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Modal parameter estimation of the coupled moving-mass and beam
time-varying system
Z.-S. Ma1, L. Liu1, S.-D. Zhou1, W. Yang1 1Beijing Institute of
Technology, School of Aerospace Engineering Zhongguancun South
Street 5, Haidian District, Beijing, 100081, China e-mail:
[email protected]
Abstract The coupled moving-mass and beam system exhibits
time-dependent characteristics, requiring time-varying dynamic
models and corresponding modal analysis methods. The dynamic model
of the coupled moving-mass and beam time-varying system under
arbitrary excitation was firstly built; the influence of the
moving-mass velocity and acceleration on the modal parameters was
analyzed; and the modal parameters of the coupled time-varying
system were estimated based on the non-stationary responses
obtained through the state space representation in numerical
simulation. An experimental system consisting of a simply supported
beam and a moving mass sliding on it was set up. The responses of
the experimental system under random excitation were measured and
the modal parameters of the experimental system were estimated
afterwards. The estimated results from the numerical simulation and
the experimental system validate the time-varying dynamic model and
indicate the effectiveness of the modal parameter estimation.
1 Introduction
The linear time-varying systems commonly used to represent many
variable dynamic systems which are important in engineering
practice. During the last years, many efforts have been spent in
studying time-varying systems. Within this topic, an important
class of time-varying systems is the case of moving loads: if a
structure is travelled by a load whose mass is not negligible with
respect to the structure mass, then the dynamic properties of the
system change with time. Typical examples include railway bridges
with crossing vehicles, crane bridges with moving weights, guide
rails with moving sliders and many more. The coupled moving-mass
and beam system is often used as the simplified model of such
engineering structures during their preliminary design process. In
the past, the modelling and analysis of the coupled moving-mass and
beam system were given many attentions. For example,
Michaltsos[1-3] discussed the effects of the moving vehicle,
including the centripetal force, the Coriolis force, the rotatory
inertia and the variable speeds of the vehicle, on the dynamic
response of the simply supported beam. The dynamic response of
beams subjected to moving loads is a problem commonly classified
into three main types: the moving force problem, the moving mass
problem and the moving oscillator problem. Biondi[4] presented the
motion equations of the coupled beam-oscillator system and took
into account the gravitational, inertial and damping effects due to
the moving oscillators. In the study of the mechanical vibrations
caused by moving loads, the coupled moving-mass and beam system
actually has been modelled as a linear time-varying system. With
recent advances in analysis of time-varying systems, the time
varying nature of the coupled moving-mass and beam system is
receiving renewed attention now. On the one hand, the accurate
modelling of the coupled moving-mass and beam system offers
possibilities for structural damage detection and vibration
control. On the other hand, the experimental set-up of this
time-varying system is often built to validate some new
identification methods[5], and the time-dependent dynamic
characteristics of this system are focused on.
587
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The goal of this study is to present the complete modelling of
the coupled moving-mass and beam time-varying system, and to
validate the dynamic model through the comparison of the modal
parameters obtained from the numerical simulation and the
experimental estimation. The remainder of the paper is organized as
follows: Section 2 introduces the dynamic model of the coupled
moving-mass and beam time-varying system, Section 3 analyzes the
influence of the moving-mass motion parameters on the modal
parameters and estimates the modal parameters of the coupled
time-varying system using numerical simulation, Section 4 describes
the experimental system including the experimental set-up and the
“frozen-time” experiment, Section 5 presents the experimental
estimation results, and Section 6 summaries the study.
2 Dynamic model
Consider the straight beam, shown in figure 1, of length L ,
having a uniform cross-section with constant mass per unit length m
, the coefficient of viscous damping c , and flexural stiffness EI
, made from linear, homogeneous and isotropic material. The
transverse displacement response ( , )y x t is a function of
position x and time t , ( , )Q x t is the transverse loading which
is assumed to vary arbitrarily with position x and time t , ( , )P
x t is the force acting on the beam by the moving mass. The
end-support conditions for the beam are arbitrary, although they
are pictured as simply supports for illustrative purposes.
0M
Figure 1: Structural system: beam crossed by a moving mass
For above coupled moving-mass and beam system, the equation of
motion[6] can be written as
2 4
2 4
( , ) ( , ) ( , ) ( , ) ( , )y x t y x t y x tm c EI Q x t P x
tt t x
(1)
The influence of the moving-mass rotatory inertia can be
neglected only for the wheelbase d of the moving mass much lower
than the length L [2]. After neglecting the effects of the
moving-mass rotatory inertia, the force acting on the beam by the
moving mass is
2
0 2( )
( , )( , ) ( ) ( ( )) ( ( ))x s t
d y x tP x t M g s t x s tdt
(2)
where 0M is the mass of the moving mass, g is the gravitational
acceleration, ( )s t is the moving-mass instantaneous position on
the beam, ( ( ))x s t is the Dirac’s delta function, ( ( ))s t is
the window function, defined as follows
1 0 ( )
( ( ))0 ( ) 0 ( )
s t Ls t
s t or s t L
(3)
For the transverse displacement response ( , )y x t , we have
that
2 2 2 2 2
22 2 2 2
( ) ( )
( , ) ( , ) ( , ) ( , ) ( , )2 ( )x s t x s t
d y x t y x t x y x t x y x t x y x tdt t t x t x t t x
(4)
Introducing equation (2) and (4) into equation (1) leads to
"" ' ' 2 "0( , ) [ ( 2 )] ( ) ( )my cy EIy Q x t M g y sy sy s y
s x s (5)
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where the prime and dot over a variable denote space and time
derivative, respectively. A series solution of equation (5) in
terms of linear normal modes can be sought in the form
1
( , ) ( ) ( )N
i ii
y x t x q t
(6)
where ( ) ( 1,2, )i x i N is the ith eigenfunction of the
unloaded and undamped beam, and these enginfuctions satisfy the
boundary conditions and following orthogonality conditions
0
""
0
0( ) ( )
0( ) ( )
L
i ji
L
i ji
i jm x x dx
M i j
i jEI x x dx
K i j
(7)
where iM and iK are the ith modal mass and modal stiffness of
the beam, and 2
i i iK M , i is the ith eigenfrequency of the beam. Introducing
(6) into (5) produces
""
1 1 1
' ' 2 "0
( ) ( ) ( ) ( ) ( ) ( )
( , ) [ ( ( , ) ( , ) 2 ( , ) ( , ))] ( ) ( )
N N N
i i i i i ii i i
m x q t c x q t EI x q t
Q x t M g y s t sy s t sy s t s y s t s x s
(8)
The space and time derivatives of 1
( , ) ( ) ( )N
i ii
y s t s q t
have following formulas
' '
1
" "
1
' " '
1
' ' 2 "
1
( , ) [ ( ) ( )]
( , ) [ ( ) ( )]
( , ) [ ( ) ( ) ( ) ( )]
( , ) [ ( ) ( ) 2 ( ) ( ) ( ) ( ) ( ) ( )]
N
i iiN
i iiN
i i i iiN
i i i i i i i ii
y s t s q t
y s t s q t
y s t s s q t s q t
y s t s q t s s q t s s q t s s q t
(9)
Introducing (9) into (8) and multiplying this expression by ( )j
x , considering the orthogonality conditions of eigenfunctions and
integrating the expression form 0 to L gives
200
' ' 2 "0
1
( ) ( ) ( ) ( ) ( , ) ( ) ( )
[ ( ) ( ) 4 ( ) ( ) 2 ( ) ( ) 4 ( ) ( )] ( )
L
i i i i i i i i i
N
j j j j j j j j ij
M q t c m M q t M q t Q x t x dx M g s
M s q t s s q t s s q t s s q t s
(10)
Furthermore, the matrix motion equation of the coupled system
has the following general form [ ( )]{ ( )} [ ( )]{ ( )} [ ( )]{ (
)} { ( )}M t q t C t q t K t q t F t (11)
where
0'
0
2 ' 2 "0 0
1 2 N 0 1 20
[ ( )] { } { ( )}[ ( )]
[ ( )] ( ) { } 4 { ( )}[ ( )]
[ ( )] { } 2 { ( )}[ ( )]+4 { ( )}[ ( )]
{ ( )} ( , ){ ( ), ( ) , ( )} { ( ), (
i i
i i
i i i iL T
M t diag M M diag s sC t c m diag M M sdiag s sK t diag M M
sdiag s s M s diag s s
F t Q x t x x x dx M g s
， N) , ( )}Ts s，
(12)
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In above equation, [ ( )]s is the eigenfunctions matrix
evaluated at ( )x s t , '[ ( )]s and "[ ( )]s are the first and
second order partial derivative of [ ( )]s with respect to x
evaluated at ( )x s t . From the matrix motion equation of the
coupled moving-mass and beam system, equation (11), it can be found
that the coupled system is a time-varying system because of the
time-dependent matrix [ ( )]M t , [ ( )]C t and [ ( )]K t .
The boundary conditions for the beam are arbitrary in above
process, that means equation (11) is applicable to all cases as far
as the eigenfunctions are known. For the simply supported beam, we
have
( ) sin( ) ( 1,2, )iix x iL (13)
3 Numerical simulation
3.1 Influence of motion parameters on modal parameters
The matrix [ ( )]M t , [ ( )]C t and [ ( )]K t of the coupled
time-varying system are related to the motion parameters of the
moving mass, as depicted in equation (11). For example, the
velocity of the moving mass affects both the matrix [ ( )]C t and [
( )]K t , while the moving-mass acceleration only affects the
matrix [ ( )]K t . In this section, the influence of the velocity
and acceleration of the moving mass on modal parameters of the
coupled time-varying system are discussed based on above dynamic
model. The coupled time-varying system consisting of a simply
supported beam and a moving mass sliding on the beam is considered
here. The moving mass slides on the simply supported beam with
uniformly variable speed, with the motion form 20( ) 2s t v t at ,
where, 0v is the initial velocity, a is the acceleration. The
parameters of the coupled time-varying system, including the length
L , the mass per length m , the flexural stiffness EI , the
coefficient of viscous damping c , the mass of the moving mass 0M
and the gravitational acceleration g , are given by table 1, as
follows,
L m EI c 0M g
2m 4.71kg m 21050Nm 0 4.8658kg 29.8m s
Table 1: Parameters of the coupled time-varying system
The influence of the velocity of the moving mass is discussed
firstly. The acceleration is set as 0a , the initial velocity of
the moving mass is set as 01 0.05v m s , 02 0.10v m s and 03 0.20v
m s , respectively. The duration is 40s , 20s , and 10s for the
mass to move from one end of the beam to the other end in above
three situations. Figure 2 shows the first four modal parameters
(natural frequency and damping ratio) of the coupled time-varying
system during the movement of the mass. As shown in figure 2, the
velocity of the moving mass has less influence on the natural
frequencies in comparison with the damping ratios. The modal
parameters of the coupled time-varying system exhibit symmetrical
variation during the mass’ movement because of the symmetrical
boundary condition of the simply supported beam. The influence of
the acceleration of the moving mass is discussed here. The initial
velocity is set as
01 0.05v m s , the acceleration of the moving mass is set as 1
0a , 2
2 0.005a m s and 2
3 0.03a m s , respectively. The duration is 40s , 20s , and 10s
for the mass to move from one end of the beam to the other end in
these three situations. Figure 3 shows the first four modal
parameters of the coupled time-varying system during the movement
of the mass. As shown in figure 3, the acceleration of the moving
mass has less influence on the natural frequencies in comparison
with the damping ratios. Due to the interconnection between the
acceleration and the velocity of the moving mass, the modal
parameters of the coupled time-varying system don’t exhibit
symmetrical variation during the mass’ movement.
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(a) (b)
(c) (d) Figure 2: Modal parameters influenced by moving-mass
velocity. (a) Mode 1; (b) Mode 2; (c) Mode 3; (d)
Mode 4
(a) (b)
(c) (d) Figure 3: Modal parameters influenced by moving-mass
acceleration. (a) Mode 1; (b) Mode 2; (c) Mode 3;
(d) Mode 4
It is important to note that the coefficient of viscous damping
is artificially set as 0c , and the damping of the coupled
time-varying system is totally induced by the motion of the moving
mass. In this way, the influence of the motion parameters on the
damping can be clearly captured. If the initial damping of the beam
is low, the damping of the coupled time-varying system may be
negative due to the motion of the moving mass. However, the induced
damping sometimes can be neglected when the initial damping of the
structure is much higher than the induced component.
0 0.5 1 1.5 24
4.5
5
5.5
6
x/mFr
eque
ncy/
Hz
0.05m/s0.10m/s0.20m/s
0 0.5 1 1.5 2-0.01
0
0.01
x/m
Dam
ping
Rat
io
0 0.5 1 1.5 215
20
25
x/m
Freq
uenc
y/H
z
0.05m/s0.10m/s0.20m/s
0 0.5 1 1.5 2-5
0
5x 10
-3
x/m
Dam
ping
Rat
io
0 0.5 1 1.5 240
45
50
55
x/m
Freq
uenc
y/H
z
0.05m/s0.10m/s0.20m/s
0 0.5 1 1.5 2-5
0
5x 10
-3
x/m
Dam
ping
Rat
io
0 0.5 1 1.5 270
80
90
100
x/m
Freq
uenc
y/H
z
0.05m/s0.10m/s0.20m/s
0 0.5 1 1.5 2-2
0
2x 10
-3
x/mD
ampi
ng R
atio
0 0.5 1 1.5 24
4.5
5
5.5
6
x/m
Freq
uenc
y/H
z
0m/s2
0.005m/s2
0.03m/s2
0 0.5 1 1.5 2-0.02
-0.01
0
0.01
x/m
Dam
ping
ratio
0 0.5 1 1.5 215
20
25
x/m
Freq
uenc
y/H
z
0m/s2
0.005m/s2
0.03m/s2
0 0.5 1 1.5 2-10
-5
0
5x 10
-3
x/m
Dam
ping
ratio
0 0.5 1 1.5 240
45
50
55
x/m
Freq
uenc
y/H
z
0m/s2
0.005m/s2
0.03m/s2
0 0.5 1 1.5 2-5
0
5x 10
-3
x/m
Dam
ping
ratio
0 0.5 1 1.5 270
80
90
100
x/m
Freq
uenc
y/H
z
0m/s2
0.005m/s2
0.03m/s2
0 0.5 1 1.5 2-4
-2
0
2x 10
-3
x/m
Dam
ping
ratio
DAMPING 591
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3.2 Modal parameters estimation of the dynamic model
In this section, the varying modal parameters of the coupled
time-varying system are estimated based on the responses obtained
from the numerical examples. In the simulation, the following
numerical quantities, including the length L , the mass per length
m , the flexural stiffness EI , the mass of the moving mass
0M and the gravitational acceleration g , are same as those
given in table 1. The coefficient of viscous damping c is not set
as zero, but 30c N s m here. To simplify the motion-induced damping
effect, the motion form of the moving mass is set as uniform
motion, i.e. 0( )s t v t , where 0 0.20v m s is constant speed. The
duration for the mass to move from one end of the beam to the other
end is 10s . A white noise input is generated to excite the system
and the location of the excitation is 0.5714( , ) x mQ x t .
In the actual complementation, the responses of the coupled
time-varying system are computed by numerical integration using
Runge-Kutta method. Because of the time-dependent characteristics
of the dynamic model, the responses of the coupled moving-mass and
beam system are non-stationary. Based on these non-stationary
responses, the first four modal parameters of the coupled
time-varying system are estimated by the subspace-based
algorithm[7, 8]. The estimated results of the modal parameters are
depicted by black circles and the theoretical modal parameters are
depicted by the yellow lines, as shown in figure 4,
(a) (b)
(c) (d) Figure 4: Modal parameters of the coupled time-varying
system. (a) Mode 1; (b) Mode 2; (c) Mode 3; (d)
Mode 4
4 Experimental system
4.1 Experimental set-up
The experimental system is composed of the test structure,
exciter systems, force and motion transducers, measurement and
analysis systems, control systems and boundary conditions. The test
structure is the coupled time-varying system consisting of a simply
supported beam and a moving mass sliding on it. The dimensions of
the beam is 2000 60 10mm ( L W H ) and the weight of the moving
mass is 4.8658kg .
0 2 4 6 8 102
4
6
8
Freq
uenc
y/H
z
0 2 4 6 8 10-0.2
0
0.2
0.4
Dam
ping
Rat
io
Time/s
0 2 4 6 8 1015
20
25
Freq
uenc
y/H
z
0 2 4 6 8 10-0.04-0.02
00.020.040.060.08
Dam
ping
Rat
io
Time/s
0 2 4 6 8 1040
45
50
55
Freq
uenc
y/H
z
0 2 4 6 8 10-0.02
0
0.02
0.04
Dam
ping
Rat
io
Time/s
0 2 4 6 8 1075
80
8590
95100
Freq
uenc
y/H
z
0 2 4 6 8 10-0.02
0
0.02
0.04
Dam
ping
Rat
io
Time/s
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The exciter systems consist of an exciter ( 2025ETMModalshop )
and a power amplifier ( TMSmartAmp2100 21-400E ). The TMPCB 288D01
impedance head and the TMPCB 333B30 accelerometer are used as the
force transducer and the motion transducer, respectively.
Measurement and acquisition module is
TMLMS SCADAS III system. Control systems consist of a
TMFaulhaber DC motor and its motion controller. Figure 5 shows the
schematic diagram of the experimental system and its set-up.
(a) (b) Figure 5: Schematic diagram of the experimental system
and its set-up
4.2 “Frozen-time” experiment
For the time-varying systems, the frozen approximation depends
on the assumption that the systems are slowly varying[9, 10]. This
approach is an attempt to apply the results of time-invariant
systems to slowly varying systems. Obviously, the closer the
operating points at which the frozen approximations are made, the
better the accuracy. However, such an approximation is meaningful
only in a limited sense, and the stability of the time-varying
systems cannot be directly predicted by the eigenvalues or
characteristic roots obtained from the frozen approximation[11].
The coupled time-varying system is studied using the frozen
technique here. For the experimental system, its time-dependent
dynamic characteristics are function of the position of the moving
mass, while the position of the moving mass is the function of the
time. In other words, if the motion form of the moving mass is
known, we can determine the instantaneous position of the moving
mass in arbitrary instant of time. The duration for the mass to
move from one end of the beam to the other end can be partitioned
into many discrete segments. When the moving mass stays at a
certain segment, the experimental system can be considered as a
time-invariant system and its modal parameters can be estimated by
the time-invariant system identification techniques. The modal
parameters of the “frozen-time” experiment are usually used as the
reference of the time-varying case in reality. During the actual
complementation of the “frozen-time” experiment, the midpoint of
the beam is selected as the starting position of the moving mass,
and the end position is away from the starting position at a
distance of 0.8m . We equally divide this duration into 80 segments
and the length of each segment is 0.01m . The mass is placed at the
starting edge of each segment and 81 times of “frozen-time”
experiment are carried out. In the experiment, a random excitation
is generated to excite the system at the location
0.5714( , ) x mQ x t , and fifteen accelerometers are used to
measure the acceleration of the beam at fifteen uniformly
distributed positions along the axial direction of the beam, as
shown in figure 5. The least squares complex frequency domain
(LSCFD) method [12] is used to estimate the modal parameters of the
“frozen-time” experimental system and the first four modal
parameters are depicted in figure 6. The horizontal axis is the
position of the moving mass; the vertical axis is the natural
frequency in figure 6(a) and the damping ratio in figure 6(b).
Based upon the comparison of the estimated results (black circles)
and the theoretical results (solid line) in figure 6(a), we find
that the dynamic model of the coupled time-varying system and the
experimental system are consistent in terms of the natural
frequencies. The estimated damping of the “frozen-time”
DAMPING 593
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experiment should be considered as the initial damping of the
experimental system, and the real damping of the experimental
system during the movement of the moving mass can be obtained by
adding the induced damping to the initial component.
(a) (b) Figure 6: Modal parameters of the “frozen-time”
experimental system
5 Experimental estimation
5.1 Time-frequency analysis of response signals
In this section, the same experimental set-up as the
“frozen-time” experiment is used, but the experimental system
exhibits time-dependent characteristics due to the continuous
movement of the moving mass. The speed of the moving mass is 0.2v m
s and the duration is four seconds. 50 tests are repeatedly carried
out, and the coupled time-varying system undergoes the same
variation in each test. Of course, the random excitation in every
test is different from each other. The accelerations from fifteen
different positions of the beam and the input excitation are
measured, and these response signals also form the original data
set for the modal parameter estimation of the coupled time-varying
system. Due to the time-dependent dynamic characteristics of the
systems, the responses of the time-varying systems are
non-stationary, requiring time-frequency analysis[13] methods to
obtain their time-dependent spectra. Here, the smoothed pseudo
Wigner-Ville distribution[14] is used to process the acceleration
signals measured from the tests and the averaged time-dependent
spectra of these signals is depicted in figure 7(a). As shown in
figure 7(a), four peaks with respect to frequency can be found,
which indicates there are four modes in the bandwidth from 0 to
110Hz .
(a) (b) Figure 7: Averaged time-dependent spectra of
acceleration signals
The first four theoretical natural frequencies are drawn in
figure 7(b) with the blue line, and the background of figure 7(b)
is filled by the spectra from figure 7(a). It is obvious that the
peaks of the time-
1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.80
10
20
30
40
50
60
70
80
90
100
110
x/m
Freq
uenc
y/H
z
1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.80
0.01
0.02
0.03
0.04
0.05
0.06
x/m
Dam
ping
Rat
io
Mode 4Mode 3Mode 2Mode 1
0 0.51 1.5
2 2.53 3.5
4
010
2030
4050
6070
8090
100110-50
0
50
Time/s
Frequency/Hz
Am
plitu
de/d
B
Time/s
Freq
uenc
y/H
z
0 0.5 1 1.5 2 2.5 3 3.5 40
10
20
30
40
50
60
70
80
90
100
110
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dependent spectra are consistent with the theoretical natural
frequencies of the experimental system, which also validate the
dynamic model of the coupled moving-mass and beam time-varying
system.
5.2 Modal parameter estimation of the experimental system`
Based on the ensemble of the input and output data, the
subspace-based algorithm is used here to estimate the modal
parameters of the experimental system. From previous results and
analysis, we regard the first four natural frequencies of the
experimental system as known parameters, and select those modes
which are close to the theoretical natural frequencies as physical
modes. The grouping method put forward in reference [8] is not used
here, but the damping of the modes is considered during the
selection of the physical modes. Those modes with surprisingly high
levels of damping ratio (e.g. higher than 10% of critical damping)
are abandoned because they are often a strong indication of
computation modes. Figure 8 shows the first four estimated modal
parameters of the experimental system. The estimated results of the
modal parameters are depicted by black circles and the theoretical
modal parameters are depicted by the yellow lines. It should be
noted that the theoretical damping consists of two parts: the
initial damping of the experimental system and the induced damping
caused by the motion of the moving mass.
(a) (b)
(c) (d) Figure 8: Modal parameters of the experimental system.
(a) Mode 1; (b) Mode 2; (c) Mode 3; (d) Mode 4
As depicted in figure 8, the estimated results of the damping
are not good due to many possible reasons. First, the
subspace-based algorithm is sensitive to measurement noise, and
there are some approximations in this algorithm which also
influence the accuracy of the estimated results. Second, damping
estimation is much more difficult than natural frequency
estimation, especially for time-varying systems, because the
responses of the time-varying systems are non-stationary and the
damping level of the systems is also varying with time. Third, the
damping mechanism of the experimental system is not well understood
and the presence of nonlinearity between moving mass and the beam
is not considered either.
6 Conclusions
The time-varying systems have been frequently used to model
systems that have time-dependent or non-stationary properties, and
the identification of time-varying systems has received increasing
attention. As
0 0.5 1 1.5 2 2.5 3 3.5 43
4
5
6
Freq
uenc
y/H
z
0 0.5 1 1.5 2 2.5 3 3.5 40
0.05
0.1
Dam
ping
Rat
io
Time/s
0 0.5 1 1.5 2 2.5 3 3.5 415
20
25
Freq
uenc
y/H
z
0 0.5 1 1.5 2 2.5 3 3.5 40
0.05
0.1
Dam
ping
Rat
io
Time/s
0 0.5 1 1.5 2 2.5 3 3.5 435
40
4550
5560
Freq
uenc
y/H
z
0 0.5 1 1.5 2 2.5 3 3.5 40
0.05
0.1
Dam
ping
Rat
io
Time/s
0 0.5 1 1.5 2 2.5 3 3.5 470
80
90
100
Freq
uenc
y/H
z
0 0.5 1 1.5 2 2.5 3 3.5 40
0.05
0.1
Dam
ping
Rat
io
Time/s
DAMPING 595
-
an important class of time-varying systems, the moving mass
problem is studied in the paper. The accurate dynamic model of the
coupled moving-mass and beam system is presented and verified
through the numerical simulation and experimental validation. The
influence of the moving-mass velocity and acceleration on the modal
parameters is analyzed and other effects of the moving mass can
also be further studied based on the dynamic model presented in
this paper. Modal parameters of the numerical model and the
experimental system are estimated by the subspace-based algorithm,
and the estimated results indicate the consistency between them. In
this paper, the damping estimation is not as good as the natural
frequency estimation, especially in the experimental example.
Possible reasons have been analyzed before and more efforts should
be spent to improve the accuracy of the damping estimation.
Besides, the induced damping caused by the motion of the moving
mass should be also paid more attentions, and more precise
experimental systems are required to acquire the reliable
information on the damping.
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