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Weidong Zhu, Nengan Zheng, and Chun-Nam Wong Department of Mechanical Engineering University of...
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Transcript of Weidong Zhu, Nengan Zheng, and Chun-Nam Wong Department of Mechanical Engineering University of...
Weidong Zhu, Nengan Zheng, and Chun-Nam Wong
Department of Mechanical Engineering
University of Maryland, Baltimore County (UMBC)
Baltimore, MD 21250
A Novel Stochastic Model for the Random Impact Series Method in Modal Testing
Shaker Test
Advantages: Persistent excitation – large energy input and high signal-to-noise ratio Random excitation – can average out slight nonlinearities, such as those arising from opening and closing of cracks and loosening of bolted joints, that can exist in the structure and extract the linearized parameters
Disadvantages:Inconvenient and expensive
Single-Impact Hammer Test
Advantages: Convenient Inexpensive Portable
Disadvantages: Low energy input Low signal-to-noise ratio No randomization of input –
not good for nonlinear systems
Development of a Random Impact Test Method and a Random Impact Device Combine the advantages of the two excitation method
s Increased energy input to the structure Randomized input – can average out slight nonlinearities th
at can exist in the structure and extract the linearized parameters
Convenient, inexpensive, and portable
Additional advantages A random impact device can be designed to excite very larg
e structures It cab be used to concentrate the input power in a desired fr
equency range
Novel Stochastic Models of a RandomImpact Series
Random impact series (RIS) (Zhu, Zheng, and Wong, JVA, in press)
Random arrival times and pulse amplitudes, with the same deterministic pulse shape
Random impact series with a controlled spectrum (RISCS)
Controlled arrival times and random pulse amplitudes, with the same deterministic pulse shape
Previous Work on the Random Impact Test
Huo and Zhang, Int. J. of Analytical and Exp. Modal Analysis, 1988
Modeled the pulses as a half-sine wave, which is usually not the case in practice
Analysis essentially deterministic in nature The number of pulses in a time duration was modeled as a
constant No stochastic averages were determined
The mean value of a sum involving products of pulse amplitudes were erroneously concluded to be zero
N - total number of force pulses y(∙) - shape function of all force pulses - arrival time of the i-th force pulse - amplitude of the i-th force pulse - duration of all force pulses
Mathematical model of an impact series
1
( )N
i ii
x t y t
ii
1
2 N
… …
i | |
x(t)
t
The amplitude of the force spectrum for the impact series
0
5
10
15
20
(a)
|X(j)| o
r E
(|X
(j
)|)
(N
s)
0
10
20
30
(b)
E(|
X(j)|)
(dB
)
20N
i normally distributed
random variable around i
20N
i i
1i
Dotted line
Solid linei normally distributed
random variable
1i
Dotted line
Solid linei normally distributed
random variable
Deterministic arrive times with deterministic or random amplitudes
Random arrive times with deterministic or random amplitudes
N(T) - total number of force pulses that has arrived within the time interval (0,T]y(∙) - arbitrary deterministic shape function of all force pulses - random arrival time of the i-th force pulse - random amplitude of the i-th force pulse - duration of all force pulses
Time Function of the RIS*
( )
1
( )
(0, ] and (0, ]
N T
i ii
i
x t y t
T t T
ii
Challenge: A finite time random process with stationary and non-stationary parts
1
2 N(T)
… …
i T+
| |
x(t)
t
T
*Zhu et al., JVA, in press
Probability Density Function (PDF) of the Poisson Process N(T)
n=N(T) - number of arrived pulses
λ - constant arrival rate of the pulses
{ } ,
!
nT
N
e Tn T
n
PDF of the Identically, Uniformly Distributed Arrival Times
1
0
0 elsewherei
Tp T
where 1, 2, , ( )i N T
PDF of the Identically, Normally Distributed Pulse Amplitudes(Used in numerical simulations)
- amplitude of the force pulses
- mean of
- variance of
2
2210
2ip e
2i
i
Mean function of x(t)
1
1
1
0E W t t
E x t E W t T
E W W t T T t T
const when [ , ]E x t t T
0( ) ( )
tW t x u duwhere
Autocorrelation Function of x(t)
x(t) is a wide-sense stationary random process in
1 2
2 2 2 21 10
,
xx
k
R k E x t x t
E y u y u k du E W
[ , ]t T
where
When
2 1k t t
[ , ]T
Average Power Densities of x(t)
[0, ]t T
2
[0, ]1( ) TX j
ST
1 0
2 2 21 12
1( ) cos( )
1 cos( )2
kE S x v k x v dv k dk
T
TE TE
[ , ]t T
Non-stationary at the beginning and the end of the process
Wide sense stationary
2
[ , ]2 ( ) TX j
ST
Comparison between x(t) in and [0, ]T [ , ]T
Average power densities
The expectations of average power densities
22 1 0
2 2 21 2
1 cos
1 cos2
k kE S E dk x u k x u kdu
T
TE W
T
where [ ( )]X j F x t
Averaged, Normalized Shape Function
0.00000 0.03125 0.06250 0.09375 0.12500 0.15625
0.0
0.2
0.4
0.6
0.8
1.0
y(t)
t (s)
4.14 / s 8T s 0.15625 s
1[ ] 0.8239E N 2 21[ ] 0.7163E N
Comparison of Analytical and Numerical Results for Stochastic Averages
Mean function of x(t)
0 1 2 3 4 5 6 7 80.00
0.01
0.02
0.03
0.04
0.05
Analytical Numerical
E [
x(t)
] (N)
t (s)
Comparison of Analytical and Numerical Results for Stochastic Averages (Cont.)
0 10 20 30 40 50
10-3
10-2
1x10-1
Analytical =1.0
Analytical =4.14 Numerical =4.14
10
lo
g | E
[S 1(j
)]
| (d
B)
/2 (Hz)Expectation of the average power density of in( )x t [0, ]T
Increasing the Pulse Arrival Rate Increases the Energy Input
0 10 20 30-21
-18
-15
-12
-9 single random, burst random, continuous exact
|G
(j)|
2
/2 (Hz)
NoiseImpact force
A single degree of freedom system under single and random impact excitations
Sampling time
Excitation time
Ts=16 s
T=16 s (Continuous)
T=11.39 s (Burst)
Arrival times
it=Uniformly distributed
random variables over T
0 10 20 30-21
-18
-15
-12
-9 multiple, deterministic arrival times multiple, random exact
|G(j
)|2
/2 (Hz)
A single degree of freedom system under multiple impact excitations (cont.)
NoiseImpact force
Sampling time
Arrival times
Ts=16 s
it=0.2424i s
(Deterministic)
it=Uniformly distributed
random variables
(Random)
Excitation time
T=11.39 s
Where i=1, 2,…, 66
Random Impact Series with a Controlled Spectrum (RISCS)
RISCS can concentrate the energy to a desired frequency range. For example, if one wants to excite natural frequencies between 7-13 Hz. The frequency of impacts can gradually increase from 7 Hz at t=0 s to 13 Hz at t=8 s.
0 2 4 6 80.0
0.3
0.6
0.9
1.2
1.5
1.8
Am
plitu
de
Time
Uniform Amplitude
0 2 4 6 80.0
0.4
0.8
1.2
1.6
2.0
Am
plitu
de
Time
Controlled arrival times and uniform amplitudesRandom Amplitude
Random Impact Series with a Controlled Spectrum (RISCS) (cont.)
0 20 40 60 80 100-90
-80
-70
-60
-50
-40
-30
-20 Uniform Amplitude Random Amplitude
Po
we
r S
pe
ctra
lD
en
sity
Est
ima
tion
(d
B)
f (Hz)
RISCS Can Concentrate the Input Energy in a Desired Frequency Range and the Randomness of the Amplitude Can Greatly Increase the Energy Levels of the Valleys in the Spectrum
Conclusions
Novel stochastic models were developed to describe a random impact series in modal testing. They can be used to develop random impact devices, and to improve the measured frequency response functions.
The analytical solutions were validated numerically.
The random impact hammer test can yield more accurate test results for the damping ratios than the single impact hammer test.