khối lượng đỡ bởi lò xo.
Khối lượng chuyển động theo hướng bán kính (lên và xuống) và nó gây ra lực Coriolis theo hướng
chuyển động tròn và sẽ hạn chế sự lắc lư của cabin. Nếu tần số riêng của bộ hấp thụ được điều
chỉnh gấp 2 lần tần số của cabin thì bộ hấp thụ sẽ chuyển động với biên độ lớn do cộng hưởng. Bộ
hấp thụ này càng hiệu quả nếu được đặt ở vị trí càng thấp. Thí nghiệm với một mô hình thu nhỏ
và một cabin thật dành cho 10 hành khách đã được thực hiện và các kết quả phù hợp tốt với các
dự đoán lý thuyết.


Dynamic absorber for ropeway gondola using Coriolis force

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Dynamic absorber for ropeway gondola using Coriolis force

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