A stochastic model for localized disturbances and its applications

Zhikun Houa,*, Adriana Heraa, Mohammad Noorib

aDepartment of Mechanical Engineering, Worcester Polytechnic Institute, Worcester, MA 01609, USAbMechanical and Aerospace Engineering, North Carolina State University, USA

Received 15 November 2003; revised 15 December 2003; accepted 2 February 2004

Abstract

A stochastic model for local disturbances, particularly for a temporal harmonic with random modulations in amplitude and/or phase, is

proposed in this paper. Results for the second moment responses of a linear single-degree-of-freedom system to this type of stochastic

loading demonstrate a significant change in response characteristics due to a small uncertainty. A local phenomenon may last much longer

and resonance may be smeared to a broad range. Integrated with wavelet transform, the proposed approach may be used to model a random

process with non-stationary frequency content. Especially, it can be effectively used for Monte Carlo simulation to generate large size of

samples that have similar characteristics in time and frequency domains as a pre-selected mother sample has. The technique has a great

potential for the case where uncertainty study is warranted but the available samples are limited.

q 2004 Elsevier Ltd. All rights reserved.

Keywords: Random vibration; Stochastic modeling; Random process; Wavelet analysis; Monte Carlo simulation; Earthquake engineering; Structural dynamics

1. Introduction

Vibration analysis is often required to evaluate

performance of engineering structural systems in a

dynamic environment where sources of vibration are

specified by time-varying inputs [3,8]. When uncertainty

of the input is involved, a stochastic model for the input

is often required to describe its random nature appro-

priately. Popular stochastic models are exemplified by the

white noise process, the Poisson process, and their

descendants associated with various filters to describe

the frequency content of the input and various envelope

functions to introduce non-stationarity into the forcing

function [1,10,11]. Once the stochastic model for the

excitation is selected, a close-form solution based on

probabilistic theory or the numerical Monte Carlo

simulation technique based on statistics can be employed

to find random response of the dynamic system of

concern. Consequently, performance of structural systems

can be evaluated in a probabilistic sense and reliability of

these structures in a random environment can be

assessed.

This study was motivated by certain needs as how to

describe time-varying frequency content of a random

process and how to obtain samples for a random process

for which only limited samples are available. In the former

case, modulated random processes were introduced and

usage of associated envelope functions may partially

address the issue. However, it is challenge to find simple

envelope function to match the specified time-varying

frequency content. For the latter case, to improve the

accuracy of the Monte Carlo simulation, large size of

samples are required for a random process but often there

are only a limited number of samples available for a

practical application. Sometimes there are extreme cases

where only one sample is available, such as a ground motion

record at a specific site for a specific earthquake event. This

paper proposes a wavelet-based approach to address the

issue based on the ability of wavelet analysis for multi-level

time–frequency domain analysis.

As one of the popular wavelets in wavelet analysis, the

Morlet or Garbor wavelet plays an important role in this

study. First, the wavelet has an explicit frequency content

as expressed by its central frequency, which facilitates a

physical interpretation of the associated wavelet transform

of a given signal regarding its time-varying frequency

content. Second, the Morlet wavelet with random

0266-8920/$ - see front matter q 2004 Elsevier Ltd. All rights reserved.

doi:10.1016/j.probengmech.2004.02.004

Probabilistic Engineering Mechanics 19 (2004) 211–218

www.elsevier.com/locate/probengmech

* Corresponding author. Tel.: þ1-508-831-5705; fax: þ1-508-831-5680.

E-mail address: hou@wpi.edu (Z. Hou).

disturbances falls into a category of stochastic model for a

harmonic process with random disturbances that has been

extensively studied in the literature [4–7,9,12]. However,

due to its nearly finite support, a scaled and shifted Morlet

wavelet-based model may be useful for a localized random

disturbance.

In this paper, a stochastic model based on the Morlet

wavelet is proposed to describe a localized harmonic

process with random disturbances in both amplitude and

phase and then a technique is proposed to generate samples

of a random process using one sample from its limited

available sample database, referred as a mother sample.

Two applications are given: one addresses effects of

uncertainties in a localized disturbance on second moment

response of a linear single-degree-of-freedom system and

the other illustrates how to use the proposed technique to

generate a set of ground motion samples from the 1940 El

Centro ground acceleration record, S00E component.

Promising results were obtained in both cases.

2. Morlet wavelet-based stochastic model for localized

disturbances

The proposed methodology in the present paper is

based on the Morlet wavelet. A complex Morlet wavelet is

given by:

cðtÞ ¼Affiffip

ps

exp 2t2

s2

!expðivctÞ ð1Þ

where vc and s are both real and positive. By the standard

operations of shifting and scaling in wavelet analysis, the

Morlet wavelet represents a localized and oscillatory signal.

In the frequency domain, the complex Morlet wavelet is

characterized by its Fourier transform as follows:

cðvÞ ¼ A exp 2ðv2 vcÞ

2

4=s

!ð2Þ

Eq. (2) shows that the energy of the wavelet is concentrated

around the center frequency vc and with a window size that

depends on s: Therefore, vc and s are often referred as the

center frequency and the bandwidth parameter of the

Morlet wavelet, respectively. As shown in Eqs. (1) and (2),

the complex Morlet function has localized Gaussian-type

windows in both time and frequency domains.

A Morlet wavelet-based stochastic model is proposed in

the following form:

f ðtÞ ¼ ð1 þ haðtÞÞe2t2=2 cosðvct þ hpðtÞÞ ð3Þ

where vc is the central frequency of the Morlet wavelet and

haðtÞ and hpðtÞ represent random disturbances in amplitude

and phase of the harmonic part, respectively. Both haðtÞ

and hpðtÞ are modeled as a stationary white noise process of

a normal distribution, satisfying:

E½ha1ðtÞ� ¼ E½hpðtÞ� ¼ 0

E½haðtÞhaðt þ tÞ� ¼ DadðtÞ

E½hpðtÞhpðt þ tÞ� ¼ DpdðtÞ

E½haðtÞhpðt þ tÞ� ¼ DapdðtÞ

ð4Þ

where Da and Dp represent, respectively, the intensities of

haðtÞ and hpðtÞ and Dap is the intensity of their cross

correlation. As a preliminary study, Dap is assumed to be

zero. If all D’s are zero f ðtÞ is reduced to a deterministic

case, i.e. the real part of the complex Morlet wavelet. The

exponential term introduces temporal localization into the

model. Alternative expressions may be used for the purpose

but the form in Eq. (3) is consistent with future wavelet

formulation.

Two representative deterministic signals, localized in the

time domain, are illustrated in Fig. 1. Both signals are

obtained by shifting the Morlet wavelet in Eq. (3) to t ¼ 5

and 15 s, respectively. The Morlet wavelets are also scaled

such that two signals have different decay rates and

therefore exhibit different localization properties. The signal

in the right figure is more localized due to its higher decay

rate characterized by a great s:

Fig. 2 plots samples generated from the proposed

stochastic model with different noise intensities of random

disturbances in phase. In Fig. 2, solid lines represent

samples generated from the proposed model and the dashed

lines are for the original signals without random disturbance

for comparison. All signals are localized in the neighbor-

hood of t ¼ 6 s. Fig. 2 plots results for random disturbances

in phase only. From the top to the bottom, the disturbance

intensity gradually increased from 0.01, 0.05, 0.1, up to 0.5.

For the top two plots with smaller disturbances, the samples

Fig. 1. Two representative shifted and scaled Morlet wavelets localized in

the neighborhood of t ¼ 5 and 15 s, respectively. The signal in the right

figure has a higher decay rate.

Z. Hou et al. / Probabilistic Engineering Mechanics 19 (2004) 211–218212

still demonstrate localized harmonic waveform as that of the

original deterministic counterpart but distorted by the

random phase modulation. With an increasing noise level,

the waveform becomes more and more irregular, as shown

in the two lower figures. It is expected that the waveform

will eventually be destroyed by significantly large random

phase modulation. Note that since these results are only for

the disturbance in phase, all samples are bounded.

Fig. 3 demonstrates samples generated from the proposed

stochastic model with different noise intensities of random

disturbances in magnitude. Again, the original signals

without random disturbance are plotted as the dashed lines

for comparison. All signals are still localized in the

neighborhood of t ¼ 6 s. From the top to the bottom, the

disturbance intensity in amplitude is gradually increased

from 0.01, 0.05, 0.1, up to 0.5. Like in Fig. 2, the top two

plots with smaller disturbances clearly show distorted

localized harmonic waveform but the waveform becomes

less recognized in the two lower figures. In comparison to

the bounded samples in Fig. 2, results in Fig. 3, caused by

disturbances in the amplitude only, are unbounded and show

a quite clear pattern in the frequency content.

Samples generated from the proposed stochastic model

with different noise intensities of the random disturbances in

both magnitude and phase are illustrated in Fig. 4. Results

exhibit similar features as those in the previous two figures

but the original harmonic waveform can be recognized only

for very small disturbances. In general, all these samples are

Fig. 2. Samples generated from the proposed model with various intensities

of random disturbances in phase. All signals are localized in the

neighborhood of t ¼ 6 s.

Fig. 3. Samples generated from the proposed model with various intensities

of random disturbances in amplitude. All signals are localized in the

neighborhood of t ¼ 6 s.

Fig. 4. Samples generated from the proposed model with various intensities

of random disturbances in both amplitude and phase. All signals are

localized in the neighborhood of t ¼ 6 s.

Z. Hou et al. / Probabilistic Engineering Mechanics 19 (2004) 211–218 213

neither bounded due to the disturbance in amplitude nor

having the original frequency pattern due to the disturbance

in phase.

3. Application: random response of linear SDOF systems

to a localized random disturbance

The proposed model has been applied to a linear single-

degree-of-freedom oscillator to study the effects of a

temporally local disturbance of uncertainties. The govern-

ing equation of motion of the dynamic system is:

md2x

dt2þ c

dx

dtþ kx ¼ f ðtÞ ð5Þ

where xðtÞ is displacement response of the system; m; c and k

are mass, viscous damping and stiffness coefficients of

the system, respectively; and f ðtÞ; described by Eq. (3), is

a localized excitation with random disturbance to take the

load uncertainty into account. Assume that the system is

originally at rest.

Monte Carlo simulation was adopted at the current stage

to find the second moment responses, i.e. auto-covariance of

the displacement and velocity, CxxðtÞ and CvvðtÞ and their

cross-covariance CxvðtÞ: The numerical results are shown in

Figs. 5–7. Since the mean values of the displacement and

velocity responses are zero in the studied case, the auto-

covariance responses are in fact also the mean square

responses of the displacement and velocity. Note that m ¼ 1

unit was consistently used for all results and the central time

of the local disturbance is at t ¼ 5 s. Two hundred samples

were used in the simulation and the fourth-order Runge-

Kutta numerical scheme was employed for calculation of

dynamic response of the system for each sample. Parameter

Fig. 5. Second moment responses of a linear single-degree-of-freedom oscillator subjected to a localized harmonic excitation with random phase modulation of

various intensities.

Z. Hou et al. / Probabilistic Engineering Mechanics 19 (2004) 211–218214

studies were performed for the intensity of the random

disturbance and the natural frequency and the damping ratio

of the system.

Fig. 5 presents the second moment responses of the

system to a localized harmonic input with random phase

disturbance for four intensity levels of 0, 0.001, 0.005 and

0.01. While the local input shows only a local influence on

the system responses for the deterministic case, i.e. the noise

level is zero, as shown in Fig. 5a, a small uncertainty may

significantly change the response characteristics, as

observed in Fig. 5b–d. The response tends to last much

longer than expected after the localized input applied.

Fig. 6 illustrates effects of the system damping on the

second moment responses of a single-degree-of-freedom

system subjected to a localized harmonic with random phase

modulation. The critical damping ratio was set to 0, 0.01,

0.05 and 0.2, respectively. The existence of system damping

will in general help to keep the local nature of a physical

phenomenon. More damping the system has, more localized

the system response is due to the local input.

Fig. 7 investigates how a system with different natural

frequencies responds to a local harmonic with random

phase modulation. The frequency parameter varies from 1 to

9 rad/s and only four sets of results for 1, 4, 5 and 6 rad/s are

plotted. The peak response may not necessarily appear at its

resonance frequency, i.e. 5 rad/s. The random phase

modulation smears the resonance peak of the deterministic

case, as observed.

In general, amplitude of the second moment responses

may be determined by a few competing factors. Among

them, higher damping will accelerate the decay of the

response caused by the temporal impact, frequency ratio

may increase or decrease the system response depending

whether moving into or away from the resonance, and

Fig. 6. Effects of the system damping on the second moment responses of a single-degree-of-freedom system subjected to a localized harmonic with random

phase modulation.

Z. Hou et al. / Probabilistic Engineering Mechanics 19 (2004) 211–218 215

the higher intensity of the random phase modulation may

bring more ‘averaging’ effect into secondary moment

responses. Therefore, amplitude of the second moment

responses in the above figures does not show a monotonic

change with a selected physical parameters.

4. Application: sample generation of general random

processes

The proposed model effectively works with wavelet

transform to generate a sample set of similar time–

frequency characteristics based on a given signal, referred

as the mother sample of a random process. Using a selected

analyzing or mother wavelet function CðtÞ; wavelet trans-

form of a signal, f ðtÞ; is defined as:

ðWf Þða; bÞ ¼1ffiffia

pðþ1

21f ðtÞ �C

t 2 b

a

� �dt ð6Þ

where a and b are the dilation/scaling and translation/

shifting parameters, respectively. Both are real and a must

be positive. The bar over CðtÞ indicates its complex

conjugate. The original signal may be recovered or

reconstructed by an inverse wavelet transform of ðWf Þða; bÞ

as defined by:

f ðtÞ ¼1

Cc

ðþ1

21

ðþ1

21ðWf Þða; bÞC

t 2 b

a

� �1

a2da db ð7Þ

Fig. 7. Effects of the system natural frequency on the second moment responses of a single-degree-of-freedom system subjected to a temporal harmonic input

with random disturbances.

Z. Hou et al. / Probabilistic Engineering Mechanics 19 (2004) 211–218216

where CC is defined by:

Cc ¼ðþ1

21

lFcðvÞl2

lvldv , 1 ð8Þ

in which FCðvÞ is the Fourier transform of the mother

wavelet. Wavelet analysis may be viewed as an extension of

the traditional Fourier transform with adjustable window

location and size. Merits of wavelet analysis lie in its ability

to examine local data with ‘zoom lens having an adjustable

focus’ to provide multi-levels of details and approximations

of the original signal. Details of wavelet analysis may be

found in Chui [2].

One of the popular wavelets is the Garbor or Morlet

wavelet, expressed as:

CðtÞ ¼ e2t2=2 cosðvctÞ ð9Þ

which is a real version of Eq. (1). vc is the central frequency

of the wavelet. Without loss of generality, vc ¼ 5 rad/s is

selected. Note that the Morlet wavelet in Eq. (9)

corresponds to the deterministic case of the proposed

model. A merit of Morlet wavelet is an explicit form of its

Fourier transform given by:

FCðvÞ ¼ðþ1

21cðtÞ e2ivt dt

¼ e2v2c2v2

e1=2ð2vcþvÞ2 þ e1=2ðvcþvÞ2� � ffiffiffiffi

p

2

rð10Þ

Using a modified version of reconstruction process in

wavelet analysis, a sample can be expressed as:

~fðtÞ ¼1

Cc

ðþ1

21

ðþ1

21ðWf Þða; bÞ �C

t 2 b

a

� �1

a2da db ð11Þ

where:

~CðtÞ ¼ ð1 þ Sða; bÞhaðtÞÞe2t2=2 cosðvct þ Sða; bÞhpðtÞÞ ð12Þ

in which Sða; bÞ is a characteristic function being one for all

dominant wavelet components of scale a at time b and zero

otherwise. As a result, samples can be generated based on

wavelet transform of a given signal, used as a mother

sample of the random process.

Fig. 8 presents a sample set generated from the proposed

model using the S00E component of the 1940 El Centro

Fig. 8. Samples generated using the 1940 El Centro Earthquake ground motion records with random disturbances of various intensities. The dashed lines are the

original ground motion data.

Z. Hou et al. / Probabilistic Engineering Mechanics 19 (2004) 211–218 217

ground acceleration data as the mother sample. By applying

the random disturbance for both amplitude and phase of

some dominant wavelet components at different times,

samples with a nature in both time and frequency domains,

similar to the mother sample, can be generated. A sample set

can be obtained for a combination of selected intensities of

random disturbance in both phase and amplitude. Only one

sample per intensity set is plotted in Fig. 8 for illustration.

For the sake of convenience, only data of the first 8 s are

presented. When compared with the original El Centro

ground acceleration data represented by the dashed line, the

induced sample set exhibits similar characteristics as the

original data. By changing intensities of the random

disturbances, a large size of samples can be generated for

Monte Carlo simulation study in many practical appli-

cations where uncertainty study is warranted but the

available samples are limited.

5. Conclusion

A stochastic model for local disturbances, particularly

for a temporal harmonic with random modulations in

amplitude and/or phase, is proposed in this paper. Results

for the second moment responses of a linear single-degree-

of-freedom system to this type of stochastic input show

significant changes in response characteristics due to small

uncertainties. A local phenomenon may last much longer

and resonance may be smeared to a broad range. Integrated

with wavelet transform, the model may be effectively used

for Monte Carlo simulation to generate large size of samples

that have similar characteristics in time and frequency

domains as a pre-selected mother sample has. The technique

has a great potential for the case where uncertainty study is

warranted but the available samples are limited.

References

[1] Amin M, Ang AHS. Nonstationary stochastic model of earthquake

motions. J Engng Mech Div, ASCE 1968;94:559–83.

[2] Chui CK. Introduction to wavelets. San Diego, CA: Academic Press;

1992.

[3] Dimentberg MF. Statistical dynamics of nonlinear and time-varying

system. England: Research Studies Press; 1988.

[4] Dimentberg MF. A stochastic model of parametric excitation of a

straight pipe due to slug flow of a two-phase fluid. Proceedings of the

Fifth International Conference on Flow-induced Vibrations, Brighton,

UK; 1991. p. 207–9.

[5] Hou Z, Zhou Y, Dimentberg M, Noori M. A non-stationary stochastic

model for periodic excitation with random phase modulation. J Probab

Engng Mech 1995;(10):73–81.

[6] Hou Z, Zhou Y, Dimentberg M, Noori M. A stationary model for

periodic excitation with uncorrelated random disturbances. J Probab

Engng Mech 1996;(11):191–203.

[7] Hou Z, Zhou Y, Dimentberg M, Noori M. Random response to

periodic excitation with correlated disturbances, ASCE. J Engng

Mech 1996;122(11):1101–8.

[8] Lin YK, Cai GQ. Probabilistic structural dynamics: advanced theory

and applications. New York: McGraw-Hill; 1995.

[9] Lin YK, Li QC. New stochastic theory for bridge stability in turbulent

flow. J Engng Mech 1993;119:113–27.

[10] Shinozuka M, Deodatis G. Stochastic process models for earthquake

ground motion. Probab Engng Mech 1988;3(3):114–23.

[11] Sundaram SM, Babu KG, Aswathanarayana PA. Probabilistic model

for wave forces acting on a cylinder in random waves. Ocean Engng

1992;19(3):259–69.

[12] Zhou Y, Hou Z, Dimentberg M, Noori M. A model for general

periodic excitation with random disturbance and its application.

J Sound Vib 1997;203(4):607–20.

Z. Hou et al. / Probabilistic Engineering Mechanics 19 (2004) 211–218218