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Elevator Rope Tension Analysis with Uneven
Groove Wear of Sheave
To cite this article: Daisuke Nakazawa et al 2018 J. Phys.: Conf. Ser. 1048 012006
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1234567890 ‘’“”
The 7th Symposium on Mechanics of Slender Structures IOP Publishing
IOP Conf. Series: Journal of Physics: Conf. Series 1048 (2018) 012006 doi :10.1088/17426596/1048/1/012006
Elevator Rope Tension Analysis with Uneven Groove Wear of
Sheave
Daisuke Nakazawa
1
, Seiji Watanabe
1
, Daiki Fukui
2
, Ayaka Fujii
1
and Ken Miyakawa
1
1
Advanced Technology R&D Center, Mitsubishi Electric Corp., 811 Tsukaguchi Honmachi,
Amagasaki, Hyogo, Japan
2
Inazawa Works, Mitsubishi Electric Corp., Hishimachi, Inazawa, Aichi, Japan
Email: Nakazawa.Daisuke@df.MitsubishiElectric.co.jp
Abstract. Traction elevators are suspended by multiple ropes. Uneven sheave wears affect the rope
tension during the elevator operation. If the tension condition reaches to the critical traction ratio, a
rope slip occurs. As the rope slip also affects the rope tension, a tension calculation with rope slip is
necessary to evaluate the relation between the groove wear and the rope tension. In this paper, a
tension evaluation model is derived by including the rope slip behavior and then the influence of the
rope tension due to the groove wear is evaluated.
1. Introduction
In the traction elevator system, a car is suspended by multiple ropes. Ideally, each rope tension should be
maintained as the balanced condition. However, the tension fluctuates due to the elongation of the rope, the
variation of rope stiffness or the uneven sheave groove wear. If the tension of each rope is unbalanced
severely, the elevator system is out of condition. Therefore, an analytical method is required to evaluate the
rope tension behavior. Since the uneven groove wear of the sheave in highrise elevators causes a significant
tension change even in a small amount of wear, it is more important to establish the tension analysis with the
groove wear than other factors. Therefore, clarification of the relationship among the groove wear, tension
and traveling distance can contribute to the elevator’s system design. Concerning the influence of the groove
wear, the relationship between the amount of wear and the tension is investigated by Togawa et al [1]. While
the above work mainly focuses on the experimental study of the phenomenon, an analysis of the tension
change and the influence of groove wear have not been studied in detail. In this paper, firstly we describe the
elevator’s vertical vibration model with multiple ropes. Secondary, the mechanism of rope tension behavior
with the uneven groove wear is explained. Finally, as the proposed model also includes the rope slip, the
influence of the slip due to the unbalanced tension is evaluated by simulation and experiment.
2. Analytical Model
We derive an analytical model of the elevator that considers tension difference among the multiple ropes,
and the rope slip on the sheave. Figure 1 shows the proposed model. In the model, ropes wrapped around a
traction machine are connected to a car and a counterweight (CWT) via shackles. The ropes are modeled as
springs. To evaluate the slip of the ropes on the sheave, the parts of ropes on the sheave are modeled as
intensified masses. The equation of elevator’s vertical motion of the elevator system is derived by the
following equation.
+ + =
Mx Cx Kx F
(1)
where M, C and K are the inertia matrix of the system, the damping matrix and the stiffness matrix
respectively. F is the external force vector which contains the gravity force, the traction force and the torque
by the traction machine. The elevator’s transient motion is calculated by the numerical integration of
equation (1).
2
1234567890 ‘’“”
The 7th Symposium on Mechanics of Slender Structures IOP Publishing
IOP Conf. Series: Journal of Physics: Conf. Series 1048 (2018) 012006 doi :10.1088/17426596/1048/1/012006
Figure1. Elevator model
Figure2. Slip model
[
]
( 3) ( 3)
n n
c w
diag M M m m J
+ × +
= ∈M R
(2)
( 3) ( 3)
0 0
0 0
0 0 0
0 0 0
0 0 0
0 0 0 0 0 0
c c c c
w w w w
c w c w n n
c w c w
c w c w
nk k k k
nk k k k
k k k k
k k k k
k k k k
+ × +
− − −
− +
= ∈
− +
− +
K R
(3)
[
]
3
11 +
=
∈−= ∑
n
T
n
i sisnswc
fruffgMgM RF
(4)
[
]
3
21 +
∈=
n
T
nwc
xxxxx Rx θ
(5)
Note that the damping matrix
C
consists of the damping elements which are proportional to the stiffness ones.
Each element of
K
is quantified by the specification of the rope and the shackle spring stiffness.
C
is
determined by the comparison with the experimental results which are shown in the following sections. The
model parameters are shown in Table 1 where m is the intensified mass of the rope on the sheave. Other part
mass of the rope is included in M
c
or M
w
.
Table1. Model parameters
M
c
Car mass M
w
CWT mass
m
Rope mass on the sheave x
c
Displacement of car
x
w
Displacement of CWT x
i
Displacement of rope mass on the sheave
θ
Rotation angle of sheave r Sheave radius
J
Sheave moment of inertia g Gravity acceleration
c
k
~
Car side rope stiffness with shackle spring
w
k
~
CWT side rope stiffness with shackle spring
u
Torque n Number of the ropes
The rope slip can be checked by the ratio of the car side rope tension T
ci
and the CWT side rope tension T
wi
,
as shown in figure 2.
The ratio is compared with the critical traction ratio of the sheave Γ which is defined as
p
e
µκθ
Γ =
(6)
where µ is the coefficient of friction, κ is the coefficient of the groove, θ
p
is the wrapping angle of the rope
on the sheave. In equation (4), f
si
(i=1,…, n) is the force acting between the rope and the sheave. The force
is switched according to the condition of the slip as follows.
(i) No slip condition
The condition that the rope does not slip is written as T
ci
< ΓT
wi
where T
ci
is larger than T
wi
. On the other hand,
if T
ci
is smaller than T
wi
, the no slip condition is given by ΓT
ci
> T
wi
[2]. Therefore, by summarizing the
equations, the no slip condition is
Γ<<Γ
ciwi
TT1
(7)
k
c
k
s
c
k
~
k
w
w
k
~
M
c
M
w
x
c
x
w
m
1
m
n
x
1
x
n
J
θ
k
s
k
c
k
w
・・・
・・・・・・
・・・
Counter
weight
Car
Rope
Traction
sheave
Shackle
spring
・・・
・・・・・・
・・・
・・・
・・・・・・
・・・
θ
p
x
i
θ
T
wi
T
ci
3
1234567890 ‘’“”
The 7th Symposium on Mechanics of Slender Structures IOP Publishing
IOP Conf. Series: Journal of Physics: Conf. Series 1048 (2018) 012006 doi :10.1088/17426596/1048/1/012006
In this case, the rope and the sheave move with the same velocity, because the tension ratio of T
ci
and T
wi
is
within the critical traction ratio. Hence, the following constraint equation is given as the no slip condition.
0 ( 1,2, )
i i
C r x i n
θ= − = =
(8)
Each constraint equation can be summarized as the vector form
1
, , ,
T
i n
C C C
= =
C 0
(9)
Differentiating equation (9) with respect to time yields
0
d
t dt
∂ ∂
= = =
∂ ∂
x
C C x C x
x
(10)
where
x
C
is Jacobian of
C
which can be written as
( 3)
0 0 1 0 0 0
0 0 0 1 0 0
0 0 0 0 1 0
0 0 0 0 0 1
n n
r
r
r
r
× +
−
−
∂
≡ = ∈
−
∂
−
x
C
C R
x
(11)
Then, equation (1) and (10) can be written in the matrix equation as
T
s
− −
=
x
x
x
M C
F Cx Kx
f0
C 0
(12)
[ ]
3
0 0 ,
Tn
c w
M g M g u
+
= ∈F R
[ ]
1
, , ,
T
n
s s si sn
f f f= ∈
f R
where constraint force
f
s
is calculated as the vector of Lagrange multipliers.
(ii) Slip condition
When the tension ratio of T
ci
and T
wi
is beyond the critical traction ratio, the rope experiences the slip on the
sheave to keep the ratio equal to the critical traction ratio. Then the friction force f
si
in equation (13) acts on
the rope and the sheave, respectively.
Γ>Γ−
Γ<−Γ
=
ciwici
ciwiwi
si
TTT TTT
f/,)1( /1/,)1(
(13)
3. Tension Analysis without Rope Slip
In this section, we show the tension change mechanism under the uneven groove wear condition by
simulations and experiments. At first, we evaluate the tension behavior with respect to the car position
without the rope slip. In the following discussion, we suppose that there is no passenger in the car, and the
elevator consists of two ropes and the one of the sheave groove has a certain amount of wear. As the number
of the ropes is two, the degree of freedom in the model equation (1) is five.
The groove wear is equivalent to the decrease of the sheave radius as shown in figure 3. It causes the
difference of each rope’s winding length on the sheave when the car goes up or down. This effect to the rope
length causes the tension change as shown in figure 4. The horizontal axis shows the normalized car position.
Its origin corresponds to the bottom floor, and the top floor is set to one. On the other hand, the vertical axis
shows the tension which is normalized by the car side tension at the bottom floor. The mechanism of the
tension change in the upward motion illustrated in figure 4 can be explained as follows;
1) The length of the rope 2 on the worn groove in the car side becomes longer than the rope 1 by the car
upward motion. Also the length of the rope 2 in the CWT side becomes shorter than the rope 1, because
the amount of winding on the sheave is smaller than one of the rope1.
2) The rope length difference causes the tension imbalance. The tension of rope 2 in the car side and the
tension of rope1 in the CWT side decrease according to the car upward motion as shown in figure 4(a).
3) Since the rope length in the car side becomes shorter by the car upward motion, the tension changes
sharply. On the other hand, as the rope length in the CWT side becomes longer, the tension change is
4
1234567890 ‘’“”
The 7th Symposium on Mechanics of Slender Structures IOP Publishing
IOP Conf. Series: Journal of Physics: Conf. Series 1048 (2018) 012006 doi :10.1088/17426596/1048/1/012006
slower than the car side. In figure 4(b), the tension ratio of rope 2 becomes larger and it finally exceeds the
critical traction ratio.
The tension change mechanism described above can be also expressed by the following functions of the car
position.
δ
β
δ
β
δ
β
δ
β
+
+=
+
−=
+−
−=
+−
+=
x
x
K
gM
T
x
x
K
gM
T
x
x
K
gM
T
x
x
K
gM
T
w
w
w
w
c
c
c
c
2
,
2
,
1
2
,
1
2
2121
(14)
LEAK =
(15)
In the above equations, βx is the change of the rope length due to the groove wear where β is defined as β =
(1r
2
/r
1
)/2. δ =OH/L is the ratio of the remaining rope length OH on the top floor. x is the normalized car
position which is defined as x=(TRx
c
)/TR where TR is the elevator shaft height. E, A and L are the Young’s
modulus, the cross section area of the rope, and the distance from the bottom to the top floor, respectively. In
Eq.11, the second
term of the right hand side in all four equations represents the tension change as the curved
line in figure 4(a). Note that we ignore the stiffness of the shackle spring in the derivation of equation (14)
for the simplicity. When the traction ratio reaches the critical traction ratio, the rope2 on the worn groove
slips with respect to the sheave in the actual system.
Figure3. 2rope elevator model with groove wear
(a)Normalized tension
(b)Tension ratio
Figure4. Simulation result of rope tension without slip (upward motion)
4. Tension Analysis with Rope Slip
In this section, the rope tension is measured by an elevator testbed to evaluate the rope tension behavior with
rope slip. The configuration of the testbed is the same as figure 3. The tension sensors are attached to each
rope end. Table.2 shows the conditions of the experiments. Figure 5 shows the comparison result between
the simulation and experiment. As shown in figure 5, the simulation corresponds to the experimental data.
The tension in the car upward motion behaves according to the following steps.
A) The tension change occurs based on 1) and 2) described in the previous section. The tension ratio of the
rope2 reaches the critical traction ratio Γ. Then the rope 2 slips on the sheave.
B) Even after the tension ratio of the rope 2 reaches the critical traction ratio, the tension tries to change due
to the length difference of the rope by the groove wear. As the traction ratio can’t exceed the critical
traction ratio, the tension ratio sticks to Γ, and the rope continues to slip until arriving at the top floor.
0 0.2 0.4 0.6 0.8 1
0
1
2
3
Car position
Tension
Tc1 Tc2 Tw1 Tw2
0 0.2 0.4 0.6 0.8 1
0
2
4
6
Car position
Tension ratio
Tw1/Tc1
Tw2/Tc2
Traction capacity
Down Up
Worn
groove
Rope1 Rope2
Car
Counter
weight
T
c1
T
c2
T
w1
T
w2
Tension
sensor
Tension
sensor
5
1234567890 ‘’“”
The 7th Symposium on Mechanics of Slender Structures IOP Publishing
IOP Conf. Series: Journal of Physics: Conf. Series 1048 (2018) 012006 doi :10.1088/17426596/1048/1/012006
Table2. Experimental conditions
Elevator shaft height [m]
9.4
Car mass [kg] 292
CWT mass [kg] 457
Amount of wear [mm] 0.6
Number of ropes 2
Tension sensor (Load cell)
KYOWA LCWC
10KN25SA3
When the tension ratio reaches the critical traction ratio, the slope of tension curve changes as shown in
figure 5. Since the car and the CWT are suspended by multiple ropes, a tension change of a certain rope
affects the tension balance. Thus, the tensions of the other ropes also change accordingly. Since the tension
ratio with rope slip x
s
can be written as (T
w2
k
w
x
s
)/(T
c2
+k
c
x
s
)= Γ where k
w
=K/(x+δ) and k
c
=K(1x+δ), the
amount of rope slip x
s
can be expressed by the following equation.
0 0.2 0.4 0.6 0.8 1
0
1
2
3
Car position
Tension [kN]
Tc1(exp)
Tc1(sim)
Tc2(exp)
Tc2(sim)
T
w
1
(exp)
T
w
1
(sim)
T
w2
(exp)
T
w2(sim)
(a)Tension
0 0.2 0.4 0.6 0.8 1
0
1
2
3
Car position
Tension ratio
Γ
Γ
/1
Tw1/Tc1(exp)
Tw1/Tc1(sim)
Tw2/Tc2(sim)
Critical
t
raction
ratio
Tw2/Tc2(exp)
(b)Tension ratio
Figure5. Rope tension with rope slip behavior (Upward motion)
(Solid Line: Experiment, Dotted Line: Simulation)
+−
Γ+
+
=
Γ−
=δδ xx
KD
DTT
x
cw
s
111
,
22
(16)
Each of the rope tensions (
1
ˆ
c
T
,
2
ˆ
c
T
,
1
ˆ
w
T
,
2
ˆ
w
T
) after the slip occurrence is given by the following equations.
δδδδ +
−=
+
+=
+−
+=
+−
−=
x
x
KTT
x
x
KTT
x
x
KTT
x
x
KTT
s
ww
s
ww
s
cc
s
cc 22112211
ˆ
,
ˆ
,
1
ˆ
,
1
ˆ
(17)
The tension change is mitigated by the effect of the second term of each equation in equation (17), and that
corresponds to the slope change of the tension in figure 5(a).
In the following part, we evaluate the relationship between the tension and the amount of the groove wear by
the simulation. figure 6 shows the calculated tension during the car upward motion with various groove wear
conditions. The vertical axis shows the tension which is normalized by the car side tension at the bottom
floor. The slip starting position is shifted to the lower floor by a larger amount of groove wear as shown in
figure 6(b). As a larger amount of wear increases the length difference of each rope, the tension ratio reaches
Γ at the lower floor. Note that the tension follows the same line after the slip as shown in figure 6(a). To
evaluate the above behavior, we transform the first equation of equation (17) by using equation (14) and
equation (16). The tension after the rope slip can be written as the following equation.
)1(222
ˆ
1
δ+−
Γ−−= xD K
gMgMgM
T
cwc
c
(18)
Since equation (18) does not include β, it can conclude that the tension after the rope slip doesn’t depend on
the amount of wear, and the effect of the wear is eliminated by the rope slip.
6
1234567890 ‘’“”
The 7th Symposium on Mechanics of Slender Structures IOP Publishing
IOP Conf. Series: Journal of Physics: Conf. Series 1048 (2018) 012006 doi :10.1088/17426596/1048/1/012006
0 0.2 0.4 0.6 0.8 1
1
1.5
2
2.5
Car position
Tension
Tc1 Tc2 Tw1 Tw2
α
1
(a)Normalized tension
0 0.2 0.4 0.6 0.8 1
0
0.5
1
1.5
2
Car position
Tension ratio
Tw1/Tc1 Tw2/Tc2 Critical traction ratio
α
Γ
Γ
/1
(b)Tension ratio
Figure6. Simulation result about the influence of the groove wear
(Groove wear condition : Thick line
×
1.5, Medium line
×
1 ,Thin line
×
0.5)
Finally, we discuss the rope tension behavior under the round trip condition. Figure 7 shows the rope
tension with the rope slip during the downward motion of the return trip. When the car runs downward
from the top floor, the car side rope tension T
c2
increases and the CWT side rope tension T
w2
decreases.
This is because the amount of winding against the rope2 is smaller than the rope1. The rope1 also
indicates the tension change with respect to the tension change of the rope2. Then the rope1 begins to
slip on the sheave due to the above tension change. Figure 8 shows the tension behavior in the second
and the third round trips in a row. The tension shows a hysteresis loop, and it follows the same line in
every run. According to equation (18), the rope tension after the slip doesn’t depend on β. Thus, the
tension at the top or bottom floor is determined independently from the amount of wear. Therefore, if
the initial tension condition is the same in every run, it concludes that the tension shows the same
hysteresis loop.
Figure7. Rope tension with rope slip
(Downward motion in first round trip)
(Solid Line: Experiment, Dotted Line: Simulation)
Figure8. Rope tension with rope slip
(consecutive round trips)
(Solid Line: Experiment, Dotted Line: Simulation)
5. Conclusions
In this study, the rope tension modeling of the elevator system was described. The proposed model can
evaluate the rope slip behavior. The mechanism of the tension change due to the increase of the sheave
groove wear was explained by the simulation and the experimental results. Under the condition that
the car runs between the bottom floor and the top floor, we get the following results.
(a) The slip starting position is shifted to the lower floor.
(b) The rope on the worn groove slips when the car runs upward. On the other hand, the counterpart
rope slips in the downward motion.
(c) The relationship between the rope tension and the car position shows the hysteresis loop, which is
the same shape in every round trip.
Since the proposed model can properly simulate the tension behavior, the model can be applied to
estimate the influence of the unbalanced tension conditions for the elevator system design.
References
[1] K. Togawa, S. Arai, M. Uwatoko: Influence of Traction Sheave P.C.D. Difference on Sheave
and Rope (in Japanese), Proceedings of Elevator, Escalator and Amusement Rides Conference,
Tokyo, Japan, 2012, pp.3134.
[2] Gina Barney: Education & Training Sweet e
f
α
?, ELEVATORI November/December, pp69,
(Volpe Editore srl, Milan 2001)
0 0.2 0.4 0.6 0.8 1
0
1
2
3
Car position
Tension [kN]
0 0.2 0.4 0.6 0.8 1
0
1
2
3
Car position
Tension [kN]
Tc1(exp)
Tc1(sim)
Tc2(exp)
Tc2(sim)
T
w
1
(exp)
T
w
1
(sim
T
w2(exp)
T
w2(sim)
Tc1(exp)
Tc1(sim)
T
w
1
(exp)
T
w
1
(sim)