*Corr. Author’s Address: Jinan University,GuangDong,China, txue@jnu.edu.cn 170
Strojniški vestnik - Journal of Mechanical Engineering 70(2024)3-4, 170-180 Received for review: 2023-11-07
© 2024 The Authors. CC BY 4.0 Int. Licensee: SV-JME Received revised form: 2024-01-09
DOI:10.5545/sv-jme.2023.852 Original Scientific Paper Accepted for publication: 2024-02-26
Forced Vibration of Time-Varying Elevator Traction System
Sun, J. – Xu, P. – Chen, M.L. – Xue, J.H.
Jie Sun
1
 – Peng Xu
2,3
 – Mingli Chen
1
 – Jianghong Xue
1,*
1
Jinan University, School of Mechanics and Construction Engineering, MOE Key Laboratory  
of Disaster Forecast and Control in Engineering, China 
2
Qingdao Xinghua Intelligent Equipment Co., China 
3
South China University of Technology, School of Civil Engineering & Transportation, China
This paper proposes a theoretical model for the forced vibration of time-varying elevator traction systems caused by eccentric excitation of 
the traction machine. Based on the Hamilton principle and a variational principle with variable boundaries, the equations of motion and the 
complex boundary condition of a time-varying elevator traction system are established. A quantitative formula between the angular velocity 
of the traction machine, the diameter of the traction wheel, the elevator running distance and the running time is put forward. It is found that 
when the maximum values of acceleration and deceleration exceed 1.5 m/s², the elevator traction system may undergo resonance.
Keywords: elevator traction system, vibration, time-varying, dynamics, numerical analysis
Highlights
• 	 The equation of motion and the corresponding boundary condition for forced vibration of a time-varying elevator traction 
system is derived based on the Hamilton principle and a variational principle with variable boundaries.
• 	 A quantitative formula is proposed to describe the relationship between the angular velocity of the traction machine and the 
diameter of the traction wheel, the running distance, and the running time of the elevator.
• 	 It is found that the greater the running acceleration of the elevator, the higher the vibration amplitude and frequency of the 
elevator traction system.
• 	 To prevent the occurrence of resonance, the maximum value of acceleration and deceleration should not exceed 1.5 m/s
2
.
0  INTRODUCTION
With the continuous emergence of high-rise and 
super-high-rise buildings, the demand for high-
speed elevators continues to grow. Traction elevators 
are widely used in high-rise buildings due to their 
advantages, such as low energy consumption and 
simple structure [1]. Fig. 1 illustrates an elevator 
traction system.
Fig. 1.  Traction system structure
However, due to the increase in lifting height 
and operating speed, it is much more difficult to 
maintain the dynamic stability of the elevator traction 
system, and the elevators are more sensitive to 
external interferences and are more prone to undergo 
abnormal vibrations [2]. These abnormal vibrations 
may not only affect the working performance of the 
elevator but also accelerate the wear and fatigue of 
transmission components, shorten the service life of 
the elevator, and eventually lead to the occurrence of 
safety accidents.
Many scholars concentrate on the investigation 
of the mechanical properties of wire rope. Meng et al. 
[3] brought forward a theoretical approach to examine 
the influence of inter-wire contact on the mechanical 
performances of wire rope strands subjected to 
tension and torsion loads based on the thin rod and 
the elastic contact theories. Zhang et al. [4] proposed a 
theoretical model to investigate the dynamic torsional 
characteristics of the hoisting rope and its internal 
spiral components under tension. The variational 
trend of the twist angle of the wire rope, the lay angle, 
and the lay length of the spiral strand with the lifting 
time were discussed. By considering the double-helix 
structure in multi-strand configuration, Xiang et al. 
[5] introduced a new method to compute the local 
deformation of the curvatures and the twist of each 
wire. It was found that different friction states between 
adjacent wires can lead to a significantly different 

Strojniški vestnik - Journal of Mechanical Engineering 70(2024)3-4, 170-180
171 Forced Vibration of Time-Varying Elevator Traction System
distribution of local bending and torsion deformation 
of double-helix wire. The elastic-plastic deformation 
of metallic strands and wire ropes under axial–
torsional loads were analysed by Xiang et al. [6] based 
on the thin rod theory of Love and the frictionless 
assumption of Foti et al. [7], using analytical and 3D 
finite element methods. For wire ropes with different 
strand shapes, Stanova et al. [8] developed a 3D solid 
model by using ABAQUS software and compared 
the predicted responses for the strands with different 
shapes and constructions.
The wire ropes of the elevators are flexible with 
low damping, so they are prone to vibrations [9]. Peng 
et al. [10] studied the free vibration of elevators in the 
vertical direction under normal operating conditions 
theoretically and experimentally. They established a 
centralized mass discretization model and obtained 
the theoretical solution. Xu et al. [11] proposed a 
theoretical model for analysing the tension-torsion 
coupling vibration of an elevator traction system. 
According to the Hamiltonian principle, the equation 
of motion of the elevator traction system with tension-
torsional coupling effect is derived and is solved using 
the Newmark-β method. Based on Bayesian network 
theory, Zhang et al. [12] put forward a multi-state 
Bayesian network model for the horizontal vibration 
of an elevator. Their result can provide quantitative 
evaluation for the reliability of the multi-state 
horizontal vibration of the elevator. Much research has 
been done to investigate the vibrations of the ropes in 
the elevator system. 
Nevertheless, many external interferences, such 
as the unevenness of the guide rail, the eccentricity of 
the traction wheel, the airflow disturbance in the lift 
well and the systematic intrinsic frequency, impact 
the vibration of the elevator traction system, which 
would significantly affect riding comfortability and 
safety. Such a situation is dire for elevators used in 
ultra-high buildings where the lifting height and the 
length of wire rope are increased significantly [13] 
to [15]. By employing the Hamiltonian principle, 
Bao et al. [16] conducted theoretical analysis for 
the nonlinear transverse vibration of a flexible 
hoisting rope with time-varying length and discussed 
resonance occurring in a passage of the hoisting 
system due to certain periodic external excitation. 
Based on the parameters of a typical double-drum 
Blair multi-rope (BMR) winding plant, Kaczmarczyk 
[17] proposed a non-linear model to investigate the 
non-linear dynamic phenomena of a hoisting cable 
system. The forced vibration of the mine hoisting 
cables induced by the periodic excitation resulting 
from the crossover cable motion on the winder drum 
is analysed. Zhang et al. [18] analysed the sensitivity 
of random parameters of transverse and horizontal 
vibration of high-speed elevator lifting systems based 
on the random perturbation method and analysed the 
influence of parameters, such as guide length, weight 
per unit length and bending stiffness, on the dynamic 
characteristics of a guide. Cao et al. [19] established a 
trackway-car coupled vibration model via numerical 
analysis to explore the influence of guide parameters 
on the lateral vibration of the elevator. The results 
show that the structural parameters of the guide have 
a significant influence on the vibration of the elevator 
system. Zhang et al. [20] studied the horizontal 
vibration in elevator vertical motion caused by uneven 
(or non-uniform) elevator guide rails. As the elevator 
traction systems are installed inside buildings, the 
swaying of the builds will lead to the vibration of the 
elevator traction system. Thus, the vibration response 
of the elevator traction systems is not only dependent 
on their structural parameters but also affected by the 
operating environment and building excitation [21] to 
[23]. Except for the vibration response of the elevator 
traction system, several researchers focused on the 
vibration control of the elevator in order to reduce the 
adverse impact of vibration on the safety performance 
of the elevator traction system. Nguyen et al. [24] 
studied the vibration of high-rise elevator ropes under 
earthquake excitation and verified the effectiveness 
of the elevator rope vibration suppression controller 
through numerical simulation. Knezevic et al. [25] 
proposed a synergistic solution based on the jerk 
control and the upgrade of the speed controller with 
a band-stop filter to solve the vibration problem in 
the process of elevator operation. Zhang et al. [26] 
developed a model of a cable conveyor system with 
arbitrarily variable lengths. The Hamiltonian principle 
was applied to derive the governing equations of 
motion. The approximate numerical solution of the 
governing equation was obtained by using the assumed 
mode method. Based on the precise integration 
method and Latin hypercube sampling method, Qiu 
et al. [27] proposed a design parameter optimization 
method for reducing the horizontal vibration of high-
speed elevators.
The above-mentioned literature mainly focused 
on the free vibration caused by traction acceleration, 
the transverse vibration due to uneven guide rail, and 
the vibration control and shock absorption design. 
However, studies regarding the longitudinal vibration 
induced by the eccentric excitation of traction wheels 
are quite limited. In fact, during the operation of the 
elevator, with the increase of the angular speed of the 
traction wheel, the acceleration of the traction system 

Strojniški vestnik - Journal of Mechanical Engineering 70(2024)3-4, 170-180
172 Sun, J. – Xu, P. – Chen, M.L. – Xue, J.H.
will increase, leading to the improvement of the 
eccentric excitation. When the angular speed of the 
traction wheel reaches a certain value, the frequency 
of the longitudinal vibration generated by the eccentric 
excitation of the traction wheel will be close to the 
natural frequency of the system itself, resulting in the 
occurrence of resonance phenomenon, which greatly 
affects the safety of the elevator operation. Such a 
phenomenon has not been discussed in the existing 
literature. In this paper, the longitudinal vibration of 
elevator traction systems with and without external 
excitation is investigated by considering the time-
varying characteristics of elevator traction systems. An 
analytical model is employed to analyse the vibration 
performance of the elevator traction system subjected 
to the eccentric excitation of traction wheels. The 
relationship between the acceleration of the elevator 
traction system and the angular velocity of the traction 
wheel is proposed, and the influence of elevator 
operating acceleration on the resonance of the elevator 
is discussed. A safe range of acceleration is proposed 
in order to prevent the resonance of the elevator. 
The results from this study provide a reference for 
improving elevator safety and seismic design.
1  METHOD
1.1  Time-Varying Elevator Traction System 
An elevator traction system mainly consists of a 
traction wheel, a lifting car, and a wire rope. The 
wire rope is simplified as a variable-length cord 
moving along the axis; the lifting car is simplified as 
a rigid body suspended at the lower end of the cord 
with a mass of m. Fig. 2 is an analytical model for 
the longitudinal vibration of time-varying elevator 
traction systems. A coordinate system is established 
with the origin located at the tangent point of the wire 
rope and pulley, and the positive direction is chosen as 
the downward direction, as shown in Fig. 2. During the 
movement, the longitudinal displacement at a spatial 
position x(t) on the wire rope is described as u(x(t), t). 
The elevator traction system has an acceleration a(t) 
and an axial velocity v(t) corresponding to a wire rope 
length x(t). The analytical model for the longitudinal 
vibration of a time-varying elevator traction system 
is based on the following assumptions. The elevator 
traction system only undergoes longitudinal vibration. 
During operation, the wire rope is always in tension 
and complies with Hooke's Law, and its elastic 
modulus remains constant along the whole length.
Fig. 2.  Time-varying longitudinal vibration model  
of elevator traction system
According to the finite deformation theory of 
continuum mechanics, the displacement of the wire 
rope at time t at a spatial position x(t) is
 lx tu xt t   




() ,. (1)
The total velocity V at the spatial position x(t) of 
the wire rope is the summation of the axial velocity 
v(t) of the elevator and the vibration velocity at x(t) 
and is expressed as:
 V
dl
dt
vt ut vu t
tx
      




. (2)
where the subscript of x or t denoted the partial 
differentiation with respect to x or t. 
The kinetic energy of the elevator traction system 
is expressed as follows.
 Em vV dx
kc
xlt
lt



05 05
22
0
.. ,
()
()
 (3)
where the first term is the kinetic energy of the lifting 
car, and the second term is the kinetic energy of the 
wire rope. 
Since the lifting car is considered to be a rigid 
body, the elastic strain energy of the elevator traction 
system is that of the wire rope, which is formulated as 
follows:
 EP u EAu dx
sx
lt
x


(. ),
()
0
2
05 (4)
where E and A are Young's modulus and the nominal 
cross-section area of the wire rope, respectively. P 
is the quasi-static tension measured at the spatial 
position x(t) on the wire rope and is expressed by:

Strojniški vestnik - Journal of Mechanical Engineering 70(2024)3-4, 170-180
173 Forced Vibration of Time-Varying Elevator Traction System
 Pm lt xg  





 . (5)
In Eq. (4), the first term in the bracket denotes 
the static strain energy, and the second term represents 
the dynamic strain energy. The gravitational potential 
energy of the elevator traction system is:
 E gudx mgu
g
xlt
lt
 
 

()
()
.
0
 (6)
1.2  Dynamic Model Without External Interference
The elevator starts to run when an acceleration a(t) 
is provided by a lift machine. Furthermore, vibration 
is also generated in the elevator traction system. 
According to the Hamiltonian principle, the movement 
trajectory of the elevator in any time interval makes 
the time integral of the total energy stationary, that is:
 IE EE dt
ksg
t
t
  






1
2
0. (7)
The boundary conditions of the elevator traction 
are:
 
  ut ux tu xt
ut
00
00
12
,, ,,
,.
     
  (8)
 Using integral by parts with a variable upper 
boundary, the expression of δE
k
, δE
g
 and δE
s
 are 
further reformed as follows:
     


t
t
kt x
t
t
tx
Ed tm
t
vu vu
m
x
vv uv u
1
2
1
2




 






  






u ud t
t
vu vu udx
v
x
vu vu
xlt
lt
tx
t
t
tx




 






 
 
()
()




0
1
2
 









 
0 0
lt lt
dt
() ()
, (9)
         Ed tm gu gudx dt
g
t
t
xlt
t
tl t
1
2
1
2
0
 




 ()
()
, (10)
 
 

Ed tP u EAu u
PuEAu udx
s
t
t
x
xlt
t
t
x
lt
xx
1
2
1
2
0





[( )
()
()
()
] ]. dt (11)
Substitute Eqs. (9) to (11) into Eq. (7). By 
applying Leibniz's formula and considering the 
arbitrary property of δu, the following equations are 
derived:
 
 uv ua uv uaEAu
xl t
tt xt xx xx x
 


 
2
0
2
, 
, (12)
mu vu au vu a EAu xl t
tt xt xx xx
 

  2
2
=0,. (13)
Eq. (12) is the equation of motion for the 
longitudinal vibration of the time-varying elevator 
traction system and Eq. (13) is the corresponding 
boundary conditions of the elevator traction system.
1.3  Dynamic Model with External Interference
As one of the external interferences, the eccentricity 
of the traction wheel will produce an eccentric force 
during the operation of the elevator, which leads to the 
force vibration of the elevator traction system. Due 
to the limitations of the manufacturing technology, 
errors from the manufacture and installation cause 
a deviation of the central axis of the traction wheel 
from the rotating axis, resulting in the eccentricity 
of the traction wheel. The boundary condition in this 
situation is formulated as:
 uteu lt 00 ,,,,     (14)
where e is the eccentric excitation displacement 
dependent on time. Note that the boundary conditions 
in Eq. (14) are nonhomogeneous. To solve the 
governing Eq. (12) by satisfying the boundary 
conditions in Eq. (14), the displacement function 
u (x, t) is decomposed into two parts:
 ux tu xt ux t ,,,,     
12
 (15)
where u
1
 (x, t) satisfies the homogeneous boundary 
conditions, and u
2
 (x, t) satisfies the nonhomogeneous 
boundary conditions in Eq. (14). Since this paper 
studies the vibration of the time-varying elevator 
traction system under the influence of periodic forces, 
the expression u
2
 (x, t) is obtained according to the 
stress-strain relationship as follows:
 ux te
e
l
x
2
,.   (16)
Substituting Eq. (16) into (15) and then the 
acquired Eqs. (12) and (13), the equation of motion 
and the corresponding boundary condition for the 
forced longitudinal vibration of the time-varying 
elevator traction system are obtained as follows: 
  
D uxtu vu au vu a
EAu v
le
tt xt xx x
xx
t
(,)
,, ,,
,
 

  




11 1
2
1
1
2
2
v ve
l
ae
l
e
l
ae
l
ve
l
ve
l
x
tt t
2
22
2
3
2 2
0














 






	



 ,
0    xl t , (17)

Strojniški vestnik - Journal of Mechanical Engineering 70(2024)3-4, 170-180
174 Sun, J. – Xu, P. – Chen, M.L. – Xue, J.H.
B uxtm uv ua uv u
a EAu
EAe
l
mv
tt xt xx x
x
(,)
,, ,,
,
 


  
11 1
2
1
1
2
2
v ve le
l
ae
l
e
l
ve
l
ae
l
ve
l
x
x
t
tt t
 


 














2
22
2
3
2 2
0 + ,
l lt  . (18)
1.4  Discretization Process 
Since the equation of motion, Eq. (17), for the 
longitudinal vibration of the time-varying elevator 
system is a complex partial differential equation 
with infinite degrees of freedom and time-varying 
parameters, it is almost impossible to obtain accurate 
solutions. In this paper, the Galerkin method is 
used to discretize the equation of motion so that the 
equation of motion is converted into a set of ordinary 
differential equations. Following this work, the 
Newmark-β method is applied to obtain the numerical 
solution. The Newmark-β method is a numerical 
calculation method that linearizes and averages the 
generalized acceleration in the equation of motion for 
a dynamic problem. It is one of the most commonly 
used numerical methods in finite element analysis for 
dynamic problems and can ensure stable numerical 
solutions [28]. The solution of u1 is assumed in the 
form:
 ux tx pt
i
i
n
i 1
1
,,    


 (19)
where p
i
(t) is the generalized coordinate, and n is the 
number of included modulus, φ
i
(x) is the trial function. 
Since the wire rope is considered as a cantilever beam, 
the trial function φ
i
(x) is taken as the following form:
 
i
x
i
l
x  
 





2
21
2
sin. (20)
Although Eq. (20) satisfies the boundary 
condition in Eq. (8) at x = 0, it cannot satisfy the 
boundary condition in Eq. (18) at x = l(t). To eliminate 
the residuals generated by the trial function in the 
domain of x and at the boundary of x, the method of 
weighted residuals is adopted, which requires the total 
residual of the trial function in the domain of x and 
that at the boundary of x vanish:
DB
ji
l
ux dx
t
ji xl t
ux
ij n
,, ,|
,, ,,
     
  

  

0
0
12 .. (21)
For the convenience of analysis, a new variable 
ξ = x/l(t) is defined to normalize the variable x, 
and the time-varying domain [0, l(t)] related to x is 
transformed into a fixed domain [0, 1] of ξ. Replace x 
= ξ  l(t). After some calculus calculation, the equations 
of motion are transformed into the following form of 
the matrix:
 
Mp Cp Kp F    ,
 (22)
where
M
m
lt
C
vt
lt
ji ij ij
ji ij


 



  

 

 
11
2
1
0
1
,
'
d d
K
vt
lt
d
EA
l
ji ij
i


 
 
,
''
''



   
 


2
2
0
1
2
2
0
1
1
j j
ij
ij
d
a
l
d
v
l


 

 

   
   


1
2
1
0
1
2
2
0
1
'
'
d d
EA
l
Fv e
ve
l
ae el ae ve
ij
jt tt t
 

  
 






  



2
11
22
'
 







	
  
  




0
1
2
0
1
2
1
ve
l
da d
ma
l
me
EAe
l
jj
jt t
  

 



 
j
1, (23)
and δ
ij
 is the delta function. Eqs. (22) and (23) are 
solved by using Newmark-β method. Fig. 3 is a flow 
chart to solve Eqs. (22) and (23) by developing a 
MATLAB program.
2  MODELLING OF THE RUNNING STATUS  
OF THE ELEVATOR TRACTION SYSTEM
A complete running cycle of the elevator includes 
at least seven stages: acceleration increasing 
(0, t
1
), uniform acceleration (t
1
, t
2
), the acceleration 
decreasing to zero (t
2
, t
3
), uniform motion (t
3
, t
4
), 
deceleration increasing to the maximum (t
4
, t
5
), 
uniform deceleration (t
5
, t
6
), and deceleration 
decreasing to zero (t
6
, t
7
). The hoisting jerk j
i
(t) 
(the change of the acceleration per unit time) of the 
elevator at Stage I is taken as a quadratic function of 
time as follows:
 jt tt tt
i
i
i
i
i
i
() () ().   

60 24 6
01
2
11 2
  (24)
Assuming that the maximum acceleration, the 
maximum speed, and the maximum jerk are a
m
, v
m
, 
j
m
, respectively, the jerk j
i
(t), hoisting acceleration 

Strojniški vestnik - Journal of Mechanical Engineering 70(2024)3-4, 170-180
175 Forced Vibration of Time-Varying Elevator Traction System
Fig. 3.  A flow chart to solve Eqs. (22) and (23)
a) 
b) 
c) 
d) 
Fig. 4.  Typical running curves of an elevator:  
a) Increase of acceleration, b) acceleration of elevator,  
c) velocity of the elevator, d) length of wire rope
a
i
(t), the hoisting speed v
i
(t) and the length l
i
(t) of the 
elevator at each stage are determined by integrating 
sequential and applying the continuity conditions. Fig. 
4 shows typical running curves of an elevator when it 
goes up and down.
3  NUMERICAL RESULTS AND DISCUSSION
As an example, the typical parameters of the 
elevator traction system in a high-rise elevator are  
ρ = 0.575 kg/m, m = 1000 kg, EA = 7.02×10
7
 N. The 
length of the wire rope is l
0
 = 140 m.
According to the Newmark method, the iteration 
equation for solving Eq. (22) in the given time domain 
is derived as follows:
Kp F
p
t
pp p
tt
tt
tt
tt tt tt
 






  












,





11
2 2
11 1
2
1
2

 






   







 
pt
p
t
pp
t
p
t
tt tt tt

 

,
 
 p
t
, (25)

Strojniški vestnik - Journal of Mechanical Engineering 70(2024)3-4, 170-180
176 Sun, J. – Xu, P. – Chen, M.L. – Xue, J.H.
where 
KK
t
M
t
C
FF M
t
p
t
p
tt
tt tt



 
  









 
 
1
111
2
1
2
2



 
,
 






 




















 
p
C
t
pp pt
t
tt t





 
 1
2
1, , (26)
γ and β are the calculation parameters from the 
algorithm of the Newmark-β method. In this paper,  
γ = 1/2 and β = 1/4. 
At time t = 0, the elevator traction system is 
stationary. Thus, the initial values of the generalized 
coordinate and its velocity are zero, i.e., p
0

= 0 and 
 p
0
 = 0. As to the initial value of the acceleration of 
 p
0
 , it is obtained from Eq. (22) with the known 
values of p
0

 and  p
0
 . In the following of this 
section, the influence of the running direction, the 
acceleration, and the mass of the lifting car on the 
vibration performance of the elevator traction system 
is under investigation.
3.1  Vibration Without the Eccentric Excitation
Table 1 lists the time interval of the seven stages in a 
complete running cycle of the elevator traction system. 
Fig. 5 plots the time history for the longitudinal 
vibration of the elevator traction system without the 
eccentric excitation when v
m
 = 5 m/s and a
m
 = 1 m/s².
It can be seen from Fig. 5 that regardless of 
whether the elevator is in the ascending or descending 
stages, aperiodic displacement occurs during 
the starting and braking stages. These aperiodic 
displacements are attributed to the inertial force 
from the acceleration or deceleration, indicating that 
the elevator deviates from the initial equilibrium 
position. Another observation from Fig. 5 is that the 
vibration amplitude and frequency of the elevator 
traction system decrease during the ascending stage 
and increase during the descending stage. The reason 
for such response is that the length of the steel wire 
rope is reduced during the ascending process, which 
causes the increasing of the stiffness of the elevator 
traction system, as a result of that, the amplitude and 
frequency of the longitudinal vibration decreases. The 
situation is reversed during the descending process.
Table 1.  Time interval of seven stages in a complete running cycle 
of the elevator traction system [s]
t
1
t
2
t
3
t
4
t
5
t
6
t
7
1.5 5 6.5 16.5 18 21.5 23
Fig. 5.  The time history of longitudinal vibration of the elevator 
traction system without external interference
Fig. 6.  Extracted frequency spectrum of the elevator traction 
system without the eccentric excitation
Fig. 6 shows the frequency spectrum of the 
elevator traction system obtained by Fourier transform 
of the time history of the longitudinal vibration during 
uniform motion in both ascending and descending 
processes without eccentric excitation. As shown in 
Fig. 6, the fundamental frequencies of the elevator 
traction system during uniform motion in both 
ascending and descending processes are 4 Hz.
3.2  Vibration Response with Eccentric Excitation
The eccentric excitation distance e is applied at the top 
of the rope and is expressed as:
 et   11 0  m
-4
0
sin , (27)
where ω
0
 is the angular velocity of the lift machine 
and is closely related to the diameter d
0
 of the traction 
wheel, the maximum hoisting speed v
m
 and the 
maximum acceleration a
m
 of the elevator traction 
system. In this paper, the angular velocity of the lift 
machine ω
0
 is calculated by considering that the 

Strojniški vestnik - Journal of Mechanical Engineering 70(2024)3-4, 170-180
177 Forced Vibration of Time-Varying Elevator Traction System
drawn in Fig. 7. It can be seen from Fig. 7 that the 
amplitude of the longitudinal vibration of the elevator 
traction system with external excitation gradually 
decreases during the ascending process and increases 
during the descending process, which is consistent to 
the response for the elevator traction system without 
eccentric excitation in Section 3.1. Nevertheless, the 
time histories shown in Fig. 7 for the elevator with 
eccentric excitation are quite different from those in 
Fig. 5 for the elevator without eccentric excitation. 
To facilitate explain the reason for the difference, 
frequency spectra for the time history in Fig. 7 are 
extracted and plotted in Fig. 8. As shown in Fig. 8, the 
leading frequency is 1.5 Hz, which is the frequency 
of the lift machine. Furthermore, the fundamental 
frequency 4 Hz of the elevator traction system is 
also captured. The spectrum analysis indicates that 
the vibration response of the elevator with eccentric 
excitation is governed by eccentric excitation.
distance travelled by the elevator per unit of time must 
equal that of the traction wheel and is estimated as 
follows:
 ω
0
=
2
07
l
dt
T
, (28)
where l
T
 is the distance travelled by the elevator in a 
running cycle, and t
7
 is the time period of a running 
cycle.
3.2.1  Influence of the Running Direction
In this subsection, the diameter of the traction wheel, 
the maximum hoisting speed and the maximum 
acceleration of the elevator traction system are chosen 
as d
0
 = 0.4 m, v
m
 = 2.5 m/s, a
m
 = 0.5 m/s², and ω
0
 = 
3π. The time intervals of the seven stages are the same 
as those in Table 1 of Section 3.1. The time history 
of the longitudinal vibration of the elevator traction 
system in the process of ascending and descending is 
a)           b) 
Fig. 7.  The time history of the elevator traction system caused by external eccentric excitation.; a) ascending stage, and b) descending stage
a)             b) 
Fig. 8. Extracted frequency spectrum of the elevator traction system with eccentric excitation; a) ascending stage, and b) descending stage

Strojniški vestnik - Journal of Mechanical Engineering 70(2024)3-4, 170-180
178 Sun, J. – Xu, P. – Chen, M.L. – Xue, J.H.
3.2.2  Influence of the Acceleration of the Elevator
To clarify the influence of the acceleration of the 
elevator traction system on its vibration performance, 
the following three cases in which the elevator goes 
down are investigated: 
Case I   v
m
 = -3.75 m/s, a
m
 = -0.75 m/s2, ω
0
 = 9/2π;
Case II   v
m
 = -5.0 m/s, a
m
 = -1.0 m/s2,  ω
0
 = 6π;
Case III v
m
 = -7.5 m/s, a
m
 = -1.5 m/s2,  ω
0
 = 9π.
Fig. 9 demonstrates the influence of the 
acceleration on the time history of the descending 
elevator traction system. It can be seen the vibration 
amplitude of the elevator traction system increases 
for all three cases during the descending process. 
Furthermore, the vibration amplitudes of the elevator 
for Case II are less than those for Case III but greater 
than those for Case I. In other words, the greater 
the values of vm and am, the larger the vibration 
amplitude of the elevator. However, Fig. 9c also 
shows that the amplitude of the elevator increases 
significantly at t = 13 s. Such a response indicates 
that the resonance occurs in the elevator traction 
system, as the frequency of the eccentric excitation 
of the traction machine is close to the fundamental 
frequency of the elevator traction system. Therefore, 
in order to prevent the occurrence of resonance and 
the maximum values of acceleration and deceleration 
should not exceed 1.5 m/s².
4  CONCLUSIONS
In this paper, the longitudinal vibration of the time-
varying elevator traction system is studied by means 
of theoretical analysis and numerical simulation. The 
equation of motion and the corresponding boundary 
condition of the time-varying elevator traction system 
with and without external interference is derived 
based on the Hamiltonian principle and is solved by 
employing Galerkin's weighted residual method to 
take into consideration the boundary condition. As 
an application of the proposed analytical approach, 
the force vibration of the elevator traction system 
induced by the eccentric excitation of the traction 
wheel is analysed. Depending on the running status 
of the elevator, a mathematical formula is proposed 
to estimate the angular velocity of the traction wheel. 
A MATLAB program is developed to obtain the 
numerical solutions of the elevator traction system 
under different running statuses. The following 
conclusions can be drawn:
a) 
b) 
c) 
Fig. 9.  The influence of the acceleration on the time history  
of the descending elevator traction system;  
a) a
m
 = -0.75 m/s², b) a
m
 = -1 m/s², and c) a
m
 = -1.5 m/s²
1. During the ascending stage, the equivalent 
stiffness of the elevator traction system increases 
with the decreasing length of the wire rope, 
resulting in a decrease in the vibration amplitude 

Strojniški vestnik - Journal of Mechanical Engineering 70(2024)3-4, 170-180
179 Forced Vibration of Time-Varying Elevator Traction System
and vibration frequency. The situation is reversed 
during the descending stage.
2. As far as the structural parameters of the elevator 
are determined, the vibration response of the 
elevator traction system is determined by its 
running state. In particular, the greater the running 
acceleration of the elevator, the larger the angular 
velocity of the traction wheel, and the higher the 
vibration amplitude and frequency of the elevator 
traction system.
3. When the acceleration reaches am = 1.5 m/s², the 
frequency of eccentric excitation of the traction 
machine is close to the fundamental frequency 
of the elevator traction system, which leads to 
the resonance of the elevator traction system. 
Therefore, in order to prevent the occurrence of 
resonance, the maximum value of acceleration 
and deceleration should not exceed 1.5 m/s².
In practical engineering applications, elevators 
are installed within buildings, so the vibration of 
elevators is affected by the vibration of buildings. 
Further experimental and theoretical studies can 
be conducted on the coupled vibration between the 
elevators and the buildings subjected to wind loads 
and earthquakes, which may provide a reference for 
seismic design of elevator traction system.
5  ACKNOWLEDGEMENTS
This work is supported by the Guangdong Natural 
Science Foundation (No. 2021A1515012037).
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