J R Maurer

R L Barnoski

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R. L. BARNOSKI 
J. R. MAURER 
Computer Sciences Corporation, 
Mountain View, Calif. 
Transient Characteristics of Simple Systems 
to Modulated Random Noise 
Discussed are the mean-square response exceedance characteristics of a single-tuned 
system to amplitude modulated noise. The results bear on the accuracy of spectral esti-
mates of nonstationary data, and subsequently, relate directly to the design, analysis, 
and testing of structural systems in environments as gusts, earthquakes, and ignition 
transients. For noise correlated as an exponentially damped cosine, the nonstationary 
response may exceed its stationary value by a factor in excess of two. A time-varying 
shaping filter explanation is offered for this behavior. For white noise, such exceedances 
do not occur. 
Introduction 
L 1 RANSIENT response properties of unimodal systems 
to modulated random noise are fundamental to understanding 
and subsequently solving statistical problems common not only to 
structural design, but to data processing as well. Such results 
are applicable directly to response predictions in nonstationary 
environments as gusts, earthquakes, and ignition transients. 
Since this work relates to the filtered output of a "weighted" sig-
nal or, correspondingly, the results of a Fourier analysis of a 
weighted data stream, our results are relevant in a more general 
sense to methods of time series analysis. 
The form of the input considered is amplitude modulated 
random noise. For a more rapidly varying, nonrepetitive 
modulation function of limited time duration, the resultant noise 
frequently is called a random shock pulse. The system is 
a mechanical single-degree-of-freedom system or, equivalently, 
a single-tuned bandpass filter. Interest here focuses on the time-
varying character of the system mean-square response. 
The basic formulation for this class of problems has been estab-
Contributed by the Applied Mechanics Division for publication 
(without presentation) in the JOT/BNAL OF APPLIED MECHANICS. 
Discussion on this paper should be addressed to the Editorial De-
partment, ASME, United Engineering Center, 345 East 47th Street, 
New York, N. Y. 10017, and will be accepted until April 20, 1973. 
•Discussion received after this date will be returned. Manuscript re-
ceived by ASME Applied Mechanics Division, September, 1971; final 
revision, March, 1972. Paper No. 72-APM-lfFF. 
lished previously by the authors [ I ] 1 and others [3-7], and much 
of the mathematics carried out for both white noise and noise 
correlated in the form of a damped harmonic. A leading result 
was that the output response of the system for the correlated 
noise may exceed its stationary value. Except for two studies of 
somewhat specialized interests [2, 6], explanations for this be-
havior and plots to predict its occurrence have not been ad-
vanced. I t is the intent of this paper to fill these voids. 
Problem Definition 
We seek the variance of y{t) given the equation of motion 
y(t) + 2fw„j/(«) + co„2j/(i) = - / ( 0 
m 
(1) 
where the input excitation is the modulated noise expression 
/(*) = e{t)n{t) (2) 
with n(t) assumed Gaussian with a mean value of zero. For 
conciseness, we concentrate only on the modulation function 
e(t) = u(t) 
and the two noise correlation functions 
R„(T) = 2 X « 0 S ( T ) 
JR»(T) = i?oe~ a | r | cos pr 
(3) 
(4) 
L Numbers in brackets designate References at end of paper. 
Journal of Applied Mechanics MARCH 1 9 7 3 / 73 Copyright © 1973 by ASMEDownloaded From: https://appliedmechanics.asmedigitalcollection.asme.org on 06/29/2019 Terms of Use: http://www.asme.org/about-asme/terms-of-use
where uit) denotes the unit step function and T is the time lag of 
the autocorrelation function. The delta function identifies white 
noise and the exponentially decaying harmonic expression defines 
the correlated noise. 
Solution Formulations 
Procedures frequently used to compute the expectation 
E[y2it)] for a modulated noise input are founded upon some form 
of either unit impulse formulations or/and spectral formulations. 
We concentrate upon the latter. Since the detailed mathematics 
are outlined elsewhere [1, 2],2 we omit many intermediate steps 
and quote expressions essential to an understanding of our solu-
tion. However, for continuity and completeness in this discus-
sion, expressions previously reported may be repeated here. Im-
plicit in what follows are the assumptions of system linearity, arid 
separability of the input excitation with e(i) a real function. 
The desired mean-square response is given by 
ffy'it) = E[yKt)\ = 
/ : 
Syit, C0)&0 
where the time-varying spectral density is of the form 
Sv(t,u) = S„(«)|/(«, co)|
a 
and 
2»J-
lit, to) = — I ff(co)Fe(co - oi)e
iMd(b 
2ir 
(5) 
(6) 
(7) 
with 
We make 
where 
and 
use 
F.iu 
mcon
2 co2 co 
1 h i2f — 
co„2 con 
/ • c o 
- to) = I eity-'te-^dt 
U — co 
of the fact that 
h'(t) <-> #(to) 
h(t) = — e _ M s i n o « 
am 
a = to„[l - H ' / s 
b = f«„ 
(8) 
(9) 
(10) 
(11) 
2 The serious reader is urged to examine reference [2] as many use-
ful integral evaluations are listed in the Appendix. 
The quantity hit) is the unit impulse response of the system 
with H(co) as the corresponding frequency response function 
F,iw — to) defines a transformation associated with the modula-
tion function and £„(«) is the (two-sided) spectral density func-
tion of the input noise. 
Defining 
1 27 r J - co ff(") 
Equation (7) is reduced to the form 
| /0 , to) | 2 = \H{u)\*Kit, co) 
and 
Sy(t,U) = Kit, 0))SyiU>) 
where 
S,((d) = |ff(to)|2&(to) 
"(dto (12) 
(13) 
(14) 
(15) 
Note the product |ff(to)|2jSn(to) corresponds to the integrand for 
stationary response so that 
/ : 
Syiu)dw. (16) 
The term Kit, co) is dependent only on the system and the shape 
of the modulation function. I t is dubbed a "shaping" filter as it 
acts to alter (in time) the spectral content of what otherwise 
represents a stationary response. This function ultimately 
governs the time variation of <r,,(S) and, subsequently, response 
overshoot. 
Unit Step Modulation 
For the unit step modulation function, 
K(f,w) = 1 + Ait) + Bit) 
/ b 2 - a2 + to2\ 
V «2 / 
2co 
- 2 C ( 0 cos coS 
a 
with the time-varying coefficients 
Ail) = e~2bt(l + — sm2at) 
Bit) = e~ 2 M (sin2 at) 
dt) = e~u 
Bit) = e~u ism at) 
(Dit) sin <at) (17) 
cos at + — sin at 
a ») 
08) 
a -
b = 
Co = 
eit) -
fit) = 
fn -
hit) -
Hico) = 
#o(co) = 
= co„(l - f 2 ) ' / 2 = damped 
natural frequency of system 
= fco„ = exponential decay coef-
ficient in system response to 
a unit impulse 
= stationary coefficient 
= envelope modulation function 
= input excitation 
= system natural frequency in 
Hertz 
= system response to a unit im-
pulse 
= frequency response function of 
the system 
= mcon
2ff(to) 
Kit,w) 
m 
nit) 
Q 
Rnir) 
Ro 
Sl 
s3 
Sniw) 
<S„(co) 
-So 
shaping filter for unit step 
modulation 
mass of system 
Gaussian noise with zero mean 
l / 2 f = measure of system 
damping 
autocorrelation function of nit) 
autocon'elation constant 
— Si* = a + ib 
— St* = p + ia 
two-sided power spectral den-
sity function of input noise 
(o/iSo)S„(co) 
R0/ir — spectral magnitude for 
white noise 
uit) = unit step function 
a = exponential decay coefficient 
for correlated noise 
f = damping factor of system 
p = frequency of autocorrelation 
function for correlated noise 
ay = stationary root-mean-square 
response 
<rvit) = nonstationary response 
Cpk = maximum value of <rvit) 
co„ = 2irf„ = system natural fre-
quency in rad per sec 
74 / M A R C H 1973 Transactions of the ASME Downloaded From: https://appliedmechanics.asmedigitalcollection.asme.org on 06/29/2019 Terms of Use: http://www.asme.org/about-asme/terms-of-use
where the quantities a and 6 are those defined earlier. 
When &,(&)) ->- So, 
5 0 0 
flV '(0 = 2froW 1 
1 - e -2bt 
X I 1 H sin 2at + 2 f})2^)} 
while for the correlated noise, 
VvKt) = —. {RiTi - XtT* + R3T3 - X,Ti] 
TO2 
(19) 
(20) 
where 
fli = Re (Zi) 
ft3 = Be {Z,) 
Z i = Im (Zi) 
X3 = Im (Z„) 
(21) 
with 
1 = S(; Zi = 
z3 = 
p2 + a2 + _Si» \ 
- ^)) .81(81' - S 3 2 ) (Si 2 
1 
(22) 
si2 
The remaining terms are given by 
Ti - - b [1 - A(t>] 
r2 = - 5 ( 0 
Tt = 
ft = 
\ + A{t) + ^ - o » + p . - ^ B ( i ) 
(C(«) + — #(*)) e_ a < cos pt-2 — D{t)e-"1 sin pt 
2 f ~ J 5 (0 - 2 ( C{t) + — D(0) e-" ( sin p« 
2 — D{t)e-at cos (23) 
where A{t), B{t), C{t), and D(t) are those defined previously. 
For the stationary response, equation (20) reduces to 
-Bo 
where 
-[fe)*+*r 
with the normalized values 
Ri = a*Ri 
Rs = a*R, 
(24) 
(25) 
(26) 
When si = s3, the solution offered by equation (20) becomes un-
bounded. Under such circumstances, the mean-square response 
reduces to 
«r.«(0 = 
Ro (/a2 A 
3TO2CO„4 \\b*+ J 
/ / a 2 + 3&2 + 2co„2bA -91, 1 1 a2 62 „  , 2 „ „ , . „ 
- - (a2 - b2 - 2bfco„2) sin2 at ) \ (27) , ) ) 
a i / a — • -
Fig, 1 Normalized spectral plots of |H0(o)| for Q = 5 0 and S0(w) for 
p /a = 0.5 
and has the stationary value 
a 
8OT2G>, * ( 3 + *) 
(28) 
Results 
For purposes of display, it is convenient to define the functions 
So(co) = - S„(co) 
ffo(co) = munW{u) 
(29) 
Plots of these normalized quantities are shown in Fig. 1. The 
effect on So{u) of a change in the center frequency is a cor-
responding translation with no alteration in shape. Thus, for 
p/a = 1.5, <S0(w) would appear but translated to co/o = 1.5. 
The stationary, mean-square response for any /So(o>) and 
Ho{u) is given by the integral 
Ro 
; / : 
|H0(w)|
2So(w)cfcj (30) 
which can be reduced to equation (24). The coefficient, Co, is 
proportional to the foregoing integral and is plotted for Q = 50 in 
Fig. 2. The maximum response of av occurs at "resonance," that 
is, where co/a = 1 and p/a = 1. 
Typical response time histories are shown in Fig. 3. Since 
arguments of 2a, {a + p), and {a — p) are embedded in the ex-
pressions for ay{t), the oscillatory nature of the response is not 
surprising. For n{t) where a/b > 5, the response oscillation is 
twice the damped natural frequency of the system which is typical 
of a„{t) when n{t) is white noise. The "arrowed" values repre-
sent stationary response magnitudes for the indicated a/b ratios. 
For a spectral input with broadband characteristics as for a/b = 
10, exceedance of the stationary value is not observed. Over-
shoot of stationary values is noted for the smaller values of a/b. 
In such cases, the spectral character of the input is highly peaked 
and prominent oscillations in the response are particularly notice-
able. Such oscillations are governed by the center frequency 
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100 
Fig. 2 Normalized stationary value for correlated noise inputs, Q 
associated with either So(w) or -Ho(co), or sums and differences 
thereof. 
A requisite of overshoot, therefore, is a highly peaked spectral 
input. A lightly damped system is catalytic but not essential 
as exceedances do occur for systems with moderate damping 
(Q = 5, for example). As iSo(cu) and H0(w) collectively cannot 
dictate overshoot, we are left with K(t, co). 
The spectral character of Kit, co) is determined by the modula-
tion and system functions, and is clearly time-dependent. It is 
highly selective (many peaks and valleys) very early in the re-
sponse time history and gradually resolves to a constant as the 
system achieves stationarity. A plot of this function at one in-
stant in time is shown in Fig. 4. For interest, the product of 
K{t, co) and the system function is shown in Fig. 5. 
At any instant in time, the value of <rv(t) is computed by in-
tegration over co of the integrand \Ho(co)\2Si)(co)K(t, co), a triple 
product. For a system tuned to resonance, the peaks of <S0(co) 
and .ffo(co) are coincident so that the nonstationary integrand is 
affected mainly by the character of K(t, co) near resonance. Such 
is notched for all /„£ and appears as nearly a constant to the 
product |.Ho(co)|2iSo(co), albeit a different magnitude at each value 
of fnt- When stationarity is achieved, K(t, co) reduces to a single 
constant value. The notching effect (magnitude) is most pro-
nounced for the early values of fnt and gradually becomes less 
severe with increasing time. At resonance, therefore, the response 
is governed by K(t, co) evaluated over an extremely narrow fre-
quency band; cry(t) thus gradually builds up (in time) to its 
stationary value and overshoot is not experienced. 
For So(co) away from ffo(co), it is mainly the interplay of the 
notches and peaks of K{t, co) with a peaked »So(co) which produces 
response spectra that fluctuate widely in time. Upon integra-
tion, such spectra yield fluctuating response values which may 
exceed the stationary value. The conditions under which <ry(t) 
exceeds its stationary value for a system with Q = 50 are sum-
marized in Fig. 6. For a/b > 5, no overshoot is noted over the 
range 0.1 < p/a < 10 and, as expected, no overshoot occurs at 
Fig. 3 Normalized system response to correlated noise modulated by 
unit step function, Q — 50, p/a = 0.5 
Fig. 4 Shaping filter K{t, a) with fnt = I 
500 
1 0 0 -
v~« 
fnt = 3 
0.5 1.0 1.5 2.0 2.5 
CU/CUr, — 
Fig. 5 mw„2|/(f,«)| = K(&>)K(r, a) with Q = 50 
76 / M A R C H 1 9 7 3 Transactions of the AS ME Downloaded From: https://appliedmechanics.asmedigitalcollection.asme.org on 06/29/2019 Terms of Use: http://www.asme.org/about-asme/terms-of-use
Fig. 6 Response overshoot for correlated noise inputs modulated by e(f) 
= u(f), Q = 50 
resonance. Exceedance values greater than two are found over 
1 < pM < 10 and for 0.1 < a/b < 0.50. 
Concluding Remarks 
The mathematics is reviewed for computing the nonstationary 
response of a single-tuned system to amplitude modulated noise 
of damped harmonic correlation. Previously established is that 
the system response does not exceed its stationary value for white 
noise. For correlated noise inputs where a/b > 5, a lightly 
damped system perceives the noise spectrum as nearly white and 
overshoot of the stationary value does not occur. 
For highly peaked spectral inputs, overshoots of the stationary 
value commonly occur. This phenomenon is dependent upon 
the relative interaction of the system parameters, the noise 
parameters, and the modulation function. I t is governed mainly 
by the properties of the function K(t, u>), dubbed a time-varying 
shaping filter. Stationary response properties are summarized 
completely in Fig. 2, with Fig. 3 needed for clarity. 
The formulation presented is general and may be applied, in 
theory, to any real modulation function. The rectangular step 
and damped exponential modulations have been studied in detail 
elsewhere [2, 5]. For rectangular step modulation, the residual 
response is sensitive to the step duration as might be expected. 
The maximum response may occur after termination of the pulse 
and may be higher than the peak response for a unit step modula-
tion. I t may not only exceed the stationary value, but the peak 
value during the primary response as well. For exponentially 
damped modulation, response overshoot may be controlled by 
varying the decay coefficient, the system damping, or both. 
For some envelope functions, it may be useful to modify K(t, ca) 
as defined by equation (12). This was done for the damped ex-
ponential modulation [2] in order to make a direct comparison 
with the results for a unit step modulation. Such resulted in a 
fictitious system function which, interestingly, was subject to 
ready interpretation. 
Although the work shown is analytically precise, the functions 
are not necessarily convenient to evaluate for all practical appli-
cations. Much attention must be given to mathematical detail, 
particularly for modulation functions of finite duration. Accord-
ingly, approximate formulations have been the subject of investi-
gation by various authors [3, 5, 7], each with varying degrees of 
success. 
Acknowledgments 
The contractual support by the NASA Marshall Space Flight 
Center for an initial s tudy is acknowledged here with thanks. 
Additionally, appreciation is due the NASA Ames Research 
Center for support of extensions in this work relevant to process-
ing data from aircraft flight simulations and wind tunnel tests. 
References 
1 Barnoski, R. L., and Maurer, J. R., "Mean-Square Response of 
Simple Mechanical Systems to Nonstationary Random Excitation," 
JOT/BNAL OP APPLIED MECHANICS, Vol. 36, No. 2, TBANS. ASME, Vol. 
91, Series E, June 1969, pp. 221-227. 
2 Barnoski, R. L., and Maurer, J. R., "Mean-Square Exceedance 
Characteristics of a Single-Tuned System to Amplitude Modulated 
Random Noise," NASA CR-61352, National Aeronautics and Space 
Administration, Washington, D. C , Apr. 1971. 
3 Bucciarelli, L. L., and Kuo, C , "Mean-Square Response of a 
Second-Order System to Nonstationary Random Excitation," JOTJE-
NAL OF APPLIED MECHANICS, Vol. 37, No. 3, TBANS. ASME, Vol. 92, 
Series E, Sept. 1970, pp. 612-616. 
4 Caughey, T. K., and Stumpf, H. J., "Transient Response of a 
Dynamic System Under Random Excitation," JOTJBNAL OF APPLIED 
MECHANICS, Vol. 28, No. 4, TBANS. ASME, Vol. 83, Series E, Dec. 
1961, pp. 563-566.. 
5 Hasselman, T. K., "An Analytical Basis for Time-Modulated 
Random Vibration Testing," NASA CR-66770, National Aeronau-
tics and Space Administration, Washington, D. C , Jan. 1969. 
6 Holman, R. E., and Hart, G. C , "Response of Simple Mechani-
cal Systems to Segmented Nonstationary Random Excitation," 
UCLA Technical Report 71-15, School of Engineering and Applied 
Science, University of California at Los Angeles, July 1970. 
7 Roberts, J. B., "The Covariance Response of Linear Systems 
to Nonstationary Random Excitation," Journal of Sound and Vibra-
tion, Vol. 14, No. 3, Feb. 1971. 
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Transient Characteristics of Simple Systems to Modulated Random Noise

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Transient Characteristics of Simple Systems to Modulated Random Noise

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