*Corr. Author’s Address: University of Novi Sad, Faculty of Technical Sciences, Trg D. Obradovića 6, 21000 Novi Sad, Serbia, djokic@uns.ac.rs 287
Strojniški vestnik - Journal of Mechanical Engineering 67(2021)6, 287-301 Received for review: 2021-03-21
© 2021 Journal of Mechanical Engineering. All rights reserved.  Received revised form: 2021-05-06
DOI:10.5545/sv-jme.2020.7179 Original Scientific Paper Accepted for publication: 2021-05-07
Dynamic Modelling, Experimental Identification and Computer 
Simulations of Non-Stationary Vibration in High-Speed Elevators
Đokić, R. – Vladić, J. – Kljajin, M. – Jovanović, V . – Marković, G. – Karakašić, M.
Radomir Đokić
1,*
 – Jovan Vladić
1
 – Milan Kljajin
2
 – Vesna Jovanović
3
 – Goran Marković
4
 – Mirko Karakašić
5
1 
University of Novi Sad, Faculty of Technical Sciences, Serbia 
2 
University North, University Center Varaždin, Croatia 
3 
University of Niš, Faculty of Mechanical Engineering, Serbia 
4 
University of Kragujevac, Faculty of Mechanical and Civil Engineering in Kraljevo, Serbia 
5 
University of Slavonski Brod, Mechanical Engineering Faculty, Croatia
Modelling the dynamic behaviour of elevators with high lifting velocities (contemporary elevators in building construction and mine elevators) 
is a complex task and an important step in the design process and creating conditions for safe and reliable exploitation of these machines. 
Due to high heights and lifting velocities, the standard procedures for dynamic exploitation are not adequate. The study presents the method 
of forming a dynamic model to analyse nonstationary vibrations of a rope with time-varying length with nonholonomic boundary conditions 
in the position where the rope is connected with the cabin (cage) and in the upcoming point of its winding onto the pulley (drum). A unique 
method was applied to identify the basic parameters of the dynamic model (stiffness and damping) based on experimental measures for a 
concrete elevator. Due to the verification of this procedure, the experiment was conducted on a mine elevator in RTB Bor, Serbia. Using the 
obtained computer-experimental results, the simulations of the dynamic behaviour of an empty and loaded cage were shown. In addition, the 
study shows the specific method as the basis for forming a control program that would enable the decrease in vertical vibrations during an 
elevator starting and braking mode.
Keywords: high-speed elevators, dynamic analysis, a rope with time-varying length, mechanical characteristics of steel ropes, 
longitudinal oscillations, control program
Highlights
• 	 The complexity of the dynamic analysis of elevators is because these are systems for lifting (lowering) people and load to great 
heights (depths) with high velocities and variable parameters.
• 	 Determining the critical hoisting velocity of the elevator car can be performed in the function with mechanical characteristics of 
ropes, such as the elasticity modulus and damping and loads in steel ropes.
• 	 Based on the theory of free harmonic damping oscillations, the mechanical characteristics of steel ropes can be determined 
through the oscillation diagrams obtained by measurement.
• 	 By defining the basis for the driving mechanism control program, it is possible to provide minimum dynamic loads of elevators 
based on adequate models and simulations of their operation in real conditions.
0  INTRODUCTION
Mines with underground exploitation and operating 
levels up to 2500 m, along with the rising number of 
exceptionally tall buildings, with heights reaching up 
to 850 m nowadays - such situations require electric 
elevators with specific characteristics, whose velocity 
reaches up to 20 m/s, and load capacity up to over 
ten tons. The elevator quality is estimated according 
to several important indicators. Safety, comfort, 
and reliability are especially important features [1]. 
These indicators depend on, first of all, vibrations 
occurring while the elevator is in motion. Vibrations 
are a consequence of driving parameters, inertial 
characteristics and elasticity of the binding elevator 
elements. Fig. 1 shows the elements with the greatest 
impact.
An elevator can be divided into two basic parts 
according to the dynamic impact on the vibration 
values. The first part is a driving mechanism (engine, 
reductor, brake, and couplings), while the second 
part is a cabin lifting system, mostly made of steel 
ropes for lifting the counterweights on one end and 
cabin (cage) on the other end and their guide rails. 
The driving mechanism comprises elements that are 
much more rigid (c
1
, c
2
, Fig. 1) and have a smaller 
mass than the cabin lifting system (c, Fig. 1), which 
in turn causes the oscillations in smaller amplitudes 
and higher frequencies. As well as that, bearing in 
mind that the oscillations are indirectly transferred 
to the cabin (cage) via ropes, it can be deduced that 
the lifting system has a much bigger influence on the 
comfort during the motion than driving mechanism 
elements does.
Deep shaft mines require special mining ropes to 
hoist personnel and materials safely and efficiently. 
They are made of round wires that must be either 
bright or galvanized. The values of the rope safety 

Strojniški vestnik - Journal of Mechanical Engineering 67(2021)6, 287-301
288 Đo kić,	R.	–	Vladić,	J.	–	Kljajin,	M.	–	Jo v ano vić,	V .	–	Mar k o vić,	G.	–	K arak ašić,	M.
factors for hoisting in mines depend on shaft depth 
and rope number and are higher in the case of 
personnel hoisting. According to ISO standards for 
the area of mine elevators, safety factors vary between 
4 and 8 for new ropes and between 3.6 and 6.4 for 
ropes to be discarded, depending on shaft depth. The 
highest value of 13 is required by special regulations 
for hoisting people to depths of up to 600 m.
Fig. 1.  Elements with the biggest impact  
on vibration occurrence in elevators
While the elevator is in operation mode, the 
hoist ropes increase and decrease their free length, so 
the parameters, such as rope stiffness and damping, 
are constantly changing [2] to [4]. In high-speed 
elevators, dynamic instability may occur during 
lifting (reducing free length) due to increased relative 
deformation. This instability seriously impacts the 
safety of the passengers. Since classic models are 
based on elastic body (rope) oscillations with constant 
dynamic parameters (mass, stiffness and damping), it 
is necessary to form dynamic models that will enable 
the analysis and definition of the dynamic behaviour 
of elevators with variable parameters [5] and [6].
Because special attention has to be paid to the 
accuracy of installing and making of cabin guide 
rails and counterweight in high-speed elevators, the 
following conclusion arises. Without the addition 
of external influences, it can be concluded that 
longitudinal oscillations are dominant, as opposed to 
transversal oscillations [7] and [8].
Taking into consideration that up until this 
moment, the problems in driving mechanism 
vibrations were the subject of a great number of 
scientific and research papers, with standard analyses 
as the most frequently applied method, it seems logical 
that the main focus of dynamic research of elevators 
should be pointed towards innovative methods for 
analysing the longitudinal oscillations with variable 
parameters [9].
1  DYNAMIC MODELS FOR THE ANALYSIS  
OF ELEVATOR LONGITUDINAL OSCILLATIONS
1.1  Standard Models
Many researchers are interested in studying 
longitudinal oscillations, and their studies have been 
based on the general theory about the application of 
oscillation of elastic bars with constant parameters 
(mass, stiffness, and damping). These are the so-
called standard models [8] and [10]. Those models 
are acceptable for analysing elevators with low lifting 
velocities and heights. Figs. 2a and b show models 
with one and two degrees of freedom and a rope of 
constant lengths, represented here as Hook’s, i.e., 
Calvin‘s body.
A certain improvement has been made with 
the analysis of high-lift elevators (≥ 35 m) and low 
velocities (till 3 m/s) by using the model represented 
in Fig. 2c. The model represents a bar of a constant 
length with an equally spread mass q (kg/m), i.e., it is 
a model of an elastic body with an unlimited number 
of degrees of freedom and a concentrated mass at the 
bottom end as the boundary condition.
Based on the analysis which was shown in detail 
in [11] to [14], in the case when the free rope length 
is small compared to the cabin mass, it is possible to 
significantly simplify the dynamic model analysis.
Fig. 2.  S tandar d	dynamic	mo dels;	а)	with	o ne	degree	o f	freedo m;	
b) with two degrees of freedom; c) a “heavy” constant length bar
Fig. 3 shows an oscillation diagram for the first 
three harmonics. Due to the very small oscillation 
amplitudes in higher harmonics, their influence can 
be neglected. Thus, the total oscillation process with 
infinite degrees of freedom, whose total oscillation 

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289 Dynamic	Mo delling,	Experimental	Id entif icatio n	and	Co m put er	Simulatio ns	o f	N on-S tatio nar y	Vibratio n	in	High-Speed	Ele v at o r s
form is shown in Fig. 3 with a dashed line (d), can 
be replaced with a straight line (a), with satisfactory 
accuracy. In other words, it is replaced with a system 
with one degree of freedom and constant dilatation (ε) 
along the free rope end.
Fig. 3.  Oscillation shapes (forms) of the first three harmonics (a, b 
and c), and the summary oscillation form for α = 0.1
In accordance with the above, regarding high-lift 
elevators (≥ 35 m) and low velocities (to 3 m/s), it is 
roughly possible to create load oscillation models with 
one degree of freedom, with a “heavy” spring, which 
was studied in general literature. Also, it is necessary 
to replace the total mass (of both load and rope) with 
an equivalent mass M
e
 = M + (1/3) ∙ qL, reduced in the 
cabin place, [15] and [16].
1.2  Dynamic Elevator Models with Dynamic Variable 
Parameters
Fig. 4a shows the most common solutions of the 
lifting systems for high-speed elevators with a driving 
pulley, while the corresponding dynamic model is 
shown in Fig. 4b.
Fig. 4.  Ele v at o r	mo dels;	a)	Kö epe	sys t em,	 
b) a dynamic model of a high-speed elevator
In order to secure comfort during the motion, 
control programs are used in contemporary elevators. 
They define the circumferential velocity of the pulley. 
Thus, they also define the cabin motion velocity (a 
kinematic condition), as opposed to the previous period 
when the motion velocity depended a great deal on the 
driving electromotor’s mechanical characteristics and 
brake system (the dynamic equilibrium condition). 
In the earlier periods, replacing one-speed engines 
with two-speed engines was observed as a significant 
improvement. This improved the motion comfort in 
braking instances and aided the accuracy of stopping 
the cabin. As for the process of a regular elevator, in 
cases in which there is no slipping of the steel rope on 
the driving pulley and when the driving characteristic 
is represented via a rope velocity at the meeting point 
of the rope and pulley, the elevator model can be 
simplified and represented in the form shown in Fig. 
5.
Upon observing just the upcoming rope end, 
the model can be represented as a system with an 
unlimited number of degrees of freedom; at one end 
it is rolling onto the pulley at a v(t) velocity, while on 
the other it is burdened with concentrated mass. Due 
to the variable rope length during the motion, the 
stiffness (c = EA/L) changes. This is a characteristic of 
parametric oscillations and contributes to the possible 
occurrence of resonance. To this end, it is necessary 
to complete certain steps in the analysis of dynamic 
behaviour. The critical lifting velocity, during which 
the unstable motion occurs, i.e., the rope strain is 
increased when its free length is reduced, needs to be 
determined.
Fig. 5.  Elevator model with a rope of a variable length  
with boundary conditions
The deformation of an arbitrary cross-section 
represents the function of the position x, and the time 
t, i.e.:
 uf xt = (,). (1)
Upon observing the equilibrium of the elementary 
part (dx), it can be deduced that:

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290 Đo kić,	R.	–	Vladić,	J.	–	Kljajin,	M.	–	Jo v ano vić,	V .	–	Mar k o vić,	G.	–	K arak ašić,	M.
 
qd x
g
uxt
t
SxtS xt
S
x
dx qd x
qd x
g
a




 



 


2
2
(,)
(,)( ,)
. (2)
By representing the rope as Calvin’s model, 
where the influence of internal friction can be taken 
into consideration via the so-called rope resistance 
force factor (b
f
), the dependence of the inner force in 
the rope on its deformation can be noted in this form:
 SxtE A
x
uxtb
uxt
t
(,)( ,)
(,)
. 


 








f
 (3)
If the Eq. (2) is divided with (q·dx) / g and a 
replacement for S(x,t), this is obtained:
 

















2
2
2
2
uxt
t
gEA
q x
uxtb
uxt
t
ga
(,)
(,)
(,)
.
f
 (4)
In Eqs. (2) and (3), (E) is the rope elasticity 
modulus whose magnitude depends on the elasticity 
modulus of wires (E
r
 = 2.1 ∙ 10
5
 MPa) and the 
construction of the rope [17] and [18], whose value can 
be twice smaller (stranded wire ropes).
By using the differential equation, (Eq. (4)) and 
the equilibrium condition of moments on the driving 
pulley, it is possible to form a system of equations that 
describes a dynamic equilibrium on the driving pulley 
in the case of a model shown in Fig. 4b in this form 
[12] and [19]:
q
g
ux t
t
EA
x
ux tb
ux t
t
q






 








 
2
1
2
2
2 1
1
1
(,)
(,)
(,)
f
a a
g






, (5)
q
g
ux t
t
EA
x
ux tb
ux t
t
q






 








 
2
2
2
2
2 2
2
1
(,)
(,)
(,)
f
a a
g






, (6)
M
R
i
EA
x
ultu lt b
t
ultu lt
m
f








 

11 22 11 22
(,)( ,) (,)( ,)
 







J
ai
R
r
. (7)
1.3  Boundary Conditions
In order to solve a partial differential equation, (Eq. 
(7)), i.e., an equation system (Eqs. (5) to (7)), it is 
necessary to define the boundary conditions in the 
incoming point of the rope to the pulley in the meeting 
point of hoist ropes and the elevator cabin.
Boundary conditions at point C, where the rope 
makes the first contact with the pulley, are dependent 
on whether the winding is with or without rope 
slipping. Fig. 6a shows the distribution of the forces 
on the wrap angle of the pulley (α) for a quasistatic 
case of elevator operation (without the influence of 
dynamic forces). The regular elevator operations 
must not allow for the rope to slip in the whole wrap 
angle (α), which is regulated with a safety degree 
against slipping, i.e., with the existence of a suitable 
angle (α
M
) with the so-called relative abeyance of 
the rope on a driving pulley. It should be noted that, 
in quasistatic conditions, the zone with the elastic 
slipping of the rope on a driving pulley (α
K
), Fig. 
6a, always occurs on the descending side. However, 
due to the oscillation, the force in the incoming 
end changes, so slipping is probable in the pulley’s 
incoming zone. The change of force in the wound rope 
part can only be maintained if this change is smaller 
than the adhesive force making it possible. Fig. 6d 
shows different cases of force distribution over the 
wound rope length as a function of elevator velocity. 
With high-speed elevators, it should be expected that 
the force change in the wound part of the rope is 
smaller than the friction force change in the incoming 
zone (curve v
3
, Fig. 6d). Basically, in this case, it can 
be accepted that there is no slipping in the rope and 
rope pulley meeting point. The problem is analysed in 
great detail in [12] and briefly summarized in [20].
Fig. 6.  Boundary conditions; a) force distribution on the wrap 
angle of the pulley, b) driving pulley without slipping,  
c) cage (cabin), d) different cases of the force distribution

Strojniški vestnik - Journal of Mechanical Engineering 67(2021)6, 287-301
291 Dynamic	Mo delling,	Experimental	Id entif icatio n	and	Co m put er	Simulatio ns	o f	N on-S tatio nar y	Vibratio n	in	High-Speed	Ele v at o r s
The boundary condition at the meeting place 
of the rope and the pulley without slipping, Fig. 6b 
(point C), is presented in this form:
 
0
(,) d
(, ) d.
d
t
ult l
ult t
xt
∂ 
=

∂

∫
 (8)
The boundary condition in the meeting point of 
the rope and elevator cabin (mine elevator cage) or 
counterweight is given in this form:
 
QE A
x
uLtb
uLt
t
Q
g
uLt
t
a


















(, )
(, )
(, )
f
     
2
2
 



F
f
, (9)
where is FFv
fN
sign    () friction force between 
sliding guide shoes and guide rails.
1.3.1  Estimation of Critical Velocity
The non-integrated boundary condition (Eq. (8)) 
prevents the solution of the partial differential equation 
(Eq. (5)) so that the solution can be sought through 
the formation of integral equations, which contain 
both the differential equations and the corresponding 
boundary conditions. Simplifying the mathematical 
model, friction within the boundary condition is 
omitted in Eq. (9).
In Fig. 7, a weightless fibre is shown, loaded at 
point (S) by the force S
i
, where the displacement of 
the rope point (without sliding over the drive pulley) 
in the region s<x, shows a linear increase from zero to 
the boundary value u(s), while the displacement below 
point S is stable and equal to the boundary value u(s). 
Therefore, for the elemental force (S
i
 = 1), there is a 
deformation:
 Kx s
xl
EA
sx
sl
EA
sx
,,  













    for  
    for  
 (10)
and displacement u(x) = K(x,s)∙S
i
.
For the case with several acting forces with acting 
points at x=s
i
 there is a displacement:
 ux KxsS
ii
i
n
() (, ). 


1
 (11)
For the load case with evenly distributed rope 
load (rope’s weight), there is the expression:
 ux Kxsqs
l
L
() (,).  

d (12)
Fig. 7.  Deformation of a rope
By applying this procedure to the differential 
equation (Eq. (5)), replacing the floating argument 
with (s), multiplying by the function K(x,s,l) 
and performing the necessary mathematical 
transformations, the rope deformation in the form is 
obtained [20]:
uxtK xslq s
u
t
gas
ult
x
v
l
L
t
(,)( ,,)( )
(,)
(
 














2
2
0
d
t ttEA Kxsl
u
st
s
bE A
uLt
xt
KxLl
l
L
)( ,,)
(,)
(,,
dd
f








3
2
2
) ). (13)
Using the method of particular integrals, taking 
into account only the first form of oscillations of the 
rope and that the length of the free rope (L – l) changes 
“slowly” with time, and if a new variable v(x,t) is 
introduced, so that:
 uxtv xt ult (,)( ,) (,)  , 
the deformation of rope free side in the phase of load-
lifting will be:
vxtx lh
Ll
Ll
el t
xl
EA
Q
m
t
(,)( )c os ()  









 




 0
0
1
4
0
 d
 
 












qL xl a
g
()
,
2
2
1 (14)
where:
h
Q
EA
a
g
l
gEA
Q
qLl
Ll
m
b
l
0
2
2
3
2






 






 
,
()
()
()
,
()


f
 
dt
t
0
.

Strojniški vestnik - Journal of Mechanical Engineering 67(2021)6, 287-301
292 Đo kić,	R.	–	Vladić,	J.	–	Kljajin,	M.	–	Jo v ano vić,	V .	–	Mar k o vić,	G.	–	K arak ašić,	M.
h
Q
EA
a
g
l
gEA
Q
qLl
Ll
m
b
l
0
2
2
3
2






 






 
,
()
()
()
,
()


f
 
dt
t
0
.
Analysing Eq. (7), it can be concluded that during 
the reduction of the free length of the rope (load-
lifting), its deformation can increase under certain 
conditions, which causes a permanent increase in the 
dynamic load of the hoist rope. Such a phenomenon, 
which is usually described as unstable lifting, will 
not occur if the lifting velocity is less than the critical 
velocity, which is defined by the expression:
 vb
gEA
Q
qLl
b
EA
M
kr ff
e






2
3
2
()
, (15)
where MQ
qLl
g
e

 





()
3
 is the reduced mass 
of the cabin and the rope part as a “heavy spring”.
The level of critical velocity depends on the 
damping characteristic of the wire rope and the 
elevator-s basic characteristics. When it comes to 
elevators with high load capacity, high velocity 
and lifting height (high-speed elevators and mine 
elevators), the lifting velocity can exceed the critical 
velocity, which is why it is necessary to check the 
stability of the lifting process already in the design 
phase. It should be taken into account that Eq. (15) 
does not contain the friction in guide rails, which 
adds to the oscillation reduction, and simultaneously 
increases the critical velocity value.
2 EXPERIMENTAL DETERMINATION  
OF DYNAMIC PARAMETERS
Experimental studies shown in this paper represent the 
sequel to the research in [13]. The measuring was done 
on a mine elevator in RTB Bor, Serbia; the maximum 
projected cage lifting velocity was 16 m/s, and the 
projected lifting height was 523 m in the first phase 
and 763 m in the second phase of mining. The driving 
mechanism is shown in Fig. 8. The other significant 
characteristics of this exploitation machine can be 
seen in Tab. 1.
A mine shaft is like a round cross-section with 
a 10 m diameter. The transfer of the driving moment 
to the hoist ropes is done through friction (Köepe 
system) from the grooved drum with a 2.5 m diameter.
     
Fig. 8.  Driving machine of a mine elevator and the connecting 
tools for connecting the hoist ropes and the cage
Table 1.  Mine elevator technical characteristics
Driving electric 
motor
Power: 1500/2860 kW, 
number of revolutions: 122.2 rpm, 
torque: 117.2/233.4 kNm
Cage (cabin)
Mass: 13 t (it includes the cage and the connecting 
tools for connecting the hoist ropes), Fig. 8
Counterweight Mass: 21 t
Hoist ropes
z = 6 pieces,
d = 27 mm (150 wires per cross-section).
Lang’s lay ropes, three right-hand Lang’s lay ropes, 
and three left-hand Lang’s lay ropes.
Breaking force: 561 kN, tensile strength of the 
wires: 1700 MPa.
Unit mass: 3.02 kg/m.
Compensation 
ropes
z = 2 pieces,
d = 50 mm (222 wires per cross-section).
Connected with the cage via rotating hooks to 
prevent unwinding. Unit mass: 9.64 kg/m.
2.1  Equipment Used and the Method for Measuring Data 
Acquisition
The following measuring equipment was used for the 
mine elevator experiment:
• Universal 8-channel measuring amplifier (2 
pieces), QUANTUM X MX480B, pos. 10a and 
10b (Fig. 9),
• A computer for storing the measuring signals, 
pos. 13 (Fig. 9),
• Antennas for the wireless transfer of the 
measuring signal (2 pieces) NanoStation loco 
NS2L, made by IBIQUITI NETWORKS; 5 km 
range with the antennas optical visibility, pos. 12a 
and 12b (Fig. 9),
• Incremental encoder - the number of revolutions 
sensor AINS 41, made by Meyer Industrie-
Electronic GmbH – MEYLE; measuring range up 

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293 Dynamic	Mo delling,	Experimental	Id entif icatio n	and	Co m put er	Simulatio ns	o f	N on-S tatio nar y	Vibratio n	in	High-Speed	Ele v at o r s
to 6000 rpm and with an allowed axial/radial load 
at the output shaft of 30/20 N, pos. 9 (Fig. 9),
• Optical measuring device for the number of 
revolutions ROS type, made by Monarch 
Instrument; measuring range varies from 1 to 250 
000 rpm, pos. 1 (Fig. 9),
• Accelerometer HBM B12, pos. 2 (Fig. 9),
• Strain gauges HBM LY41-6/120, pos. 3 to 8 (Fig. 
9),
• Software for acquisition and processing of 
measuring signals, HBM catmanEasy-AP.
Fig. 9.  Schematic layout of the measuring places
The experiment was done according to the 
principle of summary data acquisition via a wireless 
data transfer by using the NanoStation loco NS2L 
antennas (pos. 12a and 12b, Fig. 9). In this way, a 
complete synchronization of the collected data on the 
cage roof and machine room was achieved. The drum 
number of revolutions was measured with an AINS 41 
incremental encoder (pos. 9, Fig. 9). It was set on the 
adaptive carrier with a 127 mm diameter measuring 
wheel, which was directly leaned onto the rim of the 
brake disc, Fig. 10a. The encoder measuring signal 
was led to the amplifier (pos. 10b, Fig. 9), and then 
via a LAN switch (pos. 11b, Fig. 9) and the antenna 
(pos. 12b, Fig. 9) wirelessly connected to the other 
radio antenna positioned on the cage (pos. 12a, Fig. 
9). The measuring signal was stored in a common 
computer file (pos. 13, Fig. 9) placed on the cage. It 
was done via another LAN switch (pos. 11a, Fig. 9).
The lifting and lowering of the velocity of the 
cage was measured via a roller guide, whose change 
in the number of revolutions was registered with an 
optical sensor (pos. 1, Fig. 9), positioned on a small 
magnet table, Fig. 10b. The change in the cage’s 
acceleration during the measuring process was 
registered by an HBM B12 accelerometer (pos. 2, Fig. 
9), set on the connecting tools of the cage through 
a magnetic holder, Fig. 11a. The force changes in 
the hoist ropes were monitored by measuring the 
deformation of the connecting tools (pos. 3 to 8, 
Fig. 9). The deformations were measured on each 
connecting tool (out of 6 in total) by using the HBM 
LY41-6/120 strain gauges, Fig. 11b.
a)    b) 
Fig. 10.  Measuring places; a) the incremental encoder connected 
with the measuring wheel positioned on the rim of the brake disc, 
and b) the optical sensor (number of revolutions sensor) positioned 
on the cage
a)     b) 
Fig. 11.  Measurements on the cage connecting tools; a) cage 
acceleration, and b) deformations
All measuring signals were led to the other eight-
channel measuring amplifier (pos. 10a, Fig. 9), and 
they were recorded on the computer (pos. 13, Fig. 9) 
via a LAN switch (pos. 11a, Fig. 9).
2.2  Protocols and Results of the Experiment
In Fig. 12, a schematic layout of a mine shaft elevator 
is shown with labelled levels, whose positions were 
set with altitude, using plus and minus. The ground 
level was at an altitude of +436 m, while the machine 
room was placed at +465 m.
Experiment results and determination of the 
dynamic model parameters in this paper are shown in 
four characteristic motion cases (lifting and lowering) 
of the cage, with and without load. A significantly 
higher number of measurements was completed for 
different cases (different positions for starting and 
stopping the cage motion etc.), and the results are 
presented in [11]. Here, the results of the lowering 
and lifting of an empty cage and a loaded cage 
(traction machine) are presented. According to those, 
the system oscillation parameters are defined. The 
characteristics of experimental cases are as follows:
I.  Lowering an empty cage from position 
+104 m to position +56 m (48 m). The action of 

Strojniški vestnik - Journal of Mechanical Engineering 67(2021)6, 287-301
294 Đo kić,	R.	–	Vladić,	J.	–	Kljajin,	M.	–	Jo v ano vić,	V .	–	Mar k o vić,	G.	–	K arak ašić,	M.
stopping the cage was performed by suddenly 
switching off the driving electromotor (at a high 
velocity, Fig. 13a).
II. Lifting of an empty cage from position 
+56 m to a position +199 m (143 m). Stopping 
the cage was done just like in the case above, 
with the sudden switching off the power to the 
driving electromotor, at a high velocity, Fig. 13b.
III. Lifting a loaded cage (the traction machine 
for wagons, mass ~9.35 t) from position a 
-71 m, to position +52 m (123 m). Stopping the 
cage was done at a low speed, Fig. 13c.
IV . Lowering the loaded cage (the traction machine 
for wagons, mass ~9.35 t) from position +52 m to 
-71 m (123 m). Stopping the cage was done at a 
low speed, Fig. 13d.
The results of the conducted experimental 
research can be seen in the following figures, which 
are showing the changes in the winding velocities 
of hoist ropes that were wound onto the drum; cage 
acceleration and changes in the forces in the hoist 
ropes, i.e., on the elements for connecting the ropes 
to the cage. The results are valid for all four chosen 
examples of motion. The diagram shows the character 
of the changes of the said parameters. It shows the 
cage oscillation amplitudes that occur after stopping 
the driving machine. This part of the diagram (after 
stopping the driving machine) is used to set the 
mechanical characteristics of steel ropes (harmonic 
oscillations).
Fig. 14.  Diagram with the results obtained during lowering an 
empty cage and sudden stopping, at a “high” velocity  
(I motion case)
Fig. 15.  Diagram with the results obtained during lifting the empty 
cage and sudden stopping at a “high” velocity (II motion case)
Fig. 12.  Sc hematic	la y o ut	o f	a	v er tical	mine	shaf t	in	“Jama” 	mine	
of RTB Bor
Fig. 13.  Mine elevator parameters that are relevant for the 
analysis; a) lowering an empty cage, b) lifting of an empty cage, c), 
lifting a loaded cage, d) lowering the loaded cage

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295 Dynamic	Mo delling,	Experimental	Id entif icatio n	and	Co m put er	Simulatio ns	o f	N on-S tatio nar y	Vibratio n	in	High-Speed	Ele v at o r s
Fig. 16.  Diagram with the results obtained during the loaded cage 
lifting process (III motion case)
Fig. 17.  Diagram with the results obtained during loaded cage 
lo w ering	pr o cess	(IV	mo tio n	case)
2.3  Determination of Dynamic Parameters Based on 
Measurement Results
The stiffness and damping parameters can be 
determined, and the elasticity modulus as mechanical 
characteristics of a steel rope can be indirectly 
determined too, by using the above diagrams and 
determining the oscillation period, i.e., the frequency 
and logarithm decrement of the damping. In [13], and 
especially in [17], the said characteristics are described 
in detail, so this study only mentions the important 
parts for explaining the results obtained with new 
measurements.
Stiffness and elasticity modulus. Rope stiffness 
is defined with an expression [13]:
 c
EA
Ll t


 ()
. (16)
The stiffness depends on the change in the rope’s 
free length, but it also depends on the elasticity 
modulus, usually taken as a constant parameter in 
dynamic behaviour analyses. However, due to the 
complex construction of a steel rope (consisting of 
wires, strands, and a core), its change depending on 
the strain magnitude and loading character (loading-
unloading) cannot be neglected. According to [18], a 
distinction can be made between the so-called tangent 
elasticity modulus (E
t
) and an average (medium) 
elasticity modulus between two stresses (E
S
), Fig. 18. 
The tangent elasticity modulus represents a theoretical 
angle of the inclination of the tangent to the σ-ε 
curve for the current stress value (σ
Z
). The constant 
variability in the elasticity modulus magnitude can be 
observed, and significant differences occur in loading 
and unloading the rope. The average (median) value 
of elasticity modulus can be set for boundary working 
stresses - E
S
 (σ
lower
, σ
upper
).
Fig. 18.  Tangent and average rope elasticity modulus
Damping. In many practical systems, oscillation 
energy is gradually transformed into heat or sound, 
the effect mostly known as damping. Although 
the quantity of such energy is relatively small, it is 
important to dedicate some attention to damping to 
predict the oscillatory system’s response, such as the 
elevator systems considered in this paper.
Damping at the elevator is complex, and it 
happens because of the rope’s inner friction (viscous 
and hysteretic damping), Coulomb’s friction on guide 
rails and damping due to the airflow around the cabin 
(cage) in the elevator shaft [13] and [15].
Since the presented experiment results (Tab. 
2) refer to the moment of stopping the cage (v = 0), 
damping as a consequence of airflow around the cage 
can be neglected.
Fig. 19.  An oscillating system with Coulomb’s friction;  
a) motion with the load, b) a model with Coulomb’s damping
The friction on the elevator guide rails creates the 
damping force (Coulomb’s friction), which is constant 
in its magnitude, but it is of the opposite direction 
compared to the motion of the oscillating load. Since 
this type of elevator requires special attention to be 
paid to the guiding accuracy and reducing the guide 

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rails friction, in the example with centrical cabin 
load, the friction forces can be neglected in relation 
to the total load. Nevertheless, the influence can be 
of importance for cabin oscillation analyses [21]. 
A typical diagram of an oscillator with Coulomb’s 
friction is presented in Fig. 19a. The declining of 
amplitudes, as opposed to inner friction oscillation, is 
linear in its nature [15] and [17].
As it was explained in [13], the total force of 
Coulomb’s friction on guide rails is:
 Fn nF
cv tN
  . (17)
For the most part, the literature presents the 
inner friction in the rope as viscous friction, like in 
homogenous bodies, Figs. 2a and b. Nevertheless, 
while the rope is being deformed, the energy 
dissipation also appears due to the friction between 
wires and strands, which slide against one another 
during the deformation process. This creates the 
hysteresis loop (Fig. 20a). The effect causes the 
damping known as hysteretic or structural damping. 
The loss of energy per unit of material volume in one 
loading and unloading cycle is equal to the area closed 
by the hysteresis loop. Experiments have confirmed 
that the energy loss per cycle is approximately 
proportional to the square of the oscillating amplitude. 
A model with hysteretic damping is presented in 
Fig. 20b. Similarly to the above, the rope’s inner 
friction (b
r
) is a combination of viscous and hysteretic 
damping, Fig. 20c.
Fig. 20.  Hys t ere tic	dam ping;	a)	h ys t eresis	lo o p,	 
b) a model with hysteretic damping, and  
c) an equivalent model with internal rope damping
The total damping of the considered system is 
an overall influence of the damping due to the rope’s 
internal friction and the damping caused by the friction 
in guide rails, so it can be represented by the parallel 
connections in the oscillator model, Fig. 21b. This is 
seen in the diagrams shown in Figs. 14 to 17, which 
show the cumulative influence of rope’s internal 
friction and the guide rails’ friction is generally of a 
viscous damping character.
The size of the impact caused by rope damping 
and damping due to guide rails friction can be defined 
via the “overlapping” of the diagram measurement 
results and simulation results, as presented in Fig. 22.
Based on the theory about free harmonic 
damping oscillations (Fig. 21a), it is concluded that 
for determining the dynamic parameters, one should 
take the part of the cage oscillation diagram after the 
driving machine is stopped.
Fig. 21.  Oscillating system with viscous integral damping;  
a) damped oscillations b) a lifting system model
Fig. 22.  The impact of inner and Coulomb’s friction on oscillating 
systems when b = 5161 Ns/m (Table 2)
By measuring the amplitudes and oscillating 
periods of free damping oscillations (changes in cage 
acceleration), the damping coefficient (δ) can be 
determined via a logarithmic decrement, and based on 
this, a resistance force coefficient (b):
 D
x
xn
x
x
T
D
T
i
i
i
in
  

ln ln ,
1
1



  (18)
 bM  2 
e
. (19)
Determining the stiffness coefficient (c) and 
elasticity modulus (E) of hoist ropes can be performed 

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297 Dynamic	Mo delling,	Experimental	Id entif icatio n	and	Co m put er	Simulatio ns	o f	N on-S tatio nar y	Vibratio n	in	High-Speed	Ele v at o r s
Table 2. V alues	o btained	b y	measuring	and	parame t er s	o f	ho is t	r o pes
L [m] M
e
 [kg] δ [s
–1
] ω [rad/s] c [N/m] b [Ns/m] E [MPa] δ/ω
I (Fig. 13a) ↓ 409 18 797 0.33 4.67 409 163 12 222 98 440 0.070
II (Fig. 13b) ↑ 266 20 525 0.21 5.69 663 329 8718 103 792 0.037
III (Fig. 13c) ↑ 413 28 099 0.09 4.26 510 803 5161 124 095 0.022
IV (Fig. 13d) ↓ 536 26 613 0.18 3.88 401 206 9639 126 498 0.047
with the measured oscillation parameters (oscillating 
period, oscillating amplitudes, etc.):
 cM 
e

2
, (20)
 E
cL
A


. (21)
Fig. 23 is an illustration of determining the said 
mechanical characteristics of hoist ropes based on the 
cage acceleration changes diagram (after the driving 
machine was stopped) in the III motion case.
Fig. 23.  The measured values of cage acceleration  
for III motion case
Based on the diagram in Figs. 14 to 17 and 
the expressions (Eq. (18) to (21)), mechanical 
characteristics of steel ropes are defined, as shown in 
Table 2. Upon analysing the measured results, it can 
be concluded that the data for elasticity modulus are in 
agreement with the data from the literature [18]. The 
higher values of elasticity modulus for loaded cabins 
are seen as a consequence of the rope constructed 
by putting the lays of wires in strands and strands 
into a rope. This confirms the validity of the applied 
procedure, enabling us to define the real (exploitation) 
values in mine elevators. The damping coefficient 
values, for which there is no significant comparative 
data, are not constant in size but appear to be different 
in the analysed experiments. Regarding the limitations 
in conducting the experiments in working conditions, 
it was concluded that the experiment should be 
prepared in a laboratory setting in the future.
2.4 Computer Simulations of Mine Elevator Dynamic 
Behaviour and Result Correlation
In order to simulate the mine elevator operation 
process, the study has set the change in hoisting rope 
velocity at the point of winding onto the driving drum 
as the change (diagram), obtained by direct measuring 
on the driving machine via an incremental encoder 
(pos. 9, Fig. 9).
An oscillation diagram was separated with the 
purpose of verifying the dynamic model during the 
whole period while the cage was moving, as well 
as after the complete stop in the driving machine 
moving. The acceleration diagram in the III motion 
case is shown, “overlapping” the diagram with the 
measured results, Fig. 24.
Judging by the diagram in Fig. 24, it can be 
concluded that the dynamic model with dynamic 
parameters, determined by the measuring with 
satisfactory accuracy, shows the real behaviour of 
a mine elevator. As a result, this makes computer 
simulations possible, which will analyse the real loads 
of the exploitation facility [11] and [22]. The described 
method is a new approach that will ensure the analysis 
of dynamic behaviour in systems for vertical hoisting, 
which can be found in exploitation.
Fig. 24.  Cage acceleration diagrams in the motion case III  
(Fig. 13c)
After comparing the results of numerical analysis 
with measurement results, a conclusion arises that 
it would be useful to have some form of numerical 
value for result correlation. In the literature, this value 
is known as Pearson’s correlation coefficient. This 

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298 Đo kić,	R.	–	Vladić,	J.	–	Kljajin,	M.	–	Jo v ano vić,	V .	–	Mar k o vić,	G.	–	K arak ašić,	M.
study did not determine the correlation coefficient for 
individual motions, so that it could be taken as one of 
the ideas for future research.
As an illustration of the possibility of dynamic 
analysis, the following section shows diagrams that 
represent changes in individual values. They were 
created in specialized software for dynamic analysis 
using the experimental data shown in Table 2.
Fig. 25.  Diagram showing the changes in the cage position, 
velocity, and acceleration, for I motion case
Fig. 26.  Comparative diagrams showing the changes in the 
elongation, stiffness coefficient, and resistance force coefficient  
of hoist ropes, for I motion case
Fig. 27.  Diagram showing the changes in cage position, velocity, 
and acceleration, for II motion case
Fig. 28. Comparative diagrams showing the changes in the 
elongation, stiffness coefficient, and resistance force coefficient of 
hoist ropes, for II motion case
Fig. 29.  Diagram showing the changes in cage position, velocity, 
and acceleration, for III motion case
Fig. 30.  Comparative diagrams showing the changes in the 
elongation, stiffness coefficient, and resistance force coefficient of 
hoist ropes for III motion case
Fig. 31.  Diagram showing the changes in cage position, velocity, 
and	acceleratio n	f o r	IV	mo tion	case
Fig. 32.  Comparative diagrams showing the changes in the 
elongation, stiffness coefficient, and resistance force coefficient of 
ho is t	r o pes	f or	IV	mo tio n	case
3  AN ANALYSIS OF THE HOISTING VELOCITY IMPACT AS THE 
BASIS FOR CONTROL SYSTEM
The material of this paper confirms that there are 
different analyses of individual parameters for vertical 
hoist systems. One part is of special interest, and that 
is the research on the changes in hoist velocities, 
i.e., defining the control system [23]. The above 
method can be used for determining an optimal form 

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299 Dynamic	Mo delling,	Experimental	Id entif icatio n	and	Co m put er	Simulatio ns	o f	N on-S tatio nar y	Vibratio n	in	High-Speed	Ele v at o r s
for changing the hoist velocity, to reach maximum 
capacity, i.e., the shortest time for the motion, along 
with securing a high level of motion, especially in 
modern passenger elevators, whose velocities reach 
up to 20 m/s. As stated above, the dynamic parameters 
during the elevator or exploitation facility design 
process can only be determined with limited accuracy, 
so the exploitation deviation can be significant. The 
method shown in this paper can be used as a basis, 
which shows that it is possible to apply straightforward 
methods and form an optimal elevator facility. 
The application of the illustrated method makes it 
possible to determine real parameters and perform 
dynamic analysis even after the installation process. 
Therefore, the method makes it possible to adjust 
and correct the exploitation facility and improve the 
previously defined optimal operating conditions. This 
particularly concerns defining the appropriate control 
program, which ensures the shortest motion time with 
the maximum level of comfort, i.e., with the control of 
the dynamic loads.
The following figures present some of the possible 
ways of impacting driving kinematic characteristics 
through a control program. Fig. 33 shows how the 
changes in velocity form characteristics can impact 
the cabin (cage) oscillating amplitudes, as well as 
the motion comfort. As it appears, it is important to 
note that by controlling the drive in the acceleration 
period, and especially by choosing the right moment 
to switch from acceleration to stationary velocity, the 
system can be significantly “relieved”, i.e., the motion 
comfort can be bettered (the amplitudes can be quite 
decreased), pos. 4 in Fig. 33.
Installing an acceleration transducer 
(accelerometer) on the points where ropes connect 
with the cabin (cage) has acceleration change as a 
response at any moment. This is vital with high-speed 
elevators, in cases in which nominal velocity is not 
reached between two neighbouring floors, but the 
acceleration is followed by braking, Fig. 34.
The right control is key for braking at the most 
convenient moment, as shown by line (2) in the 
Fig. 33.  The influence of driving system kinematic characteristics on the motion comfort;  
a) change of velocity, b) change of acceleration
Fig. 34.  The layout of the influence of controlling the magnitude of oscillating system amplitudes;  
a) change of velocity, b) change of acceleration

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velocity change diagram, Fig. 34a, which causes much 
smaller oscillating amplitudes, Fig. 34b. The distance 
the cabin (cage) has passed and the accuracy of station 
landing must be taken into account when creating a 
control system.
4  CONCLUSION
The dynamic analysis and scientific approach shown 
in this paper are of special importance taking into 
account the nature of the systems; they are meant for 
lifting (lowering) people and load to great heights 
(depths) at velocities up to 20 m/s. The complexity of 
the dynamic analysis can also be seen because these 
are oscillating processes with variable parameters and 
boundary conditions at the incoming/outgoing points 
of the steel rope and the driving pulley. Furthermore, 
the stiffness and elasticity modulus of steel ropes 
are not constant in magnitude but rather change 
depending on the changes in cage position, as well as 
construction, stress, and rope exploitation time.
The dynamic model that is suitable for the 
analysis can be formed by observing a particular 
system and by simplifying (i.e., omitting) the small 
values of higher order.
The presented approach on forming the 
differential motion equation for the incoming end 
of the rope onto the driving pulley helps determine 
the critical hoisting velocity in the function with 
mechanical characteristics (elasticity modulus and 
damping) and stress (load) in steel ropes. Moreover, 
this method also makes it possible to execute 
computer simulations and dynamic behaviour analysis 
during the motion by using the appropriate specialized 
software for the dynamic analysis of mechanical 
systems.
The paper presents an experimental method that 
can be used to determine dynamic parameters such as 
stiffness, elasticity modulus, and damping in exploited 
elevator steel ropes. Specific elasticity modulus values 
presented in Table 2 reveal the noticeable dependence 
on the load, i.e., rope stress. In addition, the damping 
coefficient in ropes is not a constant size, but it 
depends on the position of a cage. It can be deduced 
that ropes experience a combination of viscous 
damping and hysteretic damping, which should be 
further investigated in these systems.
By forming adequate dynamic models, 
determining real values for dynamic parameters, and 
simulations of dynamic behaviour for real facilities, it 
is possible to define the basis for a program that would 
control the driving mechanism. That would ensure 
minimal dynamic loads, which is especially relevant 
for comfort during the motion, while simultaneously 
securing the optimal motion time, i.e., the efficiency 
of high-speed passenger elevators, especially in 
transient operation regimes.
As for mine elevators, strict safety conditions are 
required in order for the facility to operate, especially 
regarding elevators for the transport of people. It is 
necessary to secure special conditions at the moment 
of power transfers and friction motion to avoid the 
slipping of the rope as a whole along the driving 
pulley. Finally, this is vital from the point of view of 
defining suitable boundary conditions for a dynamic 
model at the points where the ropes get on or off the 
driving pulley.
5  NOMENCLATURES
u
1
, u
2
 elastic deformation of the rope on the  
incoming and outgoing rope end, [mm]
E rope elasticity modulus, [MPa]
A rope’s cross-section area, [mm
2
]
a acceleration of the driving mechanism, [m/s
2
]
M
m
 driving motor torque, [Nm]
i gear ratio, [-]
η driving mechanism efficiency, [-]
J
r
 moment of inertia of rotational masses, reduced 
to the shaft of a driving pulley, [kgm
2
]
R driving pulley radius, [m]
q rope weight per meter, [kg/m]
l wound rope length, [m]
v (v = dl / dt) circumferential velocity, [m/s]
F
f
 friction force between sliding guide shoes and 
guide rails, [N]
l
0
 length of winded rope at time t = 0, [m]
n
v
 (n
v
 = 4) number of the roller guide groups, [-]
n
t
 (n
t
 = 3) number of rollers in the guide group, [-]
µ rolling resistance of the roller guide on the guide 
rail, [-]
F
N
  force of the roller guide pressure on the guide 
rail for centric load, which depends on the spring 
tightening during installation, [N]
x
i
 (also, x
i+1
 and x
i+n
) measured oscillating 
amplitudes, [mm]

T measured oscillating period of free damping 
oscillations, [s]
M
e
 reduced oscillating mass (M
e
 = M + (q · L(t)) / 3), 
[kg]
M total mass hanging on hoist ropes (loaded cage 
and compensation ropes), [kg]
L(t) hoist ropes’ free length ( Lt Lv tt () () 

d ), [m]
v(t) circumferential velocity of the pulley (drum), 
[m/s]

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301 Dynamic	Mo delling,	Experimental	Id entif icatio n	and	Co m put er	Simulatio ns	o f	N on-S tatio nar y	Vibratio n	in	High-Speed	Ele v at o r s
ω circular frequency of free oscillations  
(   
22
), [rad/s]
 ω damping oscillation frequency ( 

 2 T), 
[rad/s]
L hoist ropes length, [m]
δ damping coefficient, [s
‒1
]
c ropes stiffness coefficient, [N/m]
b resistance force coefficient, [Ns/m] 
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