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Journal of Sound and Vibration "0887# 102"3#\ 562697
IDENTIFICATION OFMULTI-DEGREE-OF-FREEDOM NON-LINEAR
SYSTEMS UNDER RANDOM EXCITATIONS BYTHE REVERSE PATH SPECTRAL
METHOD
C[ M[ RICHARDS AND R[ SINGH
Acoustics and Dynamics Laboratory\ Department of Mechanical
Engineering\The Ohio State University\ Columbus\ OH 32109!0096\
U[S[A[
"Received 00 June 0886\ and in _nal form 19 January 0887#
Conventional frequency response estimation methods such as the
{{H0|| and {{H1||methods often yield measured frequency response
functions which are contaminated by thepresence of non!linearities
and hence make it di.cult to extract underlying linear
systemproperties[ To overcome this de_ciency\ a new spectral
approach for identifyingmulti!degree!of!freedom non!linear systems
is introduced which is based on a {{reversepath|| formulation as
available in the literature for single!degree!of!freedom
non!linearsystems[ Certain modi_cations are made in this article
for a multi!degree!of!freedom{{reverse path|| formulation that
utilizes multiple!input:multiple!output data fromnon!linear systems
when excited by Gaussian random excitations[ Conditioned {{Hc0||
and{{Hc1|| frequency response estimates now yield the underlying
linear properties withoutcontaminating e}ects from the
non!linearities[ Once the conditioned frequency responsefunctions
have been estimated\ the non!linearities\ which are described by
analyticalfunctions\ are also identi_ed by estimating the
coe.cients of these functions[ Identi_cationof the local or
distributed non!linearities which exist at or away from the
excitationlocations is possible[ The new spectral approach is
successfully tested on several examplesystems which include a
three!degree!of!freedom system with an asymmetric non!linearity\a
three!degree!of!freedom system with distributed non!linearities and
a _ve!degree!of!free!dom system with multiple non!linearities and
multiple excitations[
7 0887 Academic Press Limited
0[ INTRODUCTION
The properties of multi!degree!of!freedom linear systems are
typically identi_ed using timeor frequency domain modal parameter
estimation techniques 0[ The frequency domaintechniques extract
modal parameters from {{H0|| and {{H1|| estimated frequency
responsefunctions in the presence of uncorrelated noise 1\ 2[
However\ if the system underidenti_cation also possesses
non!linearities\ these conventional estimates often
yieldcontaminated frequency response functions from which accurate
modal parameters cannotbe determined 3\ 4[ Such conventional
methods are also incapable of identifying thenon!linearities[
To accommodate the presence of non!linearities\ several
researchers have developedmethods to improve frequency domain
analysis of non!linear systems 500[ For example\the functional
Volterra series approach for estimating higher order frequency
responsefunctions of non!linear systems has gained recognition 5[
This method has been used toestimate _rst and second order
frequency response functions of a non!linear beamsubjected to
random excitation 6\ where curve _tting techniques were used for
parametric
9911359X:87:13956225 ,14[99:9:sv870411 7 0887 Academic Press
Limited
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C[ M[ RICHARDS AND R[ SINGH563
estimation of an analytical model[ However\ the method is very
computationally intensiveand estimation of third and higher order
frequency response functions has beenunsuccessful[ To alleviate
this problem\ sinusoidal excitation was used to estimate only
thediagonal second and third order frequency response functions of
the Volterra series 7[Other higher order spectral techniques have
also been employed for the analysis ofnon!linear systems 8[ For
instance\ the bi!coherence function has been used to detect
thesecond order non!linear behaviour present in a system 09[ Also\
the sub!harmonicresponses of a high speed rotor have been studied
using bi!spectral and tri!spectraltechniques 00[
An alternative approach has recently been developed by Bendat et
al[ 0104 forsingle!input:single!output systems which identi_es a
{{reverse path|| system model[ Asimilar approach has been used for
the identi_cation of two!degree!of!freedom non!linearsystems where
each response location is treated as a single!degree!of!freedom
mechanicaloscillator 05[ Single!degree!of!freedom techniques are
then used to identify systemparameters 06[ However\ this approach
requires excitations to be applied at everyresponse location and it
also inhibits the use of preferred higher dimensional
parameterestimation techniques that are commonly used for the modal
analysis of linear systems 0[
The literture review reveals that there is clearly a need for
frequency domain systemidenti_cation methods that can identify the
parameters of non!linear mechanical andstructural systems[ Also\
improvements to the frequency response estimation methods suchas
the {{H0|| and {{H1|| methods are necessary when measurements are
made in the presenceof non!linearities[ The primary purpose of this
article is to introduce an enhancedmulti!degree!of!freedom spectral
approach based on a {{reverse path|| system model[Additional
discussion is included to justify the need for spectral
conditioning andcomputational results are given to illustrate the
performance on several non!linear systems[However\ focus of this
article is on the mathematical formulation for
multi!degree!of!free!dom non!linear systems[ Speci_c objectvies
include the following] "0# to accommodate forthe presence of
non!linearities so that improved estimates of the linear dynamic
compliancefunctions can be determined from the input:output data of
multi!degree!of!freedomnon!linear systems when excited by Gaussian
random excitations^ "1# to estimate theunderlying linear systems|
modal parameters from these linear dynamic compliancefunctions
using higher dimensional modal analysis parameter estimation
techniques^"2# to determine the coe.cients of the analytical
functions which describe local ordistributed non!linearities at or
away from the locations where the excitations areapplied^ "3# to
assess the performance of this new method via three
computationalexamples with polynomial type non!linearities[
Comparison of this method with anexisting time domain method is in
progress\ and ongoing research is being conducted toconsider both
correlated and uncorrelated noise[ Issues such as the spectral
variability ofcoe.cient estimates as well as other errors are
currently being examined and will beincluded in future articles[
However\ these issues have been omitted from this article sothat
focus can be kept on introducing an analytical approach to
multi!degree!of!freedomsystems[
1[ PROBLEM FORMULATION
1[0[ PHYSICAL SYSTEMS
The equations of motion of a discrete vibration system of
dimension N with localizednon!linear springs and dampers can be
described in terms of a linear operator Lx"t# and
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NON!LINEAR IDENTIFICATION 564
a non!linear operator Nx"t#\ x "t#]
Lx"t#Nx"t#\ x "t# f"t#\ Lx"t#Mx "t#Cx "t#Kx"t#\
Nx"t#\ x "t# sn
j0
Aj yj "t#\ "0ac#
where M\ C and K are the mass\ damping and sti}ness matrices\
respectively\ x"t# is thegeneralized displacement vector and f"t#
is the generalized force vector[ Also refer to theappendix for the
identi_cation of symbols[ The non!linear operator Nx"t#\ x "t#
containsonly the non!linear terms which describe the localized
constraint forces and this operatoris written as the sum of n
unique non!linear function vectors yj "t# representing each jthtype
of non!linearity present "e[g[\ quadratic\ cubic\ _fth order\
etc[#[ Considering onlynon!linear elastic forces\ each yj "t# is
de_ned as yj "t# "Dxk "t#mj#\ where Dxk "t# is therelative
displacement across the kth junction where the jth type of
non!linearity exists\ andmj is the power of the jth type of
non!linearity[ These vectors yj "t# are column vectors oflength qj
\ where qj is the number of locations the jth type of non!linearity
exists[ Note thata single physical junction may contain more than
one type of non!linearity "e[g[\ aquadratic and cubic#^ therefore\
more than one yj "t# is necessary to describe the
non!linearconstraint force across that particular junction\ as
illustrated in the examples to follow[The coe.cient matrices Aj
contain the coe.cients of the non!linear function vectors andare of
size N by qj [ Inserting equations "0b# and "0c# into equation
"0a#\ the non!linearequations of motion take the form
Mx "t#Cx "t#Kx"t# sn
j0
Aj yj "t# f"t#[ "1#
From a system identi_cation perspective\ it is assumed that the
types of non!linearities andtheir physical locations are known[
Therefore the n non!linear function vectors yj "t# canbe
calculated^ also\ the coe.cients of yj "t# can be placed in the
proper element locationsof the coe.cient matrices Aj [ This
assumption renders limitations on the practical use ofthis method
since various types of non!linearities at each location are not
always known[Therefore\ research is currently being conducted to
alleviate this limitation[ However\ itshould be noted that this
restriction is currently true for any identi_cation scheme
whenapplied to practical non!linear systems[
Consider several multi!degree!of!freedom non!linear systems as
illustrated in Figure 0[The _rst example as shown in Figure 0"a#
possesses an asymmetric quadraticcubicnon!linear sti}ness element
which exists between the second and third masses and aGaussian
random excitation is applied to the _rst mass]
f e12 "t# k1 "x1 "t# x2 "t## a1 "x1 "t# x2 "t##1 b1 "x1 "t# x2
"t##2\
f"t# f0 "t# 9 9T[ "2a\ b#
Assuming that the form of the non!linear elastic force f e12 "t#
is known\ the non!linearoperator Nx"t#\ x "t#\ the non!linear
functions "y0 "t# and y1 "t## and their respectivecoe.cient
matrices "A0 and A1# take the form
Nx"t#\ x "t#A0 y0 "t#A1 y1 "t#\ y0 "t# "x1 "t# x2 "t##1\
y1 "t# "x1 "t# x2 "t##2\ A0 "9 a1 a1#T\ A1 "9 b1 b1#T[ "3ae#
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m1x1
c1 k1
c3 k3
f23, c2
f1
x2
x3
k5, c5
e
(a)
(c)
x1 f1
f23, c2
x2
x3
e
f3, c3e
f53, c6e
f23, c3e
f12, c1e
(b)
m1
m2 m2
m3
m4
m3
k4, c4
x4
f4f1
x3
x5
x2x1
k2, c2k1, c1
m3m1 m2
m5
C[ M[ RICHARDS AND R[ SINGH565
Notice\ since two types of non!linearities "quadratic and cubic#
exist at a single junction\yi "t# and y1 "t# both contain the same
relative displacements[ Example II of Figure 0"b#has distributed
cubic sti}ness non!linearities at every junction and a Gaussian
randomexcitation is applied to the _rst mass[ Therefore\
f e01 "t# k0 "x0 "t# x1 "t## b0 "x0 "t# x1 "t##2\
f e12 "t# k1 "x1 "t# x2 "t## b1 "x1 "t# x2 "t##2\
f e2 "t# k2 x2 "t# b2 x2 "t#2\ f"t# f0 "t# 9 9T\
Nx"t#\ x "t#A0 y0 "t#\
y0 "t# "x0 "t# x1 "t##2 "x1 "t# x2 "t##2 x2 "t#2T\ "4ag#
A0 & b0b09 9b1b1 99b2[Here a single type of non!linearity
exists at three junctions[ Therefore\ y0 "t# is a 2 by 0column
vector[ Example III of Figure 0"c# is composed of a cubic
non!linear sti}nesselement between the second and third masses and
an asymmetric non!linear sti}ness
Figure 0[ Example cases[ "a# I] three!degrees!of!freedom system
with a local asymmetric quadraticcubicnon!linearity f e12"t# and
one excitation f0"t#^ "b# II] three!degree!of!freedom system with
distributed cubicnon!linearities f e01"t#\ f e12"t#\ f e2 "t#\ and
one excitation f0"t#^ "c# III] _ve!degree!of!freedom system with a
local cubicnon!linearity f e12"t#\ a local asymmetric quadratic_fth
order non!linearity f e42"t# and two excitations f0"t# and
f3"t#[All excitations are Gaussian random with zero mean and
variance one[
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NON!LINEAR IDENTIFICATION 566
TABLE 0
Linear modal properties of example systems shown in Figure 0
Example Mode Natural frequency "Hz# ) Damping Eigenvector
0 11=3 9=6 "0=99\ 9=79\ 9=34#I\ II 1 51=7 1=9 "9=79\ 9=34\
0=99#
2 89=6 1=8 "9=34\ 0=99\ 9=79#
0 00=0 9=6 "9=12\ 9=33\ 9=50\ 9=21\ 0=99#1 29=2 0=8 "9=67\ 0=99\
9=38\ 9=28\ 9=15#
III 2 33=2 1=7 "9=46\ 9=15\ 9=34\ 0=99\ 9=98#3 48=9 2=6 "0=99\
9=64\ 9=33\ 9=48\ 9=93#4 61=2 3=5 "9=17\ 9=59\ 0=99\ 9=36\
9=95#
TABLE 1
Linear and non!linear elastic force coe.cients of example
systems
Example Linear Non!linear
I k1 099 kN:m a1 7 MN:m1\ b1 499 MN:m2
II k0 k1 k2 099 kN:m b0 b1 b2 0 GN:m2
III k2 k5 49 kN:m b2 499 MN:m2\ a5 499 kN:m1\ g5 09 GN:m4
TABLE 2
Simulation and signal processing parameters] total number of
samples103h\ Dt9=4 ms\ total period102h ms\ Hanning window\ 102
samples:average\ 1h averages
Example h Magnitude of Gaussian excitation"s#
I 04 4 kNII 09 499 NIII 04 1 kN "both excitations#
element described by a quadratic and _fth order term between the
third and _fth masses[Gaussian random excitations are applied to
masses 0 and 3 of this system[ Therefore\
f e12 "t# k2 "x1 "t# x2 "t## b2 "x1 "t# x2 "t##2\
f e42 "t# k5 "x4 "t# x2 "t## a5 "x4 "t# x2 "t##1 g5 "x4 "t# x2
"t##4\
f"t# f0 "t# 9 9 f3 "t# 9T\
Nx"t#\ x "t#A0 y0 "t#A1 y1 "t#A2 y2 "t#\
y0 "t# "x1 "t# x2 "t##2\ y1 "t# "x4 "t# x2 "t##1\ y2 "t# "x4 "t#
x2 "t##4\
A0 "9 b2 b2 9 9#T\ A1 "9 9 a5 9 a5#T\
A2 "9 9 g5 9 g5#T[ "5aj#
The modal parameters of the underlying linear systems "i[e[\
systems with Aj 9# are givenin Table 0 and the coe.cients of the
non!linear elastic forces "i[e[\ the elements of Aj # aregiven in
Table 1 in terms of a\ b and g\ where a is the coe.cient of the
quadraticnon!linearities\ b is the coe.cient of the cubic
non!linearities and g is the coe.cient of
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2
3
1
0
1
2
3
20 40 60 80 100 1200
Frequency (Hz)