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

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

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#

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[

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

2

3

1

0

1

2

3

20 40 60 80 100 1200

Frequency (Hz)