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.

Acknowledgement

Vibration-Based Structural Damage Detection: Theory and Applications, Award CMS-0600559 from the Dynamical Systems Program of the National Science Foundation

Maryland Technology Development Corporation (TEDCO)

Baltimore Gas and Electric Company (BGE)

Pratt & Whitney