Dynamic analysis of flexible hoisting rope with time-varying length
The governing equations of flexible hoisting rope are developed employing Hamilton’s principle. Experiments are performed. It is found that the experimental data agree with the theoretical prediction very well. The results of simulation and experiment show that the flexible hoisting system dissipate...
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Інститут механіки ім. С.П. Тимошенка НАН України
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| Cite this: | Dynamic analysis of flexible hoisting rope with time-varying length / Ji-hu Bao, Peng Zhang, Chang-ming Zhu // Прикладная механика. — 2015. — Т. 51, № 6. — С. 128-141. — Бібліогр.: 17 назв. — англ. |
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| author | Ji-hu Bao Peng Zhang Chang-ming Zhu |
| author_facet | Ji-hu Bao Peng Zhang Chang-ming Zhu |
| citation_txt | Dynamic analysis of flexible hoisting rope with time-varying length / Ji-hu Bao, Peng Zhang, Chang-ming Zhu // Прикладная механика. — 2015. — Т. 51, № 6. — С. 128-141. — Бібліогр.: 17 назв. — англ. |
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| description | The governing equations of flexible hoisting rope are developed employing Hamilton’s principle. Experiments are performed. It is found that the experimental data agree with the theoretical prediction very well. The results of simulation and experiment show that the flexible hoisting system dissipates energy during downward movement but gains energy during upward movement. Further, a passage through resonance in the hoisting system with periodic external excitation is analyzed. Due to the time-varying length of the hoisting rope the natural frequencies of the system vary slowly, and transient resonance may occur when one of frequencies coincides with the frequency of external excitation.
Основні рівняння гнучкого підіймального тросу отримано застосуванням принципу Гамільтона. Проведено експерименти, результати яких добре узгоджуються з теоретичним передбаченням. Результати моделювання і експеримент показують, що гнучка підіймальна система розсіює енергію при спуску і накопичує енергію при підйомі. Далі досліджувався перехід гнучкої підіймальної системи через резонанс за умови періодичного зовнішнього збудження. Якщо довжина гнучкої підіймальної системи змінюється з часом, то власні частоти системи слабо змінюються і можуть спостерігатися перехідні резонанси, коли одна з частот співпадає з частотою зовнішнього збудження.
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2015 ПРИКЛАДНАЯ МЕХАНИКА Том 51, № 6
128 ISSN0032–8243. Прикл. механика, 2015, 51, №6
J i - h u B a o 1 , P e n g Z h a n g 2 , C h a n g - m i n g Z h u 3
DYNAMIC ANALYSIS OF FLEXIBLE HOISTING ROPE
WITH TIME-VARYING LENGTH
1 School of Mechanical Engineering, Shanghai Jiaotong University, Shanghai, People’s
Republic of China; e-mail: tiger0203@163.com
2 State Key Laboratory of Mechanical System and Vibration, Shanghai Jiaotong University,
Shanghai, People’s Republic of China; e-mail: zhp_roc@sjtu.edu.cn
3 School of Mechanical Engineering, Shanghai Jiaotong University, People’s Republic of
China. e-mail: zhuchangming@sjtu.edu.cn
Abstract. The governing equations of flexible hoisting rope are developed employing
Hamilton’s principle. Experiments are performed. It is found that the experimental data
agree with the theoretical prediction very well. The results of simulation and experiment
show that the flexible hoisting system dissipates energy during downward movement but
gains energy during upward movement. Further, a passage through resonance in the hoisting
system with periodic external excitation is analyzed. Due to the time-varying length of the
hoisting rope the natural frequencies of the system vary slowly, and transient resonance may
occur when one of frequencies coincides with the frequency of external excitation.
Key words: dynamic analysis; flexible hoisting rope; transient resonance.
1. Introduction.
While rope is employed in hoisting industry such as mine hoists, elevators, cranes etc, it
is subject to vibration due to its high flexibility and relatively low internal damping
characteristics [1, 2]. Most often these systems are modeled as either an axially moving
tensioned beam or string with time-varying length and a rigid body at its lower end [3, 4]. It
was shown that the vibration energy of the rope changes in general during elongation and
shortening [5, 6]. When the rope length is being shortened, vibration energy increases
exponentially with time, causing dynamic instability [7]. The study of rope vibration
problems in flexible hoisting systems has attracted wide attention. Chi and Shu [8]
calculated the natural frequencies associated with the vertical vibration of a stationary cable
coupled with an elevator car. Terumichi and Ohtsuka et al. [9] assumed the velocity of the
string is constant and studied the transverse vibrations of a string with time-varying length
and a mass-spring system at the lower end with theoretical and experimental methods. Fung
and Lin [10] analyzed the transverse vibration of elevator rope with time-varying length and
the time-varying mass and inertia of rotor were considered. A variable structure control
scheme is proposed to suppress the transient amplitudes of vibrations. Kaczmarczyka and
Ostachowiczb [11] studied coupled vibration of deep mine hoisting cable and built a
distributed-parameter model. They found that response of the catenary-vertical rope system
may feature a number of resonance phenomena. Zhang and Agrawal [12] derived the
governing equation of coupled vibration of flexible cable transporter system with arbitrarily
varying length. Zhu and Chen [13] investigated the control of elevator cable with theoretical
and experimental methods. A novel experimental method is developed to validate the
129
uncontrolled and controlled lateral responses of a moving cable in a high-rise elevator and
shown good agreement with the theoretical predictions. Zhang [14] presented a systematic
procedure for deriving the model of a cabel transporter system with arbitrarily varying cable
length and proposed a Lyapunov controller to dissipate the vibratory energy. Zhang and Zhu
et al. [15] derived the governing equation and energy equation of longitudinal vibration of
flexible hoisting system with arbitrarily varying length.
While extensive studies focus individually on vibration characteristics of the rope with
time-varying length, the dynamic stability of the rope has also been studied by several
researchers. Kumaniecka and Niziol [16] investigated the longitudinal-transverse vibration
of a hoisting cable with slow variability of the parameters. The cable material non-linearity
was taken into account and unstable regions were identified by applying the harmonic
balance method. General stability characteristics of horizontally and vertically translating
beams and strings with arbitrarily varying length and various boundary conditions were
studied in Zhu and Ni [17]. While the amplitude of the displacement can behave in a
different manner depending on the boundary conditions, the amplitude of the vibratory
energy of a translating medium decrease and increase in general during extension and
retraction, respectively. Lee [7] introduces a new technique to analyze free vibration of a
string with time-varying length by dealing with traveling waves. When the string length is
being shortened, free vibration energy increases exponentially with time, causing dynamic
instability.
Extensive research efforts on the flexible hoisting rope with time-varying length have
been done in the last few decades as aforementioned, however, the study of most studies
were restricted to cases with constant transport speed samples. The dynamic characteristics
of flexible hoisting rope with an arbitrarily varying length are the subject of this
investigation. The governing equations are developed employing the extended Hamilton’s
principle. The derived governing equations are shown to be nonlinear partial differential
equations(PDEs) with variable coefficients. On choosing proper mode functions that satisfy
the boundary conditions, the solution of the governing equations was obtained using the
Galerkin’s method. In order to evaluate the mathematical model, an experimental set-up is
built and some experiments are conducted. Comparing the experimental data to the
simulation, a favourable result is obtained, which indicates that the proposed mathematical
model is valid for flexible hoisting rope. Further, the phenomenon of passage through
resonance in hoisting rope system is studied in this paper. Based on the proposed
fundamental dynamic analyses, further vibration control can be adopted for such the flexible
hoisting systems in the near future.
2. MODEL OF FLEXIBLE HOISTING SYSTEM.
Flexible hoisting system can be simplified as an axially moving string with time-
varying length and a rigid body m at its lower end, as shown in Fig. 1. The rail and the
suspension of the rail are assumed to be rigid. The string has Young’s modulus E, diameter
d and the density per unit length ρ. The origin of coordinate is set at the top end of string and
the instantaneous length of string is l(t) at time t. The instantaneous axial velocity,
acceleration and jerk of the string are ( ) ( )v t l t , ( ) ( )a t v t and ( ) ( )j t a t respectively,
where the overdot denotes time differentiation. At any instant t, the transverse displacement
of string is described by y(x,t), at a spatial position x, where 0 ≤ x ≤ l(t). In actual flexible
hoisting system, the rotational unbalance of the traction motor or the abnormal off-track of
rope possibly causes vibration of the hoisting system. To reproduce this phenomenon, a
transverse extrinsic disturbing excitation e(t) is applied at the upper end of the string. In this
paper, all the equations and derivations base on the following assumptions.
1. Young’s modulus E, diameter d and density ρ of the string are always constants;
2. Only transverse vibration is considered here. The elastic distortion of string arousing
from the transverse vibration is much less than the length of the string;
130
3. The bending stiffness of the string, all the damp and friction, and the influence of air
current are ignored.
Fig. 1. Schematic of flexible hoisting string with time-varying length.
2.1. Energy of flexible hoisting system. After the string is deformed, the position vector
R of a point at x can be written as:
( ) ( , )x t y x t R i j , (1)
where i and j are the unit vectors along the x-axes and y-axes, respectively. The material
derivative of R yields the velocity vector
( ) [ ]t xv t y vy V i j , (2)
where the subscript t denotes partial differentiation with respect to time, and subscript x
denotes partial differentiation with respect to space. Similarly, the position vector Rc and
velocity vector Vc of rigid body can be respectively written as:
( ) ( ( ), )l t y l t t cR i j ; (3)
( ) ( ( ), )tv t y l t t cV i j . (4)
Then, the kinetic energy of flexible hoisting system is computed by
( )
0
( )
1 1
( )
2 2
l t
k
x l t
E t m dx
cV V V Vc . (5)
The first term on the right of Eq. (5) represents the kinetic energy of rigid body, the sec-
ond term represents the kinetic energy of the string. The elastic strain energy of the string is
( ) 2
0
1
( )
2
l t
eE t P EA dx
, (6)
where P(x,t) is the quasi-static tension at spatial position x of the string at time t due to
gravity. Since the string is acted upon not only by the weight of the concentrated mass at the
lowest end but also its own weight, the tension P(x,t) is expressed as
( ( ) ) P m l t x g . (7)
And ε represents the strain measure at spatial position x of the string and can be ex-
pressed as
/ds dx dx . (8)
131
Fig. 2.
A small element of the string in a deformed position.
As shown in Fig. 2, ds can be expressed as
2 4 2
2 1 1 1
1 / 1 1
2 8 2
y y y
ds dy dx dx dx dx
x x x
. (9)
Substituting Eq. (9) into Eq. (8) yields
2
2
1
xy . (10)
2.2. Free vibration equations. According to the characteristics of top restriction of the
string, the boundary conditions at x(t) = 0 are
(0, ) 0y t , (0, ) 0ty t . (11)
On substitution of Eqs (5) and (6) in the Hamilton’s Principle,
2
1
( ( ) ( )) 0
t
k et
E t E t dt (12)
and apply the variational operation. Because the length of the string l(t) changes with time,
the standard procedure for integration by parts with respect to the temporal variable can’t
apply. Applying Leibnitz’s rule and part integration results in the following expressions
( )
0
l t
t x ty vy y dx
( ) ( )
0 0( )
l t l t
t x t x t xl t
y vy ydx v y vy y y vy ydx
t t
. (13)
Following the standard procedure for integration by parts with respect to the spatial
variable and invoking Equation (13), one obtains from Eq. (12),
2
1
31
( , ) ( , )
2
t
t x xt
m y l t Py EAy y l t d t
t
2
1
( ) 3
0
1
( ) 0
2
t l t
t x t x x xt
y vy v y vy Py EA y ydxdt
t x x x
. (14)
Setting the coefficients of δw in Eq. (14) to zero yields the governing equations in the forms
2 23
( 2 ) 0
2tt xt x xx x x xx x xxy vy vy v y P y Py EAy y ; 0 < x< l(t). (15)
132
The first four terms in Eq. (15) correspond to the local, Coriolis, tangential and centripe-
tal acceleration, respectively. The resulting boundary conditions from Eq. (14) at x = l(t) is
31
0
2tt x xmy Py EAy ; x = l(t). (16)
The energy associated with the transverse vibration of the system is
( ) ( )22 2 4
0 0
1 1 1 1
( ) ( , ) ( , ) ( , )
2 2 2 4
l t l t
v t t x x xE t my l t y x t vy x t dx Py EAy dx
. (17)
2.3. Forced vibration equations. When the external excitation occurs at the upper end
of string, the governing Eq. (15) must be adjusted. Compared with Eq. (12), the
corresponding boundary conditions at are changed into
(0, ) ( )y t e t ; ( , ) 0y l t . (18)
Obviously, the boundary conditions are non-homogeneous and difficult to be applied di-
rectly. Here, the procedure described in Reference [5] is used to transfer the governing Eq.
(15) with non-homogeneous boundary conditions into equation of motion with
homogeneous boundary conditions. The transverse displacement is expressed in the form
( , ) ( , ) ( , ),y x t w x t h x t (19)
where w(x,t) is the part that satisfies the homogeneous boundary conditions and h(x,t) is the
part that does not satisfy the homogeneous boundary conditions. Substitute Eq. (19) into Eq.
(15) yields
2 2 23
( 2 ) ( 2 );
2tt xt x xx x x xx x xx tt xt x xxw vw vw v w P w Pw EAw w h vh vh v h
2 2 23 3 3
(3 3 ) 0
2 2 2x x xx x xx x xx x x xx x x xx x xxP h Ph EA w w h w h w h w h h h h ; 0 < x < l(t), (20)
where w(x,t) is the state variables. Equation (20) describes the transverse vibration of the
flexible hoisting system under extrinsic disturbing excitation. The corresponding boundary
condition is
3 2 2 31 1
3 3 0
2 2tt x x tt x x x x x xmw Pw EAw mh Ph EA w h w h h ; x = l(t). (21)
Set the function h(x,t) to first-order polynomial,
0 1( , ) ( ) ( )
( )
x
h x t a t a t
l t
. (22)
Then, when x(t) = 0 and x(t) = l(t),
(0, ) ( )h t e t ; ( ( ), ) 0h l t t (23)
Substituting Eq. (23) into Eq. (22), the coefficients a0(t) and a1(t) can be obtained as,
0 ( ) ( )a t e t ; 1( ) ( )a t e t . (24)
Therefore,
( , ) ( ) ( )
( )
x
h x t e t e t
l t
. (25)
Once h(x,t) is known, the solutions for w(x,t) is sought from Eq. (20). y(x,t) is obtained sub-
sequently from Eq. (19). Equation (20) is a partial differential equation which describes the
dynamics of the flexible hoisting string. The equation defined over time-dependent spatial
domain rendering the problem non-stationary. Hence, the exact solution to this problem is
133
not available, and recourse must be made to an approximate analysis. In what follows,
numerical techniques are employed to obtain approximate solution for the governing
equation.
3. Discretization of the governing equation.
Equation (20) is a partial differential equation with infinite dimensions and many
parameters are time-variant. It is impossible to obtain an exact analytical solution from Eq.
(20). In this section, Galerkin’s method is applied to truncate the infinite-dimensional partial
differential equation into a nonlinear finite-dimensional ordinary differential equation with
time-variant coefficients. Then, solve them with numerical methods. In order to map Eq.
(20) onto the fixed domain, a new independent variable ζ=x/[l(t)] is introduced and the time-
variant domain [0, l(t)] for x is converted to a fixed domain [0, 1] for ζ. According to the
characteristic of taut translating string, the solution of the transverse vibration w(x,t) is as-
sumed in the form [11, 12]
1 1
( , ) ( ) ( ) ( ) ( )
n n
i i i ii i
w x t q t x l q t , (26)
where qi(t) (i = 1,2,3,…,n) is the generalized coordinates respect to w(x,t), n is the number of
included modes; ( )i is trial function [11,12],
( ) 2 sini i . (27)
Consequently, expansion Eq. (26) results in the expressions for partial derivatives of the
transverse displacement function:
'
1
1
( , ) ( ) ( )
n
x i ii
w x t q t
l
; ''
12
1
( , ) ( ) ( )n
xx i ii
w x t q t
l
;
' '' '
1 2 2
1
( , ) ( ) ( ) ( ) ( ) ( ) ( )
n
xt i i i i i ii
v v
w x t q t q t q t
l l l
;
'
1 1
2
( , ) ( ) ( ) ( ) ( )
n n
tt i i i ii i
v
w x t q t q t
l
2 2 2
' ' ''
1 1 12 2
2
( ) ( ) ( ) ( )
n n n
i i i ii i i
v a v
q t
ll l
. (28)
Substituting Eq. (28) into Eq. (20), multiplying the governing equation by φj(ζ) (j =
= 1, 2, 3, …, n), integrating it from ζ = 0 to 1, and using the boundary conditions and the
orthonormality relation for φi(ζ) yield the discretized equation of transverse vibration for the
flexible hoisting rope with time-variant coefficients
( ) MQ CQ KQ S Q F , (29)
where T
n tqtqtq )](,),(),([ 21 Q is vector of generalized coordinate, M, C, K and F are
matrixes of mass, damp, stiffness and generalized force respect to Q, respectively. S(Q) is
higher order item of generalized coordinate. The matrices are expressed as follows:
ijijM ;
1 '
0
2
(1 ) ( ) ( ) ij i j
v
C d
l
;
2
1 1 2' ' '
20 0
( ) 1 ( ) ( ) 1 ( ) ( )ij i j i j
a v
K t d d
l l
134
+ 1 ' '
0
1 ( ) ( )i j
g
d
l
1 1'' 2 ''
2 40 0
3
( ) ( ) ( ) ( ) ( )
2
i j i j
mg EA
d e t d
l l
;
21 ' ''
1 14 0
3
( ) ( ) ( ) ( ) ( ) ( )
2
n n
i i i i ji i
EA
S q t q t d
l
j Q
–
1 ' ''
1 14 0
3
( ) ( ) ( ) ( ) ( ) ( )
n n
i i i i ji i
EA
e t q t q t d
l
;
2
1 1
2 0 0
2 2
( ) ( ) ( ) ( ) 1 ( ) ( ) ( ) j j j
v a v g
e t e t e t e t d e t d
l l ll
F , (30)
where the superscript “＇” denotes partial differentiation for normalized variable ζ, δij is the
Kronecker delta defined by δij =1 if i=j and δij =0 if i≠j (i=1,2,3,…,n, j=1,2,3,…,n). Solving
the ordinary differential Eq. (29) with numerical methods may yield the instantaneous
values of Q. Substituting these values into Eq. (26) may yield the instantaneous values of the
transverse vibration of the string w(x, t). The mathematical model defined by Eq. (29)
illustrates the true dynamic nature of the flexible hoisting string, and can be used to predict
and analyze the dynamic characteristics of flexible hoisting string.
4. Experiment.
4.1. Experiment set-up.
Fig. 3.
Schematic diagram of experimental set-up for flexible hoisting system.
To validate the mathematical model, an experimental set-up of flexible hoisting system
is designed and built as Fig. 3. The set-up, simulating the hoisting system of traction
elevator, consists of traction system, guide system, excitation system and data acquisition
system. A frequency conversion motor is applied in flexible hoisting system. The rotation
speed of motor may be controlled by adjusting the output of transducer to obtain the
anticipant motion curve of hoisting system. A thin steel rope with a diameter of 3.2mm is
135
chosen as the hoisting rope. The model car and counterweight are made up of many weights.
The mass of car and counterweight is changeable by adding or reducing the number of
weight.
The hoisting rope at the car side is the main research object, whose dynamic behavior
will be studied, in this set-up. To simulate the extrinsic disturbing excitation in actual
hoisting system, a transverse vibration exciter is appied at the top of the objective rope. The
output of the exciter is decided by an adjustable signal generator. A micro-sensor with a
mass of 4g is attached at a certain position of the objective rope to acquire the transverse
vibration acceleration of rope. The signals from micro-sensor are transmitted to a computer
and saved. Fig. 4 gives the actual picture of the experimental set-up. The main parameters of
test are shown in Table 1.
Fig. 4.
Actual picture of the experimental set-up for flexible hoisting system.
Table 1. Parameters of experimental set-up of flexible hoisting system.
Items Data values
Density per unit length ρ(kg/m) 0.042
Young’s modulus E (N/m2) 1×1012
Rope diameter d(m) 3.2×103
Hoisting mass m(kg) 15
Excitation e(t)(m) 5×104sin(πt)
Minimum length of the string
lmin(t)(m)
0.8
Maximum length of the string
lmax(t)(m)
4.8
Maximum velocity vmax(m/s) 0.55
Maximum acceleration amax(m/s2) 0.4
Total travel time t(s) 8
Number of transverse modes n 4
136
4.2. Experiment procedure. Now the transverse vibration of flexible hoisting system
will be calculated with theoretical equation and tested with experimental set-up, respec-
tively. And the results will be compared. All the parameters, using in calculation and test,
are the same. A downward or upward movement of car is prescribed to be the input of
theoretical equations and experimental set-up. At the beginning, the car starts up at the top
of flexible hoisting system and goes down. When arriving at the bottom, the car pauses for a
moment and turns back to the start. Fig. 5 gives the prescribed displacement, velocity,
acceleration curves of flexible hoisting system, where the processes of acceleration,
deceleration and uniform speed downwards and upwards are included.
Fig. 5.
Movement profile of flexible hoisting system: (a) )(tl ; (b) )(tv ; (c) )(tv .
In actual flexible hoisting system, the turning disbalance of rotor or the abnormal off-
track of rope is the main origin of extrinsic disturbing excitation for flexible hoisting system.
It is supposed that a transverse extrinsic disturbing excitation e(t) is applied at the top end of
flexible hoisting system to reproduce this phenomenon. The extrinsic disturbing excitation
disturbs the dynamic behavior of flexible hoisting rope only when the rope is moving. It is
applied to the theoretical equations and experimental set-up.
During following calculation, the number of included modes in w(x,t) n is set to 4,
which was already proved to be a proper value with a great deal calculating results and
comparisons. When n = 4, the less calculating time and the satisfying veracity of results may
be simultaneously obtained.
4.3. Free vibraion responses. The numerical simulations with the exact experiment pa-
rameters are conducted in order to compare with the experiments and the experimental re-
sults are favourably compared with the simulations, which can be seen in Figs. 6 (downward
movement) and 7 (upward movement). Comparing the results of test and calculation in Figs.
6 and 7, the extent and trend of vibration curves are similar. Therefore, it can be concluded
that the theoretical equations, proposed in this paper, may be used to evaluate the vibration
of flexible hoisting rope.
137
Fig. 6.
Free vibration responses of the flexible hoisting rope at 0.5m above the car during down-
ward movement: (a) Displacement curve; (b) Velocity curve; (c) Acceleration
curve(simulation); (d) Acceleration curve(experiment).
Fig. 7.
Free vibration responses of the flexible hoisting rope at 0.5m above the car during upward
movement: (a) Displacement curve; (b) Velocity curve; (c) Acceleration curve(simulation);
(d) Acceleration curve(experiment).
138
Fig. 6 displays reducing vibration amplitudes with increasing length of the rope during
downward movement. This is due to the energy of flexible hoisting system transfers from
the transverse vibration to the axial motion by bringing some mass into the domain of
effective length, i.e., the axially hoisting rope is dissipative during downward movement,
thus leading to a stabilized transverse dynamic reponse, as shown in Fig. 8(a). A possible
physical interpretation of the result is as follows: during downward movement negative
external work is required to maintain the prescribed axial motion which, in turn, brings
about a convection of mass in the domain of effective length. At the same time, frequencies
of the transverse vibration reduce with increasing length of the rope. This is due to the fact
that the mass of the rope increase and the stiffness of the rope decrease, i.e., the rope
becomes somewhat “softer”.
Fig. 8.
Total energy associated with the transverse vibration of the system during movement: (a)
Downward movement; (b) Upward movement.
By contrast, in Fig. 7, we observe that vibration amplitudes of the rope increase with
decreasing length of the rope during upward movement. This is due to the energy of flexible
hoisting system transfers from the axial motion to the transverse vibration by leaving some
mass out of the domain of effective length, i.e., the axially hoisting rope gains energy during
upward movement, thus leading to an unstabilized transverse dynamic reponse, as shown in
Fig. 8(b).
A possible physical interpretation of the result is as follows: during upward movement
positive external work is required to maintain the prescribed axial motion which, in turn,
brings about a convection of mass out of the domain of effective length. In the mean time,
frequencies of the transverse vibration increase with decreasing length of the rope. This is
due to the fact that the mass of the rope decrease and the stiffness of the rope is increase,
i.e., the rope becomes somewhat “stiffer”.
4.4. Forced vibraion responses. During movement of hoisting system, the flexible
hoisting system is subjected to vibration caused by various sources of excitation. They
include excitations due to the irregularities of the guiding system and rotational unbalance of
the traction motor as well as environmental phenomena such as air current.
The system parameters are changing due to the time-varying length of the rope. The rate
of variation of the length is, however, slow, and the vibrations represent waves in a slowly
varying domain. Hence, the hoisting rope is essentially a nonstationary vibration system
with slowly varying frequencies. Therefore, a passage through resonance may occur when
one of the slowly varying frequencies coincides with the frequency of the extrinsic
disturbing excitation at some critical time instant.
139
Fig. 9.
Forced vibration responses of the flexible hoisting rope at 0.5m above the car during
downward movement: (a) Displacement curve; (b) Velocity curve; (c) Acceleration
curve(simulation); (d) Acceleration curve(experiment).
Fig. 10.
Forced vibration responses of the flexible hoisting rope at 0.5m above the car during
downward movement: (a) Displacement curve; (b) Velocity curve; (c) Acceleration
curve(simulation); (d) Acceleration curve(experiment).
140
Forced vibration responses for hoisting rope with extrinsic disturbing excitation are il-
lustrated in Figs. 9(downward movement) and 10(upward movement). From Figs. 9 and 10,
it can be seen that transient resonance occurs during movement of hoisting system. The
amplitudes exhibit oscillatory behavior before the resonance, and near the resonance the
amplitudes increase rapidly and decline afterwards due to damping, developing damped beat
phenomena. This is due to one of time-varying frequencies of the hoisting rope coincides
with the frequency of the extrinsic disturbing excitation during movement of hoisting
system. It should be noted that the adverse dynamic response in hoisting system promote
large oscillations in rope tension. The phenomenon cannot be ignored, as the high amplitude
in the tension contributes directly to fatigue of rope. Fatigue often results in the hoisting
ropes being discarded after lower working cycles. Therefore, suitable strategy can be sought
to minimize the effects of adverse dynamic response of the system.
5. Conclusions.
The nonlinear dynamic characteristics for a flexible hoist rope with time-varying length
considering coupling of axial movement and flexural deformation are analyzed in this paper.
The flexible hoisting system is modeled as an axially moving string with time-varying
length and a rigid body at its lower end. The governing equations are derived by using
Leibnitz’s rule and the Hamilton’s principle. The Galerkin’s method is used to truncate the
infinite-dimensional partial differential equations into a set of nonlinear finite-dimensional
ordinary differential equations with time-variant coefficients.
To validate the theoretical model, an experimental set-up of flexible hoisting system is
built and some experiments are performed. By comparing the experimental results to the
numerical simulation, a good agreement between the simulation and experiment is obtained,
thus validating the mathematical model of flexible hoisting system. Based on the simulation
and experiment, the following conclusions can be obtained:
1. The flexible hoisting rope with time-varying length experiences instability during
upward movement, the natural frequences are increasing because of the reducing mass and
the increasing stiffness of the rope, and the energy transforms from the axial movement into
the flexible deformation.
By contrast, it is stable during downward movement, the natural frequences are decreas-
ing because of the increasing mass and the reducing stiffness of the rope, and the energy
coverts from the flexible deformation into the axial movement.
2. The flexible hoisting rope is a nonstationary oscillatory system with slowly varying
frequencies. The transient resonance may occur when one of time-varying frequencies of the
hoisting rope coincides with the frequency of the extrinsic disturbing excitation.
3. The proposed the theoretical model and analyses about the dynamic characteristics of
flexible hoisting system in this paper will be helpful for the researchers to comprehend its
dynamic behavior and develop the proper method to suppress the vibration in practice.
Acknowledgements.
This research work was supported by the State Key Laboratory of Mechanical System
and Vibration, Shanghai Jiaotong University (MSV-2010-06).
Р Е ЗЮМ Е . Основні рівняння гнучкого підіймального тросу отримано застосуванням прин-
ципу Гамільтона. Проведено експерименти, результати яких добре узгоджуються з теоретичним пе-
редбаченням. Результати моделювання і експеримент показують, що гнучка підіймальна система
розсіює енергію при спуску і накопичує енергію при підйомі. Далі досліджувався перехід гнучкої
підіймальної системи через резонанс за умови періодичного зовнішнього збудження. Якщо довжина
гнучкої підіймальної системи змінюється з часом, то власні частоти системи слабо змінюються і мо-
жуть спостерігатися перехідні резонанси, коли одна з частот співпадає з частотою зовнішнього
збудження.
141
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Поступила 27.12.2012 Утверждена в печать 26.05.2015
*From the Editorial Board: The article corresponds completely to submitted manuscript.
|
| id | nasplib_isofts_kiev_ua-123456789-141027 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 0032-8243 |
| language | English |
| last_indexed | 2025-12-07T18:11:07Z |
| publishDate | 2015 |
| publisher | Інститут механіки ім. С.П. Тимошенка НАН України |
| record_format | dspace |
| spelling | Ji-hu Bao Peng Zhang Chang-ming Zhu 2018-07-21T13:05:19Z 2018-07-21T13:05:19Z 2015 Dynamic analysis of flexible hoisting rope with time-varying length / Ji-hu Bao, Peng Zhang, Chang-ming Zhu // Прикладная механика. — 2015. — Т. 51, № 6. — С. 128-141. — Бібліогр.: 17 назв. — англ. 0032-8243 https://nasplib.isofts.kiev.ua/handle/123456789/141027 The governing equations of flexible hoisting rope are developed employing Hamilton’s principle. Experiments are performed. It is found that the experimental data agree with the theoretical prediction very well. The results of simulation and experiment show that the flexible hoisting system dissipates energy during downward movement but gains energy during upward movement. Further, a passage through resonance in the hoisting system with periodic external excitation is analyzed. Due to the time-varying length of the hoisting rope the natural frequencies of the system vary slowly, and transient resonance may occur when one of frequencies coincides with the frequency of external excitation. Основні рівняння гнучкого підіймального тросу отримано застосуванням принципу Гамільтона. Проведено експерименти, результати яких добре узгоджуються з теоретичним передбаченням. Результати моделювання і експеримент показують, що гнучка підіймальна система розсіює енергію при спуску і накопичує енергію при підйомі. Далі досліджувався перехід гнучкої підіймальної системи через резонанс за умови періодичного зовнішнього збудження. Якщо довжина гнучкої підіймальної системи змінюється з часом, то власні частоти системи слабо змінюються і можуть спостерігатися перехідні резонанси, коли одна з частот співпадає з частотою зовнішнього збудження. This research work was supported by the State Key Laboratory of Mechanical System and Vibration, Shanghai Jiaotong University (MSV-2010-06). en Інститут механіки ім. С.П. Тимошенка НАН України Прикладная механика Dynamic analysis of flexible hoisting rope with time-varying length Динамический анализ гибкого подъемного троса с переменной во времени длиной Article published earlier |
| spellingShingle | Dynamic analysis of flexible hoisting rope with time-varying length Ji-hu Bao Peng Zhang Chang-ming Zhu |
| title | Dynamic analysis of flexible hoisting rope with time-varying length |
| title_alt | Динамический анализ гибкого подъемного троса с переменной во времени длиной |
| title_full | Dynamic analysis of flexible hoisting rope with time-varying length |
| title_fullStr | Dynamic analysis of flexible hoisting rope with time-varying length |
| title_full_unstemmed | Dynamic analysis of flexible hoisting rope with time-varying length |
| title_short | Dynamic analysis of flexible hoisting rope with time-varying length |
| title_sort | dynamic analysis of flexible hoisting rope with time-varying length |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/141027 |
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