Hindawi Publishing CorporationJournal of Applied MathematicsVolume 2013 Article ID 754576 7 pageshttpdxdoiorg1011552013754576

Research ArticleA New Simultaneous Identification of the HarmonicExcitations and Nonlinear Damping of Forced DampedNonlinear Oscillations A Parametric Approach

T S Jang

Department of Naval Architecture and Ocean Engineering Pusan National University Busan 609-735 Republic of Korea

Correspondence should be addressed to T S Jang taekpusanackr

Received 27 September 2012 Revised 21 December 2012 Accepted 23 December 2012

Academic Editor George Jaiani

Copyright copy 2013 T S Jang This is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

This paper presents a novel general method that is aimed at identifying both the harmonic excitations and nonlinear damping offorced nonlinear oscillation systems An inverse formalism is developed for the identification method which is shown to be wellposedThat is the inverse formalism has a unique solution and it depends continuously on the dataThe unique solution is derivedin the closed form In addition to the mathematical analysis a numerical example involving a highly nonlinear system is examinedfor demonstrating the workability of the proposed method for simultaneously identifying both the two aforementioned excitationsand nonlinear damping

1 Introduction

Over the past decades a considerable number of studies havebeen made on forced oscillations arising in many diversefields of science and engineering This is mainly becausethe studies are closely related in the investigation of systemperformance such as the effective control and the estimationof the structure lifetime Thus as a forward problem manyresearchers investigated nonlinear responses and stabilityanalysis of the second-order nonlinear differential equationsof a general form which have a periodic forcing termHowever these studies strongly depend on how accurately wecan estimate the characteristics of forced loading conditionorand nonlinear damping This is regarded as an inverseproblem of the system identification which is not onlyessential but crucial for the forward problem

There are a variety of researches regarding the systemidentification devoted to the study about damping [1] Forexample Iourtchenko and Dimentberg [2] suggested a pro-cedure for in-service identification of the damping based onthe stochastic averaging method from a measured stationaryresponse Jang et al [3 4] proposed a new nonparametricmethod to recover the functional form of both nonlinear

damping and nonlinear restoring forces of the dynamicsystemusing the transient data As for the force identificationthere are also a lot of articles [5] Doyle [6] studied a wavelet-based deconvolution method to recover impact forces usingexperimentally measured responses Zen et al [7] proposeda method to obtain contact forces from experimental mea-surement of a single impact Jang et al [8] introduced a newtechnique for the identification of external loading of longtime duration which is nonharmonic but periodic acting ona nonlinear dynamic system

As mentioned above although a large number of stud-ies have been carried out on the system identification ofnonlinear damping or external force little is known abouttheir combinations that is the simultaneous identificationof nonlinear damping as well as external force In factmany researchers in engineering fields have wondered if theproblem of the simultaneous identification is mathematicallysolvable Motivated by this an inverse formalism is inthis present work mathematically developed for identifyingboth the nonlinear damping and external forces in a forcednonlinear system

In this study we propose a method for the simultaneousidentification which is a new procedure with remarkable

2 Journal of Applied Mathematics

improvements First based on the concept of a zero-crossingtime we develop an inverse formalism which is linear evenif the original dynamical equation of motion is nonlinearSecond we prove that the inverse formalism has a uniquesolution and its closed form solution is derived which makesthe proposed method quite simple and straightforwardThird the inverse formalism developed is shown to havestability properties of solution This means that the methodby the inverse formalism work without the use of anyregularization so that it can have a wide range of physicalapplications This contrasts the usual inverse problems ill-posed in the sense of stability [3 4 8ndash10] which require anadditional process of regularization

Finally given the model nonlinear differential equationconcerning the quadratic nonlinear damping and cubicnonlinear restoring characteristics the proposed methodrsquosworkability for simultaneously identifying both excitationsand nonlinear damping is demonstrated through a numericalexample

2 Forced Damped Nonlinear Oscillation

We consider a nonlinear damped dynamical system subjectto an external harmonic load Γ sdot cos 120596119905 of an amplitudeΓ gt 0 and frequency 120596 gt 0 The equation of motion is asecond-order nonlinear ordinary differential equation for theresponse 119902

119898 119902 + 119863 ( 119902) + 119877 (119902) = Γ sdot cos 120596119905 119905 gt 0 (1)

Here the constant 119898 gt 0 represents the mass of a particle ofan oscillator The term 119877(119902) is a general nonlinear restoringforce and the term 119863( 119902) appearing in (1) is the nonlineardamping of a system which is assumed to have a quadraticnonlinear term

119863( 119902) = 1198881119902 + 119888211990210038161003816100381610038161199021003816100381610038161003816 (2)

where 1198881and 1198882are the linear and nonlinear damping coeffi-

cients respectively The functional form (2) of nonlinearityis often encountered in the area of engineering field [11]

The sum of the inertia force 119898 119902 and restoring force 119877(119902)and the derivative of the sum will play an important role inidentifying both the excitation Γ and nonlinear damping119863( 119902)

in (1) We define the sum as

119869 (119902 119902) equiv 119898 119902 + 119877 (119902) (3)

The derivative of 119869 with respect to time then becomes

119869 =120597119869

120597119902(119902 119902) sdot 119902 +

120597119869

120597 119902(119902 119902) sdot 119902

=119889119877

119889119902(119902) sdot 119902 + 119898 119902

(4)

Noting that (1) is equivalent to

119869 = Γ sdot cos 120596119905 minus 119863 ( 119902) (5)

we can express 119869 in terms of the excitation and damping asfollows

119869 = minusΓ120596 sdot sin 120596119905 minus119889119863

119889 119902119902 (6)

by differentiating (5) with respect to timeIn what follows we assume that (1) has two dynamic

responses of 119879(= 2120587120596)-periodic limit cycle solutions 119902119894(119905)

which are forced by Γ119894for 119894 = 1 2

119898 119902119894+ 119863 ( 119902

119894) + 119877 (119902

119894) = Γ119894sdot cos 120596119905 119905 gt 0 for 119894 = 1 2

(7)

The response analysis of the nonlinear damped dynamicalsystem considered can be regarded as a forward problemwhose solution procedures are well established theoreticallyas well as numerically However an inverse problem can nowbe formulated in the form of a question given a motionresponse of the nonlinear dynamical system is it possibleto identify the excitations and nonlinear damping of thesystem at the same time The question may be interestingmathematically as well as practically In order to answer thequestion we will begin with introducing a zero-crossing timein the next section

3 Some Lemmas on Zero-Crossing Time

This section is concernedwith a zero-crossing time associatedwith motion responses of a system Thereby we derive somelemmas which are essential in the present study We startby defining a zero-crossing time in terms of 119869 for 119894 = 1 2denoted by (119905ze

119869)119894 satisfying

119869119894[(119905

ze119869)119894] = 0 for 119894 = 1 2 (8)

Here 119869119894stands for the sum (3) which corresponds to 119902

119894 that

is a function from (0infin) to R

119869119894equiv 119898 119902119894+ 119877 (119902

119894) (9)

Physically the zero-crossing time defined in (8) is a usefulconcept for finding the force balancing between the excitationand the damping in a forced damped oscillatorymotion of (7)as discussed in the following lemma

Lemma 1 Suppose that (119905ze119869)119894 119894 = 1 2 is a zero-crossing time

in terms of 119869 for the system (7) Then the following holds

Γ119894sdot cos [120596(119905ze

119869)119894] = 119863 119902

119894[(119905

ze119869)119894] 119891119900119903 119894 = 1 2 (10)

Proof A simple substitution of (9) into (7) yields

119869119894= Γ119894sdot cos 120596119905 minus 119863 ( 119902

119894) (11)

Since (11) remains valid for 0 lt 119905 lt infin it should also hold forthe zero-crossing time of (119905ze

119869)119894gt 0 However the left hand

side of (11) vanishes for the zero-crossing time (119905ze119869)119894due to

the definition (8) That is

0 = Γ119894sdot cos [120596(119905ze

119869)119894] minus 119863 119902

119894[(119905

ze119869)119894] for 119894 = 1 2 (12)

which immediately proves (10)

Journal of Applied Mathematics 3

Next we introduce a zero-crossing time in terms of accel-eration (119905zeacc)119894 for 119894 = 1 2 such that

119902119894[(119905

zeacc)119894] = 0 for 119894 = 1 2 (13)

where 119902119894denotes an acceleration response of (7) We notice

that the time rate of change of the damping is always zero ata zero-crossing time in terms of acceleration which shall beexamined in Lemma 2

Lemma 2 For the function 119869119894defined by

119869119894(119905) =

119889119869

119889119905[119902119894(119905) 119902119894(119905)] 119891119900119903 119894 = 1 2 (14)

one has119869119894[(119905

zeacc)119894] = minusΓ

119894120596 sdot sin [120596 (119905

zeacc)119894] 119891119900119903 119894 = 1 2 (15)

Here (119905zeacc)119894 refers to a zero-crossing time in terms of accelera-tion

Proof In the same way as Lemma 1 (14) should hold forthe zero-crossing times in terms of acceleration that is 119905 =

(119905zeacc)119894 for 119894 = 1 2 However (14) is rewritten as

119869119894= minusΓ119894120596 sdot sin 120596119905 minus

119889119863

119889 119902119894

119902119894 (16)

in which the second term of the right hand side becomeszero because the acceleration vanishes when 119905 = (119905

zeacc)119894 by

the definition (13) It follows that the above equation becomes(15) which completes the proof

4 Inverse Formalism

It is a high time that we develop an inverse formalism for theinverse problem raised by Section 2 based on the lemmasin the previous section We recall that the excitations andnonlinear damping are parameterized by Γ

1 Γ2 1198881 and 119888

2in

Section 2This results in that the developed inverse formalismalso involves the parameters of Γ

1 Γ2 1198881 and 119888

2as illustrated

in Lemma 3

Lemma 3 (inverse formalism) Equations (10) and (15) inLemmas 1 and 2 constitute simultaneous linear algebraicequations for the four unknowns Γ

1 Γ2 1198881 and 119888

2 These can be

written in a compact way as followsΑx = b (17)

Here Α is an operator from R4 to R4 whose matrix is

[A] equiv[[[[[[

[

cos [120596(119905ze119869)1] 0

0 cos [120596(119905ze119869)2]

minus120596 sdot sin [120596(119905zeacc)1] 0

0 minus120596 sdot sin [120596(119905zeacc)2]

minus 119902 [(119905ze119869)1] minus 119902 [(119905

ze119869)1]10038161003816100381610038161003816119902 [(119905

ze119869)1]10038161003816100381610038161003816

minus 119902 [(119905ze119869)2] minus 119902 [(119905

ze119869)2]10038161003816100381610038161003816119902 [(119905

ze119869)2]10038161003816100381610038161003816

0 0

0 0

]]]]]]

]

(18)

and x b isin R4 are represented as (one will call them a solutionvector and data vector resp in this paper)

[x] equiv[[[

[

Γ1

Γ2

1198881

1198882

]]]

]

[b] equiv[[[[

[

0

0

1198691[(119905

zeacc)1]

1198692[(119905

zeacc)2]

]]]]

]

(19)

Proof Equations (10) and (15) in Lemmas 1 and 2 can berewritten as the following

Γ1sdotcos [120596(119905ze

119869)1]minus1198881119902 [(119905

ze119869)1]minus1198882119902 [(119905

ze119869)1]10038161003816100381610038161003816119902 [(119905

ze119869)1]10038161003816100381610038161003816=0

(20)

Γ2sdotcos [120596(119905ze

119869)2]minus1198881119902 [(119905

ze119869)2]minus1198882119902 [(119905

ze119869)2]10038161003816100381610038161003816119902 [(119905

ze119869)2]10038161003816100381610038161003816=0

(21)

minusΓ1120596 sdot sin [120596 (119905

zeacc)1] =

1198691[(119905

zeacc)1] (22)

minusΓ2120596 sdot sin [120596(119905

zeacc)2] =

1198692[(119905

zeacc)2] (23)

It is straightforward to show that the above four equations areequivalent to the operator equation (17) from the definitionof the operator A the vectors x and b of (18) and (19) Thiscompletes the proof

5 Method

In the previous section we developed an inverse formalism(17) whose solution procedure is to be established in thissection The formal expression for the solution may besymbolically written as

x = Αminus1 b (24)

where the notation Αminus1 denotes the inverse operator for thelinear operatorΑ of (18) According to linear algebra it is wellknown that the inverse operator is bounded (or continuous)because it is defined on finite dimensional spaces Thus thesolution x in (24) depends continuously on the data vectorb that is the solution x is stable The construction of x isexamined through the following theorems

Theorem 4 (excitations) The excitations Γ119894for 119894 = 1 2 in (7)

are identified as an analytical form

Γ119894=

minus 119869119894[(119905

zeacc)119894]

120596 sin [120596(119905zeacc)119894](25)

provided that sin [120596 (119905zeacc)119894] = 0 for 119894 = 1 2

Proof It is easy to notice that the formula of (25) immediatelyfollows from the relations of (22) and (23) We first divideboth sides of (22) by minus120596 sdot sin[120596(119905zeacc)1] to obtain Γ

1as follows

Γ1=

minus 1198691[(119905

zeacc)1]

120596 sdot sin [120596 (119905zeacc)1] (26)

4 Journal of Applied Mathematics

Similarly we have

Γ2=

minus 1198692[(119905

zeacc)2]

120596 sdot sin [120596(119905zeacc)2](27)

from (23) This completes the proof

The formulas of the damping coefficients 1198881and 1198882in (2)

can also be obtained in an essentially simple way as to beexamined inTheorem 5

Theorem 5 (nonlinear damping) The linear damping coef-ficient 119888

1and the nonlinear damping coefficient 119888

2in (2) are

determined as

1198881=

1198692[(119905

zeacc)2] sdot cos [120596(119905

ze119869)2] sdot

10038161003816100381610038161003816119902 [(119905

ze119869)1]10038161003816100381610038161003816

120596 sin [120596(119905zeacc)2] 119902 [(119905ze119869)2]

10038161003816100381610038161003816119902 [(119905

ze119869)2]10038161003816100381610038161003816minus10038161003816100381610038161003816119902 [(119905

ze119869)1]10038161003816100381610038161003816

minus

1198691[(119905

zeacc)1] cos [120596(119905

ze119869)1] sdot

10038161003816100381610038161003816119902 [(119905

ze119869)2]10038161003816100381610038161003816

120596 sin [120596(119905zeacc)1] 119902 [(119905ze119869)1]

10038161003816100381610038161003816119902 [(119905

ze119869)2]10038161003816100381610038161003816minus10038161003816100381610038161003816119902 [(119905

ze119869)1]10038161003816100381610038161003816

(28)

1198882=

1198691[(119905

zeacc)1] cos [120596(119905

ze119869)1]

120596 sin [120596(119905zeacc)1] sdot 119902 [(119905ze119869)1]

10038161003816100381610038161003816119902 [(119905

ze119869)2]10038161003816100381610038161003816minus10038161003816100381610038161003816119902 [(119905

ze119869)1]10038161003816100381610038161003816

minus

1198692[(119905

zeacc)2] cos [120596(119905

ze119869)2]

120596 sin [120596(119905zeacc)2] 119902 [(119905ze119869)2]

10038161003816100381610038161003816119902 [(119905

ze119869)2]10038161003816100381610038161003816minus10038161003816100381610038161003816119902 [(119905

ze119869)1]10038161003816100381610038161003816

(29)

Proof It suffices to multiply (20) and (21) by the correspond-ing factors then subtract them from each other As a resultof taking (25) into account we find 119888

1of (28) and then 119888

2of

(29) which completes the proof

Remark 6 Because of the stability properties of solution xour solution procedure does not concern any kind of regu-larization which is necessary for the usual inverse problemsarising in science and engineering

As to the solvability and uniqueness condition the opera-torΑ in (18) should satisfy det A = 0 which is to be discussedin the following theorem

Theorem 7 (uniqueness) Let one denote by Δ

Δ = sin [120596(119905zeacc)1] sdot sin [120596(119905

zeacc)2] 119902 [(119905

ze119869)1] 119902 [(119905

ze119869)2]

times 10038161003816100381610038161003816119902 [(119905

ze119869)2]10038161003816100381610038161003816minus10038161003816100381610038161003816119902 [(119905

ze119869)1]10038161003816100381610038161003816

(30)

Then the excitations and nonlinear damping in (7) can beidentified in a unique way whenever Δ = 0

Proof It suffices to notice that the formulas (25) (28) and(29) are valid if and only if the corresponding denominatorsare different from zero After writing these restrictions weobtain a condition equivalent to Δ = 0 which completes theproof

119902(m

)

0 2 4 6 8 10

0

004

008

012

minus012

minus008

minus004

11990211199022

119905 (s)

(a) Displacements 1199021 and 1199022

119889119902119889119905(m

s)

0 2 4 6 8 10

0

02

04

06

minus06

minus04

minus02

1198891199021119889119905

1198891199022119889119905

119905 (s)

(b) Velocities 1199021 and 1199022

Figure 1 System responses

Remark 8 Theorem 7 tells us that the stable solution x isunique This implies that the inverse formalism developed inthe present study is a well-posed problem

Our method for the simultaneous identification consistsof the closed form solutions for the inverse formalism as inTheorems 4 and 5 However the solutions are still expressedthrough the unknown zero crossing times We thus arerequired to determine the unknown times for realizing thesolutions which is discussed in more detail in the nextsection

6 Numerical Example

In this section we will examine the workability of theproposed method to identify the excitations and nonlineardamping through numerical experiments For that let usassume that the restoring term 119877(119902) in (1) involves the cubicnonlinearity that is

119877 (119902) = 1198961119902 + 11989621199023 (31)

Journal of Applied Mathematics 5

(a) 1199021 and 1199021

0

002

004

006

008

0

01

02

03

minus008

minus006

minus004

minus002

minus02

minus01

1199022 (m)

1198891199022119889119905(m

s)

(b) 1199022 and 1199022

Figure 2 Phase diagrams for the system responses

where the values for 1198961and 119896

2are chosen as 40 and 10

respectively Then the responses for the system of (1) can beobserved as following subsection

61 System Responses With the initial conditions imposed as

119902119894(0) = 0 119902

119894(0) = 0 for 119894 = 1 2 (32)

we integrate (7) by employing Runge-Kutta schemes to seekthe system responses of (7) Here the mass is normalized asunit and the coefficients of 119888

1and 1198882are set to be 4 and 2

respectivelyThe excitation frequency120596 is taken as 4with Γ1=

1 and Γ2= 2 The calculated responses of displacement and

velocity of the system are depicted in Figure 1 and their phasediagrams in Figure 2

62 Recovering Solutions by Zero-Crossing Time As men-tioned in Section 5 even if we derived the closed formsolutions for the present identification they depend on theunknown two zero-crossing times namely (119905ze

119869)119894and (119905

zeacc)119894

in (8) and (13) respectivelyAs a first case we attempt to find the zero-crossing

times from the steady-state region By a proper numerical

(a) (119905ze119869)1 and (119905

zeacc)1

(b) (119905ze119869)2 and (119905

zeacc)2

Figure 3 Illustration of zero-crossing times in the region of steadystate

Table 1 Identified coefficients compared with the exact onessteady-state region

1198881

1198882

Γ1

Γ2

Exact 4 2 1 2Identified 39761 20195 09988 20003Error () 06 10 01 002

equation solver the results are calculated as (119905ze119869)1= 8401 sec

(119905ze119869)2= 8408 sec (119905zeacc)1 = 9182 sec and (119905

zeacc)2 = 9186 sec

each of which is indicated in Figure 3 We then substitute thezero-crossing times into (25) (28) and (29) for realizing theclosed form solutions of identification Table 1 presents thenumerical values for the solutions by the proposed methodwhich is compared with the exact ones In addition time-simulated oscillatory motions using the identified excitationsand damping by the proposed method are depicted inFigure 6 where the real (exact) time-simulated oscillatorymotions also appear for comparison

In a similar manner as a second case Figure 4 showsthe zero-crossing times selected from the transient regionwhich are substituted again into (25) (28) and (29) for theidentification The results compared with the exact ones are

6 Journal of Applied Mathematics

(a) (119905ze119869)1 and (119905

zeacc)1

(b) (119905ze119869)2 and (119905

zeacc)2

Figure 4 Illustration of zero-crossing times from the transientregion

Table 2 Identified coefficients compared with the exact onestransient region

1198881

1198882

Γ1

Γ2

Exact 4 2 1 2Identified 39120 24773 09984 20227Error () 22 2387 016 114

listed in Table 2 Finally (third case) we choose (119905ze119869)119894from

the transient region and (119905zeacc)119894 from the steady-state region

for each 119894 = 1 2 as shown in Figure 5 Their correspondingresults are listed in Table 3 For all three cases the values forthe condition numbers of Α in (18) are of 119874(10

2) meaning

that our system equation (17) is well conditioned

7 Concluding Remarks

As discussed in Section 1 of introduction many of engineershave doubted if it is possible to identify not only the externalforcing but also nonlinear damping characteristics by onlyusing system responses This is a basic idea underlying thisstudyWe have introduced the concept of zero-crossings with

(a) (119905ze119869)1 and (119905

zeacc)1

(b) (119905ze119869)2 and (119905

zeacc)2

Figure 5 Illustration of zero-crossing times (119905ze119869)119894from the transient

region and (119905zeacc)119894 from the steady-state region (119894 = 1 2)

Table 3 Identified coefficients compared with the exact ones(119905

ze119869)119894mdashthe transient region (119905zeacc)119894mdashthe steady-state region (119894 = 1 2)

1198881

1198882

Γ1

Γ2

Exact 4 2 1 2Identified 39702 20948 09988 20003Error () 075 474 01 002

regard to the system responses which becomes a crucial keyto answer this question successfully Thereby we present amathematical analysis for a systematic process of identifica-tion as well as a general method for the nonlinear simul-taneous identification Even though the numerical exampleprovided herein is a highly nonlinear system the proposedmethod yields a good approximation of the exact expressionin identifying the two nonlinear physical quantities in a stableand accurate manner

Acknowledgments

This research is supported byBasic ScienceResearchProgramthrough the National Research Foundation of Korea (NRF)

Journal of Applied Mathematics 7

(a)

ExactIdentified

0 2 4 6 8 10

0

005

01

minus01

minus005

1 3 5 7 9

119905 (s)

1199022(m

)

(b)

Figure 6 The recovered (or identified) responses compared withthe exact ones

funded by theMinistry of Education Science andTechnology(Grant no 2011-0010090) In addition this research was alsosupported by Basic Science Research Program through theNational Research Foundation of Korea (NRF) funded bythe Ministry of Education Science and Technology (Grantno K20903002030-11E0100-04610) Finally the author wouldlike to extend his special thanks to the students Miss Hye-ree Bae andMr Jinsoo Park in Naval Architecture and OceanEngineeringDepartment PusanNational University for theirhelp in the paper

References

[1] M Kulisiewicz ldquoA nonparametric method of identification ofvibration damping in non-linear dynamic systemsrdquo Interna-tional Journal of Solids and Structures vol 19 pp 601ndash609 1983

[2] D V Iourtchenko andM F Dimentberg ldquoIn-service Identifica-tion of non-linear damping from measured random vibrationrdquoJournal of Sound and Vibration vol 255 pp 549ndash554 2002

[3] T S Jang H S Choi and S L Han ldquoA new method fordetecting non-linear damping and restoring forces in non-linear oscillation systems from transient datardquo InternationalJournal of Non-Linear Mechanics vol 44 pp 801ndash808 2009

[4] T S Jang ldquoNon-parametric simultaneous identification ofboth the nonlinear damping and restoring characteristics ofnonlinear systems whose dampings depend on velocity alonerdquoMechanical Systems and Signal Processing vol 25 pp 1159ndash11732011

[5] S F Masri and T K Caughey ldquoA nonparametric identificationtechnique for nonlinear dynamic problemsrdquo ASME Journal ofApplied Mechanics vol 46 pp 433ndash447 1979

[6] J Doyle ldquoA wavelet deconvolution method for impact forceidentificationrdquo Experimental Mechanics vol 37 pp 403ndash4081997

[7] Q H Zen M Chapin and D B Bogy ldquoA force identificationmethod for sliderdisk contact forcemeasurementrdquo IEEETrans-actions on Magnetics vol 36 pp 2667ndash2670 2000

[8] T S Jang H S Baek H S Choi and S G Lee ldquoA newmethod for measuring nonharmonic periodic excitation forcesin nonlinear damped systemsrdquo Mechanical Systems and SignalProcessing vol 256 pp 2219ndash2228 2011

[9] T S Jang H S Back S L Han and T Kinoshita ldquoIndirectmeasurement of the impulsive load to a nonlinear system fromdynamic responses inverse problem formulationrdquo MechanicalSystems and Signal Processing vol 24 pp 1665ndash1681 2010

[10] T S Jang S L Han and T Kinoshita ldquoAn inversemeasurementof the sudden underwater movement of the sea-floor by usingthe time-history record of the water-wave elevationrdquo WaveMotion vol 47 no 3 pp 146ndash155 2010

[11] M Taylan ldquoThe effect of nonlinear damping and restoring inship rollingrdquo Ocean Engineering vol 27 pp 921ndash932 2000

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Journal of Applied Mathematics

improvements First based on the concept of a zero-crossingtime we develop an inverse formalism which is linear evenif the original dynamical equation of motion is nonlinearSecond we prove that the inverse formalism has a uniquesolution and its closed form solution is derived which makesthe proposed method quite simple and straightforwardThird the inverse formalism developed is shown to havestability properties of solution This means that the methodby the inverse formalism work without the use of anyregularization so that it can have a wide range of physicalapplications This contrasts the usual inverse problems ill-posed in the sense of stability [3 4 8ndash10] which require anadditional process of regularization

Finally given the model nonlinear differential equationconcerning the quadratic nonlinear damping and cubicnonlinear restoring characteristics the proposed methodrsquosworkability for simultaneously identifying both excitationsand nonlinear damping is demonstrated through a numericalexample

2 Forced Damped Nonlinear Oscillation

We consider a nonlinear damped dynamical system subjectto an external harmonic load Γ sdot cos 120596119905 of an amplitudeΓ gt 0 and frequency 120596 gt 0 The equation of motion is asecond-order nonlinear ordinary differential equation for theresponse 119902

119898 119902 + 119863 ( 119902) + 119877 (119902) = Γ sdot cos 120596119905 119905 gt 0 (1)

Here the constant 119898 gt 0 represents the mass of a particle ofan oscillator The term 119877(119902) is a general nonlinear restoringforce and the term 119863( 119902) appearing in (1) is the nonlineardamping of a system which is assumed to have a quadraticnonlinear term

119863( 119902) = 1198881119902 + 119888211990210038161003816100381610038161199021003816100381610038161003816 (2)

where 1198881and 1198882are the linear and nonlinear damping coeffi-

cients respectively The functional form (2) of nonlinearityis often encountered in the area of engineering field [11]

The sum of the inertia force 119898 119902 and restoring force 119877(119902)and the derivative of the sum will play an important role inidentifying both the excitation Γ and nonlinear damping119863( 119902)

in (1) We define the sum as

119869 (119902 119902) equiv 119898 119902 + 119877 (119902) (3)

The derivative of 119869 with respect to time then becomes

119869 =120597119869

120597119902(119902 119902) sdot 119902 +

120597119869

120597 119902(119902 119902) sdot 119902

=119889119877

119889119902(119902) sdot 119902 + 119898 119902

(4)

Noting that (1) is equivalent to

119869 = Γ sdot cos 120596119905 minus 119863 ( 119902) (5)

we can express 119869 in terms of the excitation and damping asfollows

119869 = minusΓ120596 sdot sin 120596119905 minus119889119863

119889 119902119902 (6)

by differentiating (5) with respect to timeIn what follows we assume that (1) has two dynamic

responses of 119879(= 2120587120596)-periodic limit cycle solutions 119902119894(119905)

which are forced by Γ119894for 119894 = 1 2

119898 119902119894+ 119863 ( 119902

119894) + 119877 (119902

119894) = Γ119894sdot cos 120596119905 119905 gt 0 for 119894 = 1 2

(7)

The response analysis of the nonlinear damped dynamicalsystem considered can be regarded as a forward problemwhose solution procedures are well established theoreticallyas well as numerically However an inverse problem can nowbe formulated in the form of a question given a motionresponse of the nonlinear dynamical system is it possibleto identify the excitations and nonlinear damping of thesystem at the same time The question may be interestingmathematically as well as practically In order to answer thequestion we will begin with introducing a zero-crossing timein the next section

3 Some Lemmas on Zero-Crossing Time

This section is concernedwith a zero-crossing time associatedwith motion responses of a system Thereby we derive somelemmas which are essential in the present study We startby defining a zero-crossing time in terms of 119869 for 119894 = 1 2denoted by (119905ze

119869)119894 satisfying

119869119894[(119905

ze119869)119894] = 0 for 119894 = 1 2 (8)

Here 119869119894stands for the sum (3) which corresponds to 119902

119894 that

is a function from (0infin) to R

119869119894equiv 119898 119902119894+ 119877 (119902

119894) (9)

Physically the zero-crossing time defined in (8) is a usefulconcept for finding the force balancing between the excitationand the damping in a forced damped oscillatorymotion of (7)as discussed in the following lemma

Lemma 1 Suppose that (119905ze119869)119894 119894 = 1 2 is a zero-crossing time

in terms of 119869 for the system (7) Then the following holds

Γ119894sdot cos [120596(119905ze

119869)119894] = 119863 119902

119894[(119905

ze119869)119894] 119891119900119903 119894 = 1 2 (10)

Proof A simple substitution of (9) into (7) yields

119869119894= Γ119894sdot cos 120596119905 minus 119863 ( 119902

119894) (11)

Since (11) remains valid for 0 lt 119905 lt infin it should also hold forthe zero-crossing time of (119905ze

119869)119894gt 0 However the left hand

side of (11) vanishes for the zero-crossing time (119905ze119869)119894due to

the definition (8) That is

0 = Γ119894sdot cos [120596(119905ze

119869)119894] minus 119863 119902

119894[(119905

ze119869)119894] for 119894 = 1 2 (12)

which immediately proves (10)

Journal of Applied Mathematics 3

Next we introduce a zero-crossing time in terms of accel-eration (119905zeacc)119894 for 119894 = 1 2 such that

119902119894[(119905

zeacc)119894] = 0 for 119894 = 1 2 (13)

where 119902119894denotes an acceleration response of (7) We notice

that the time rate of change of the damping is always zero ata zero-crossing time in terms of acceleration which shall beexamined in Lemma 2

Lemma 2 For the function 119869119894defined by

119869119894(119905) =

119889119869

119889119905[119902119894(119905) 119902119894(119905)] 119891119900119903 119894 = 1 2 (14)

one has119869119894[(119905

zeacc)119894] = minusΓ

119894120596 sdot sin [120596 (119905

zeacc)119894] 119891119900119903 119894 = 1 2 (15)

Here (119905zeacc)119894 refers to a zero-crossing time in terms of accelera-tion

Proof In the same way as Lemma 1 (14) should hold forthe zero-crossing times in terms of acceleration that is 119905 =

(119905zeacc)119894 for 119894 = 1 2 However (14) is rewritten as

119869119894= minusΓ119894120596 sdot sin 120596119905 minus

119889119863

119889 119902119894

119902119894 (16)

in which the second term of the right hand side becomeszero because the acceleration vanishes when 119905 = (119905

zeacc)119894 by

the definition (13) It follows that the above equation becomes(15) which completes the proof

4 Inverse Formalism

It is a high time that we develop an inverse formalism for theinverse problem raised by Section 2 based on the lemmasin the previous section We recall that the excitations andnonlinear damping are parameterized by Γ

1 Γ2 1198881 and 119888

2in

Section 2This results in that the developed inverse formalismalso involves the parameters of Γ

1 Γ2 1198881 and 119888

2as illustrated

in Lemma 3

Lemma 3 (inverse formalism) Equations (10) and (15) inLemmas 1 and 2 constitute simultaneous linear algebraicequations for the four unknowns Γ

1 Γ2 1198881 and 119888

2 These can be

written in a compact way as followsΑx = b (17)

Here Α is an operator from R4 to R4 whose matrix is

[A] equiv[[[[[[

[

cos [120596(119905ze119869)1] 0

0 cos [120596(119905ze119869)2]

minus120596 sdot sin [120596(119905zeacc)1] 0

0 minus120596 sdot sin [120596(119905zeacc)2]

minus 119902 [(119905ze119869)1] minus 119902 [(119905

ze119869)1]10038161003816100381610038161003816119902 [(119905

ze119869)1]10038161003816100381610038161003816

minus 119902 [(119905ze119869)2] minus 119902 [(119905

ze119869)2]10038161003816100381610038161003816119902 [(119905

ze119869)2]10038161003816100381610038161003816

0 0

0 0

]]]]]]

]

(18)

and x b isin R4 are represented as (one will call them a solutionvector and data vector resp in this paper)

[x] equiv[[[

[

Γ1

Γ2

1198881

1198882

]]]

]

[b] equiv[[[[

[

0

0

1198691[(119905

zeacc)1]

1198692[(119905

zeacc)2]

]]]]

]

(19)

Proof Equations (10) and (15) in Lemmas 1 and 2 can berewritten as the following

Γ1sdotcos [120596(119905ze

119869)1]minus1198881119902 [(119905

ze119869)1]minus1198882119902 [(119905

ze119869)1]10038161003816100381610038161003816119902 [(119905

ze119869)1]10038161003816100381610038161003816=0

(20)

Γ2sdotcos [120596(119905ze

119869)2]minus1198881119902 [(119905

ze119869)2]minus1198882119902 [(119905

ze119869)2]10038161003816100381610038161003816119902 [(119905

ze119869)2]10038161003816100381610038161003816=0

(21)

minusΓ1120596 sdot sin [120596 (119905

zeacc)1] =

1198691[(119905

zeacc)1] (22)

minusΓ2120596 sdot sin [120596(119905

zeacc)2] =

1198692[(119905

zeacc)2] (23)

It is straightforward to show that the above four equations areequivalent to the operator equation (17) from the definitionof the operator A the vectors x and b of (18) and (19) Thiscompletes the proof

5 Method

In the previous section we developed an inverse formalism(17) whose solution procedure is to be established in thissection The formal expression for the solution may besymbolically written as

x = Αminus1 b (24)

where the notation Αminus1 denotes the inverse operator for thelinear operatorΑ of (18) According to linear algebra it is wellknown that the inverse operator is bounded (or continuous)because it is defined on finite dimensional spaces Thus thesolution x in (24) depends continuously on the data vectorb that is the solution x is stable The construction of x isexamined through the following theorems

Theorem 4 (excitations) The excitations Γ119894for 119894 = 1 2 in (7)

are identified as an analytical form

Γ119894=

minus 119869119894[(119905

zeacc)119894]

120596 sin [120596(119905zeacc)119894](25)

provided that sin [120596 (119905zeacc)119894] = 0 for 119894 = 1 2

Proof It is easy to notice that the formula of (25) immediatelyfollows from the relations of (22) and (23) We first divideboth sides of (22) by minus120596 sdot sin[120596(119905zeacc)1] to obtain Γ

1as follows

Γ1=

minus 1198691[(119905

zeacc)1]

120596 sdot sin [120596 (119905zeacc)1] (26)

4 Journal of Applied Mathematics

Similarly we have

Γ2=

minus 1198692[(119905

zeacc)2]

120596 sdot sin [120596(119905zeacc)2](27)

from (23) This completes the proof

The formulas of the damping coefficients 1198881and 1198882in (2)

can also be obtained in an essentially simple way as to beexamined inTheorem 5

Theorem 5 (nonlinear damping) The linear damping coef-ficient 119888

1and the nonlinear damping coefficient 119888

2in (2) are

determined as

1198881=

1198692[(119905

zeacc)2] sdot cos [120596(119905

ze119869)2] sdot

10038161003816100381610038161003816119902 [(119905

ze119869)1]10038161003816100381610038161003816

120596 sin [120596(119905zeacc)2] 119902 [(119905ze119869)2]

10038161003816100381610038161003816119902 [(119905

ze119869)2]10038161003816100381610038161003816minus10038161003816100381610038161003816119902 [(119905

ze119869)1]10038161003816100381610038161003816

minus

1198691[(119905

zeacc)1] cos [120596(119905

ze119869)1] sdot

10038161003816100381610038161003816119902 [(119905

ze119869)2]10038161003816100381610038161003816

120596 sin [120596(119905zeacc)1] 119902 [(119905ze119869)1]

10038161003816100381610038161003816119902 [(119905

ze119869)2]10038161003816100381610038161003816minus10038161003816100381610038161003816119902 [(119905

ze119869)1]10038161003816100381610038161003816

(28)

1198882=

1198691[(119905

zeacc)1] cos [120596(119905

ze119869)1]

120596 sin [120596(119905zeacc)1] sdot 119902 [(119905ze119869)1]

10038161003816100381610038161003816119902 [(119905

ze119869)2]10038161003816100381610038161003816minus10038161003816100381610038161003816119902 [(119905

ze119869)1]10038161003816100381610038161003816

minus

1198692[(119905

zeacc)2] cos [120596(119905

ze119869)2]

120596 sin [120596(119905zeacc)2] 119902 [(119905ze119869)2]

10038161003816100381610038161003816119902 [(119905

ze119869)2]10038161003816100381610038161003816minus10038161003816100381610038161003816119902 [(119905

ze119869)1]10038161003816100381610038161003816

(29)

Proof It suffices to multiply (20) and (21) by the correspond-ing factors then subtract them from each other As a resultof taking (25) into account we find 119888

1of (28) and then 119888

2of

(29) which completes the proof

Remark 6 Because of the stability properties of solution xour solution procedure does not concern any kind of regu-larization which is necessary for the usual inverse problemsarising in science and engineering

As to the solvability and uniqueness condition the opera-torΑ in (18) should satisfy det A = 0 which is to be discussedin the following theorem

Theorem 7 (uniqueness) Let one denote by Δ

Δ = sin [120596(119905zeacc)1] sdot sin [120596(119905

zeacc)2] 119902 [(119905

ze119869)1] 119902 [(119905

ze119869)2]

times 10038161003816100381610038161003816119902 [(119905

ze119869)2]10038161003816100381610038161003816minus10038161003816100381610038161003816119902 [(119905

ze119869)1]10038161003816100381610038161003816

(30)

Then the excitations and nonlinear damping in (7) can beidentified in a unique way whenever Δ = 0

Proof It suffices to notice that the formulas (25) (28) and(29) are valid if and only if the corresponding denominatorsare different from zero After writing these restrictions weobtain a condition equivalent to Δ = 0 which completes theproof

119902(m

)

0 2 4 6 8 10

0

004

008

012

minus012

minus008

minus004

11990211199022

119905 (s)

(a) Displacements 1199021 and 1199022

119889119902119889119905(m

s)

0 2 4 6 8 10

0

02

04

06

minus06

minus04

minus02

1198891199021119889119905

1198891199022119889119905

119905 (s)

(b) Velocities 1199021 and 1199022

Figure 1 System responses

Remark 8 Theorem 7 tells us that the stable solution x isunique This implies that the inverse formalism developed inthe present study is a well-posed problem

Our method for the simultaneous identification consistsof the closed form solutions for the inverse formalism as inTheorems 4 and 5 However the solutions are still expressedthrough the unknown zero crossing times We thus arerequired to determine the unknown times for realizing thesolutions which is discussed in more detail in the nextsection

6 Numerical Example

In this section we will examine the workability of theproposed method to identify the excitations and nonlineardamping through numerical experiments For that let usassume that the restoring term 119877(119902) in (1) involves the cubicnonlinearity that is

119877 (119902) = 1198961119902 + 11989621199023 (31)

Journal of Applied Mathematics 5

(a) 1199021 and 1199021

0

002

004

006

008

0

01

02

03

minus008

minus006

minus004

minus002

minus02

minus01

1199022 (m)

1198891199022119889119905(m

s)

(b) 1199022 and 1199022

Figure 2 Phase diagrams for the system responses

where the values for 1198961and 119896

2are chosen as 40 and 10

respectively Then the responses for the system of (1) can beobserved as following subsection

61 System Responses With the initial conditions imposed as

119902119894(0) = 0 119902

119894(0) = 0 for 119894 = 1 2 (32)

we integrate (7) by employing Runge-Kutta schemes to seekthe system responses of (7) Here the mass is normalized asunit and the coefficients of 119888

1and 1198882are set to be 4 and 2

respectivelyThe excitation frequency120596 is taken as 4with Γ1=

1 and Γ2= 2 The calculated responses of displacement and

velocity of the system are depicted in Figure 1 and their phasediagrams in Figure 2

62 Recovering Solutions by Zero-Crossing Time As men-tioned in Section 5 even if we derived the closed formsolutions for the present identification they depend on theunknown two zero-crossing times namely (119905ze

119869)119894and (119905

zeacc)119894

in (8) and (13) respectivelyAs a first case we attempt to find the zero-crossing

times from the steady-state region By a proper numerical

(a) (119905ze119869)1 and (119905

zeacc)1

(b) (119905ze119869)2 and (119905

zeacc)2

Figure 3 Illustration of zero-crossing times in the region of steadystate

Table 1 Identified coefficients compared with the exact onessteady-state region

1198881

1198882

Γ1

Γ2

Exact 4 2 1 2Identified 39761 20195 09988 20003Error () 06 10 01 002

equation solver the results are calculated as (119905ze119869)1= 8401 sec

(119905ze119869)2= 8408 sec (119905zeacc)1 = 9182 sec and (119905

zeacc)2 = 9186 sec

each of which is indicated in Figure 3 We then substitute thezero-crossing times into (25) (28) and (29) for realizing theclosed form solutions of identification Table 1 presents thenumerical values for the solutions by the proposed methodwhich is compared with the exact ones In addition time-simulated oscillatory motions using the identified excitationsand damping by the proposed method are depicted inFigure 6 where the real (exact) time-simulated oscillatorymotions also appear for comparison

In a similar manner as a second case Figure 4 showsthe zero-crossing times selected from the transient regionwhich are substituted again into (25) (28) and (29) for theidentification The results compared with the exact ones are

6 Journal of Applied Mathematics

(a) (119905ze119869)1 and (119905

zeacc)1

(b) (119905ze119869)2 and (119905

zeacc)2

Figure 4 Illustration of zero-crossing times from the transientregion

Table 2 Identified coefficients compared with the exact onestransient region

1198881

1198882

Γ1

Γ2

Exact 4 2 1 2Identified 39120 24773 09984 20227Error () 22 2387 016 114

listed in Table 2 Finally (third case) we choose (119905ze119869)119894from

the transient region and (119905zeacc)119894 from the steady-state region

for each 119894 = 1 2 as shown in Figure 5 Their correspondingresults are listed in Table 3 For all three cases the values forthe condition numbers of Α in (18) are of 119874(10

2) meaning

that our system equation (17) is well conditioned

7 Concluding Remarks

As discussed in Section 1 of introduction many of engineershave doubted if it is possible to identify not only the externalforcing but also nonlinear damping characteristics by onlyusing system responses This is a basic idea underlying thisstudyWe have introduced the concept of zero-crossings with

(a) (119905ze119869)1 and (119905

zeacc)1

(b) (119905ze119869)2 and (119905

zeacc)2

Figure 5 Illustration of zero-crossing times (119905ze119869)119894from the transient

region and (119905zeacc)119894 from the steady-state region (119894 = 1 2)

Table 3 Identified coefficients compared with the exact ones(119905

ze119869)119894mdashthe transient region (119905zeacc)119894mdashthe steady-state region (119894 = 1 2)

1198881

1198882

Γ1

Γ2

Exact 4 2 1 2Identified 39702 20948 09988 20003Error () 075 474 01 002

regard to the system responses which becomes a crucial keyto answer this question successfully Thereby we present amathematical analysis for a systematic process of identifica-tion as well as a general method for the nonlinear simul-taneous identification Even though the numerical exampleprovided herein is a highly nonlinear system the proposedmethod yields a good approximation of the exact expressionin identifying the two nonlinear physical quantities in a stableand accurate manner

Acknowledgments

This research is supported byBasic ScienceResearchProgramthrough the National Research Foundation of Korea (NRF)

Journal of Applied Mathematics 7

(a)

ExactIdentified

0 2 4 6 8 10

0

005

01

minus01

minus005

1 3 5 7 9

119905 (s)

1199022(m

)

(b)

Figure 6 The recovered (or identified) responses compared withthe exact ones

funded by theMinistry of Education Science andTechnology(Grant no 2011-0010090) In addition this research was alsosupported by Basic Science Research Program through theNational Research Foundation of Korea (NRF) funded bythe Ministry of Education Science and Technology (Grantno K20903002030-11E0100-04610) Finally the author wouldlike to extend his special thanks to the students Miss Hye-ree Bae andMr Jinsoo Park in Naval Architecture and OceanEngineeringDepartment PusanNational University for theirhelp in the paper

References

[1] M Kulisiewicz ldquoA nonparametric method of identification ofvibration damping in non-linear dynamic systemsrdquo Interna-tional Journal of Solids and Structures vol 19 pp 601ndash609 1983

[2] D V Iourtchenko andM F Dimentberg ldquoIn-service Identifica-tion of non-linear damping from measured random vibrationrdquoJournal of Sound and Vibration vol 255 pp 549ndash554 2002

[3] T S Jang H S Choi and S L Han ldquoA new method fordetecting non-linear damping and restoring forces in non-linear oscillation systems from transient datardquo InternationalJournal of Non-Linear Mechanics vol 44 pp 801ndash808 2009

[4] T S Jang ldquoNon-parametric simultaneous identification ofboth the nonlinear damping and restoring characteristics ofnonlinear systems whose dampings depend on velocity alonerdquoMechanical Systems and Signal Processing vol 25 pp 1159ndash11732011

[5] S F Masri and T K Caughey ldquoA nonparametric identificationtechnique for nonlinear dynamic problemsrdquo ASME Journal ofApplied Mechanics vol 46 pp 433ndash447 1979

[6] J Doyle ldquoA wavelet deconvolution method for impact forceidentificationrdquo Experimental Mechanics vol 37 pp 403ndash4081997

[7] Q H Zen M Chapin and D B Bogy ldquoA force identificationmethod for sliderdisk contact forcemeasurementrdquo IEEETrans-actions on Magnetics vol 36 pp 2667ndash2670 2000

[8] T S Jang H S Baek H S Choi and S G Lee ldquoA newmethod for measuring nonharmonic periodic excitation forcesin nonlinear damped systemsrdquo Mechanical Systems and SignalProcessing vol 256 pp 2219ndash2228 2011

[9] T S Jang H S Back S L Han and T Kinoshita ldquoIndirectmeasurement of the impulsive load to a nonlinear system fromdynamic responses inverse problem formulationrdquo MechanicalSystems and Signal Processing vol 24 pp 1665ndash1681 2010

[10] T S Jang S L Han and T Kinoshita ldquoAn inversemeasurementof the sudden underwater movement of the sea-floor by usingthe time-history record of the water-wave elevationrdquo WaveMotion vol 47 no 3 pp 146ndash155 2010

[11] M Taylan ldquoThe effect of nonlinear damping and restoring inship rollingrdquo Ocean Engineering vol 27 pp 921ndash932 2000

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Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Journal of Applied Mathematics 3

Next we introduce a zero-crossing time in terms of accel-eration (119905zeacc)119894 for 119894 = 1 2 such that

119902119894[(119905

zeacc)119894] = 0 for 119894 = 1 2 (13)

where 119902119894denotes an acceleration response of (7) We notice

that the time rate of change of the damping is always zero ata zero-crossing time in terms of acceleration which shall beexamined in Lemma 2

Lemma 2 For the function 119869119894defined by

119869119894(119905) =

119889119869

119889119905[119902119894(119905) 119902119894(119905)] 119891119900119903 119894 = 1 2 (14)

one has119869119894[(119905

zeacc)119894] = minusΓ

119894120596 sdot sin [120596 (119905

zeacc)119894] 119891119900119903 119894 = 1 2 (15)

Here (119905zeacc)119894 refers to a zero-crossing time in terms of accelera-tion

Proof In the same way as Lemma 1 (14) should hold forthe zero-crossing times in terms of acceleration that is 119905 =

(119905zeacc)119894 for 119894 = 1 2 However (14) is rewritten as

119869119894= minusΓ119894120596 sdot sin 120596119905 minus

119889119863

119889 119902119894

119902119894 (16)

in which the second term of the right hand side becomeszero because the acceleration vanishes when 119905 = (119905

zeacc)119894 by

the definition (13) It follows that the above equation becomes(15) which completes the proof

4 Inverse Formalism

It is a high time that we develop an inverse formalism for theinverse problem raised by Section 2 based on the lemmasin the previous section We recall that the excitations andnonlinear damping are parameterized by Γ

1 Γ2 1198881 and 119888

2in

Section 2This results in that the developed inverse formalismalso involves the parameters of Γ

1 Γ2 1198881 and 119888

2as illustrated

in Lemma 3

Lemma 3 (inverse formalism) Equations (10) and (15) inLemmas 1 and 2 constitute simultaneous linear algebraicequations for the four unknowns Γ

1 Γ2 1198881 and 119888

2 These can be

written in a compact way as followsΑx = b (17)

Here Α is an operator from R4 to R4 whose matrix is

[A] equiv[[[[[[

[

cos [120596(119905ze119869)1] 0

0 cos [120596(119905ze119869)2]

minus120596 sdot sin [120596(119905zeacc)1] 0

0 minus120596 sdot sin [120596(119905zeacc)2]

minus 119902 [(119905ze119869)1] minus 119902 [(119905

ze119869)1]10038161003816100381610038161003816119902 [(119905

ze119869)1]10038161003816100381610038161003816

minus 119902 [(119905ze119869)2] minus 119902 [(119905

ze119869)2]10038161003816100381610038161003816119902 [(119905

ze119869)2]10038161003816100381610038161003816

0 0

0 0

]]]]]]

]

(18)

and x b isin R4 are represented as (one will call them a solutionvector and data vector resp in this paper)

[x] equiv[[[

[

Γ1

Γ2

1198881

1198882

]]]

]

[b] equiv[[[[

[

0

0

1198691[(119905

zeacc)1]

1198692[(119905

zeacc)2]

]]]]

]

(19)

Proof Equations (10) and (15) in Lemmas 1 and 2 can berewritten as the following

Γ1sdotcos [120596(119905ze

119869)1]minus1198881119902 [(119905

ze119869)1]minus1198882119902 [(119905

ze119869)1]10038161003816100381610038161003816119902 [(119905

ze119869)1]10038161003816100381610038161003816=0

(20)

Γ2sdotcos [120596(119905ze

119869)2]minus1198881119902 [(119905

ze119869)2]minus1198882119902 [(119905

ze119869)2]10038161003816100381610038161003816119902 [(119905

ze119869)2]10038161003816100381610038161003816=0

(21)

minusΓ1120596 sdot sin [120596 (119905

zeacc)1] =

1198691[(119905

zeacc)1] (22)

minusΓ2120596 sdot sin [120596(119905

zeacc)2] =

1198692[(119905

zeacc)2] (23)

It is straightforward to show that the above four equations areequivalent to the operator equation (17) from the definitionof the operator A the vectors x and b of (18) and (19) Thiscompletes the proof

5 Method

In the previous section we developed an inverse formalism(17) whose solution procedure is to be established in thissection The formal expression for the solution may besymbolically written as

x = Αminus1 b (24)

where the notation Αminus1 denotes the inverse operator for thelinear operatorΑ of (18) According to linear algebra it is wellknown that the inverse operator is bounded (or continuous)because it is defined on finite dimensional spaces Thus thesolution x in (24) depends continuously on the data vectorb that is the solution x is stable The construction of x isexamined through the following theorems

Theorem 4 (excitations) The excitations Γ119894for 119894 = 1 2 in (7)

are identified as an analytical form

Γ119894=

minus 119869119894[(119905

zeacc)119894]

120596 sin [120596(119905zeacc)119894](25)

provided that sin [120596 (119905zeacc)119894] = 0 for 119894 = 1 2

Proof It is easy to notice that the formula of (25) immediatelyfollows from the relations of (22) and (23) We first divideboth sides of (22) by minus120596 sdot sin[120596(119905zeacc)1] to obtain Γ

1as follows

Γ1=

minus 1198691[(119905

zeacc)1]

120596 sdot sin [120596 (119905zeacc)1] (26)

4 Journal of Applied Mathematics

Similarly we have

Γ2=

minus 1198692[(119905

zeacc)2]

120596 sdot sin [120596(119905zeacc)2](27)

from (23) This completes the proof

The formulas of the damping coefficients 1198881and 1198882in (2)

can also be obtained in an essentially simple way as to beexamined inTheorem 5

Theorem 5 (nonlinear damping) The linear damping coef-ficient 119888

1and the nonlinear damping coefficient 119888

2in (2) are

determined as

1198881=

1198692[(119905

zeacc)2] sdot cos [120596(119905

ze119869)2] sdot

10038161003816100381610038161003816119902 [(119905

ze119869)1]10038161003816100381610038161003816

120596 sin [120596(119905zeacc)2] 119902 [(119905ze119869)2]

10038161003816100381610038161003816119902 [(119905

ze119869)2]10038161003816100381610038161003816minus10038161003816100381610038161003816119902 [(119905

ze119869)1]10038161003816100381610038161003816

minus

1198691[(119905

zeacc)1] cos [120596(119905

ze119869)1] sdot

10038161003816100381610038161003816119902 [(119905

ze119869)2]10038161003816100381610038161003816

120596 sin [120596(119905zeacc)1] 119902 [(119905ze119869)1]

10038161003816100381610038161003816119902 [(119905

ze119869)2]10038161003816100381610038161003816minus10038161003816100381610038161003816119902 [(119905

ze119869)1]10038161003816100381610038161003816

(28)

1198882=

1198691[(119905

zeacc)1] cos [120596(119905

ze119869)1]

120596 sin [120596(119905zeacc)1] sdot 119902 [(119905ze119869)1]

10038161003816100381610038161003816119902 [(119905

ze119869)2]10038161003816100381610038161003816minus10038161003816100381610038161003816119902 [(119905

ze119869)1]10038161003816100381610038161003816

minus

1198692[(119905

zeacc)2] cos [120596(119905

ze119869)2]

120596 sin [120596(119905zeacc)2] 119902 [(119905ze119869)2]

10038161003816100381610038161003816119902 [(119905

ze119869)2]10038161003816100381610038161003816minus10038161003816100381610038161003816119902 [(119905

ze119869)1]10038161003816100381610038161003816

(29)

Proof It suffices to multiply (20) and (21) by the correspond-ing factors then subtract them from each other As a resultof taking (25) into account we find 119888

1of (28) and then 119888

2of

(29) which completes the proof

Remark 6 Because of the stability properties of solution xour solution procedure does not concern any kind of regu-larization which is necessary for the usual inverse problemsarising in science and engineering

As to the solvability and uniqueness condition the opera-torΑ in (18) should satisfy det A = 0 which is to be discussedin the following theorem

Theorem 7 (uniqueness) Let one denote by Δ

Δ = sin [120596(119905zeacc)1] sdot sin [120596(119905

zeacc)2] 119902 [(119905

ze119869)1] 119902 [(119905

ze119869)2]

times 10038161003816100381610038161003816119902 [(119905

ze119869)2]10038161003816100381610038161003816minus10038161003816100381610038161003816119902 [(119905

ze119869)1]10038161003816100381610038161003816

(30)

Then the excitations and nonlinear damping in (7) can beidentified in a unique way whenever Δ = 0

Proof It suffices to notice that the formulas (25) (28) and(29) are valid if and only if the corresponding denominatorsare different from zero After writing these restrictions weobtain a condition equivalent to Δ = 0 which completes theproof

119902(m

)

0 2 4 6 8 10

0

004

008

012

minus012

minus008

minus004

11990211199022

119905 (s)

(a) Displacements 1199021 and 1199022

119889119902119889119905(m

s)

0 2 4 6 8 10

0

02

04

06

minus06

minus04

minus02

1198891199021119889119905

1198891199022119889119905

119905 (s)

(b) Velocities 1199021 and 1199022

Figure 1 System responses

Remark 8 Theorem 7 tells us that the stable solution x isunique This implies that the inverse formalism developed inthe present study is a well-posed problem

Our method for the simultaneous identification consistsof the closed form solutions for the inverse formalism as inTheorems 4 and 5 However the solutions are still expressedthrough the unknown zero crossing times We thus arerequired to determine the unknown times for realizing thesolutions which is discussed in more detail in the nextsection

6 Numerical Example

In this section we will examine the workability of theproposed method to identify the excitations and nonlineardamping through numerical experiments For that let usassume that the restoring term 119877(119902) in (1) involves the cubicnonlinearity that is

119877 (119902) = 1198961119902 + 11989621199023 (31)

Journal of Applied Mathematics 5

(a) 1199021 and 1199021

0

002

004

006

008

0

01

02

03

minus008

minus006

minus004

minus002

minus02

minus01

1199022 (m)

1198891199022119889119905(m

s)

(b) 1199022 and 1199022

Figure 2 Phase diagrams for the system responses

where the values for 1198961and 119896

2are chosen as 40 and 10

respectively Then the responses for the system of (1) can beobserved as following subsection

61 System Responses With the initial conditions imposed as

119902119894(0) = 0 119902

119894(0) = 0 for 119894 = 1 2 (32)

we integrate (7) by employing Runge-Kutta schemes to seekthe system responses of (7) Here the mass is normalized asunit and the coefficients of 119888

1and 1198882are set to be 4 and 2

respectivelyThe excitation frequency120596 is taken as 4with Γ1=

1 and Γ2= 2 The calculated responses of displacement and

velocity of the system are depicted in Figure 1 and their phasediagrams in Figure 2

62 Recovering Solutions by Zero-Crossing Time As men-tioned in Section 5 even if we derived the closed formsolutions for the present identification they depend on theunknown two zero-crossing times namely (119905ze

119869)119894and (119905

zeacc)119894

in (8) and (13) respectivelyAs a first case we attempt to find the zero-crossing

times from the steady-state region By a proper numerical

(a) (119905ze119869)1 and (119905

zeacc)1

(b) (119905ze119869)2 and (119905

zeacc)2

Figure 3 Illustration of zero-crossing times in the region of steadystate

Table 1 Identified coefficients compared with the exact onessteady-state region

1198881

1198882

Γ1

Γ2

Exact 4 2 1 2Identified 39761 20195 09988 20003Error () 06 10 01 002

equation solver the results are calculated as (119905ze119869)1= 8401 sec

(119905ze119869)2= 8408 sec (119905zeacc)1 = 9182 sec and (119905

zeacc)2 = 9186 sec

each of which is indicated in Figure 3 We then substitute thezero-crossing times into (25) (28) and (29) for realizing theclosed form solutions of identification Table 1 presents thenumerical values for the solutions by the proposed methodwhich is compared with the exact ones In addition time-simulated oscillatory motions using the identified excitationsand damping by the proposed method are depicted inFigure 6 where the real (exact) time-simulated oscillatorymotions also appear for comparison

In a similar manner as a second case Figure 4 showsthe zero-crossing times selected from the transient regionwhich are substituted again into (25) (28) and (29) for theidentification The results compared with the exact ones are

6 Journal of Applied Mathematics

(a) (119905ze119869)1 and (119905

zeacc)1

(b) (119905ze119869)2 and (119905

zeacc)2

Figure 4 Illustration of zero-crossing times from the transientregion

Table 2 Identified coefficients compared with the exact onestransient region

1198881

1198882

Γ1

Γ2

Exact 4 2 1 2Identified 39120 24773 09984 20227Error () 22 2387 016 114

listed in Table 2 Finally (third case) we choose (119905ze119869)119894from

the transient region and (119905zeacc)119894 from the steady-state region

for each 119894 = 1 2 as shown in Figure 5 Their correspondingresults are listed in Table 3 For all three cases the values forthe condition numbers of Α in (18) are of 119874(10

2) meaning

that our system equation (17) is well conditioned

7 Concluding Remarks

As discussed in Section 1 of introduction many of engineershave doubted if it is possible to identify not only the externalforcing but also nonlinear damping characteristics by onlyusing system responses This is a basic idea underlying thisstudyWe have introduced the concept of zero-crossings with

(a) (119905ze119869)1 and (119905

zeacc)1

(b) (119905ze119869)2 and (119905

zeacc)2

Figure 5 Illustration of zero-crossing times (119905ze119869)119894from the transient

region and (119905zeacc)119894 from the steady-state region (119894 = 1 2)

Table 3 Identified coefficients compared with the exact ones(119905

ze119869)119894mdashthe transient region (119905zeacc)119894mdashthe steady-state region (119894 = 1 2)

1198881

1198882

Γ1

Γ2

Exact 4 2 1 2Identified 39702 20948 09988 20003Error () 075 474 01 002

regard to the system responses which becomes a crucial keyto answer this question successfully Thereby we present amathematical analysis for a systematic process of identifica-tion as well as a general method for the nonlinear simul-taneous identification Even though the numerical exampleprovided herein is a highly nonlinear system the proposedmethod yields a good approximation of the exact expressionin identifying the two nonlinear physical quantities in a stableand accurate manner

Acknowledgments

This research is supported byBasic ScienceResearchProgramthrough the National Research Foundation of Korea (NRF)

Journal of Applied Mathematics 7

(a)

ExactIdentified

0 2 4 6 8 10

0

005

01

minus01

minus005

1 3 5 7 9

119905 (s)

1199022(m

)

(b)

Figure 6 The recovered (or identified) responses compared withthe exact ones

funded by theMinistry of Education Science andTechnology(Grant no 2011-0010090) In addition this research was alsosupported by Basic Science Research Program through theNational Research Foundation of Korea (NRF) funded bythe Ministry of Education Science and Technology (Grantno K20903002030-11E0100-04610) Finally the author wouldlike to extend his special thanks to the students Miss Hye-ree Bae andMr Jinsoo Park in Naval Architecture and OceanEngineeringDepartment PusanNational University for theirhelp in the paper

References

[1] M Kulisiewicz ldquoA nonparametric method of identification ofvibration damping in non-linear dynamic systemsrdquo Interna-tional Journal of Solids and Structures vol 19 pp 601ndash609 1983

[2] D V Iourtchenko andM F Dimentberg ldquoIn-service Identifica-tion of non-linear damping from measured random vibrationrdquoJournal of Sound and Vibration vol 255 pp 549ndash554 2002

[3] T S Jang H S Choi and S L Han ldquoA new method fordetecting non-linear damping and restoring forces in non-linear oscillation systems from transient datardquo InternationalJournal of Non-Linear Mechanics vol 44 pp 801ndash808 2009

[4] T S Jang ldquoNon-parametric simultaneous identification ofboth the nonlinear damping and restoring characteristics ofnonlinear systems whose dampings depend on velocity alonerdquoMechanical Systems and Signal Processing vol 25 pp 1159ndash11732011

[5] S F Masri and T K Caughey ldquoA nonparametric identificationtechnique for nonlinear dynamic problemsrdquo ASME Journal ofApplied Mechanics vol 46 pp 433ndash447 1979

[6] J Doyle ldquoA wavelet deconvolution method for impact forceidentificationrdquo Experimental Mechanics vol 37 pp 403ndash4081997

[7] Q H Zen M Chapin and D B Bogy ldquoA force identificationmethod for sliderdisk contact forcemeasurementrdquo IEEETrans-actions on Magnetics vol 36 pp 2667ndash2670 2000

[8] T S Jang H S Baek H S Choi and S G Lee ldquoA newmethod for measuring nonharmonic periodic excitation forcesin nonlinear damped systemsrdquo Mechanical Systems and SignalProcessing vol 256 pp 2219ndash2228 2011

[9] T S Jang H S Back S L Han and T Kinoshita ldquoIndirectmeasurement of the impulsive load to a nonlinear system fromdynamic responses inverse problem formulationrdquo MechanicalSystems and Signal Processing vol 24 pp 1665ndash1681 2010

[10] T S Jang S L Han and T Kinoshita ldquoAn inversemeasurementof the sudden underwater movement of the sea-floor by usingthe time-history record of the water-wave elevationrdquo WaveMotion vol 47 no 3 pp 146ndash155 2010

[11] M Taylan ldquoThe effect of nonlinear damping and restoring inship rollingrdquo Ocean Engineering vol 27 pp 921ndash932 2000

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MathematicsJournal of

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical PhysicsAdvances in

Complex AnalysisJournal of

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Journal of Applied Mathematics

Similarly we have

Γ2=

minus 1198692[(119905

zeacc)2]

120596 sdot sin [120596(119905zeacc)2](27)

from (23) This completes the proof

The formulas of the damping coefficients 1198881and 1198882in (2)

can also be obtained in an essentially simple way as to beexamined inTheorem 5

Theorem 5 (nonlinear damping) The linear damping coef-ficient 119888

1and the nonlinear damping coefficient 119888

2in (2) are

determined as

1198881=

1198692[(119905

zeacc)2] sdot cos [120596(119905

ze119869)2] sdot

10038161003816100381610038161003816119902 [(119905

ze119869)1]10038161003816100381610038161003816

120596 sin [120596(119905zeacc)2] 119902 [(119905ze119869)2]

10038161003816100381610038161003816119902 [(119905

ze119869)2]10038161003816100381610038161003816minus10038161003816100381610038161003816119902 [(119905

ze119869)1]10038161003816100381610038161003816

minus

1198691[(119905

zeacc)1] cos [120596(119905

ze119869)1] sdot

10038161003816100381610038161003816119902 [(119905

ze119869)2]10038161003816100381610038161003816

120596 sin [120596(119905zeacc)1] 119902 [(119905ze119869)1]

10038161003816100381610038161003816119902 [(119905

ze119869)2]10038161003816100381610038161003816minus10038161003816100381610038161003816119902 [(119905

ze119869)1]10038161003816100381610038161003816

(28)

1198882=

1198691[(119905

zeacc)1] cos [120596(119905

ze119869)1]

120596 sin [120596(119905zeacc)1] sdot 119902 [(119905ze119869)1]

10038161003816100381610038161003816119902 [(119905

ze119869)2]10038161003816100381610038161003816minus10038161003816100381610038161003816119902 [(119905

ze119869)1]10038161003816100381610038161003816

minus

1198692[(119905

zeacc)2] cos [120596(119905

ze119869)2]

120596 sin [120596(119905zeacc)2] 119902 [(119905ze119869)2]

10038161003816100381610038161003816119902 [(119905

ze119869)2]10038161003816100381610038161003816minus10038161003816100381610038161003816119902 [(119905

ze119869)1]10038161003816100381610038161003816

(29)

Proof It suffices to multiply (20) and (21) by the correspond-ing factors then subtract them from each other As a resultof taking (25) into account we find 119888

1of (28) and then 119888

2of

(29) which completes the proof

Remark 6 Because of the stability properties of solution xour solution procedure does not concern any kind of regu-larization which is necessary for the usual inverse problemsarising in science and engineering

As to the solvability and uniqueness condition the opera-torΑ in (18) should satisfy det A = 0 which is to be discussedin the following theorem

Theorem 7 (uniqueness) Let one denote by Δ

Δ = sin [120596(119905zeacc)1] sdot sin [120596(119905

zeacc)2] 119902 [(119905

ze119869)1] 119902 [(119905

ze119869)2]

times 10038161003816100381610038161003816119902 [(119905

ze119869)2]10038161003816100381610038161003816minus10038161003816100381610038161003816119902 [(119905

ze119869)1]10038161003816100381610038161003816

(30)

Then the excitations and nonlinear damping in (7) can beidentified in a unique way whenever Δ = 0

Proof It suffices to notice that the formulas (25) (28) and(29) are valid if and only if the corresponding denominatorsare different from zero After writing these restrictions weobtain a condition equivalent to Δ = 0 which completes theproof

119902(m

)

0 2 4 6 8 10

0

004

008

012

minus012

minus008

minus004

11990211199022

119905 (s)

(a) Displacements 1199021 and 1199022

119889119902119889119905(m

s)

0 2 4 6 8 10

0

02

04

06

minus06

minus04

minus02

1198891199021119889119905

1198891199022119889119905

119905 (s)

(b) Velocities 1199021 and 1199022

Figure 1 System responses

Remark 8 Theorem 7 tells us that the stable solution x isunique This implies that the inverse formalism developed inthe present study is a well-posed problem

Our method for the simultaneous identification consistsof the closed form solutions for the inverse formalism as inTheorems 4 and 5 However the solutions are still expressedthrough the unknown zero crossing times We thus arerequired to determine the unknown times for realizing thesolutions which is discussed in more detail in the nextsection

6 Numerical Example

In this section we will examine the workability of theproposed method to identify the excitations and nonlineardamping through numerical experiments For that let usassume that the restoring term 119877(119902) in (1) involves the cubicnonlinearity that is

119877 (119902) = 1198961119902 + 11989621199023 (31)

Journal of Applied Mathematics 5

(a) 1199021 and 1199021

0

002

004

006

008

0

01

02

03

minus008

minus006

minus004

minus002

minus02

minus01

1199022 (m)

1198891199022119889119905(m

s)

(b) 1199022 and 1199022

Figure 2 Phase diagrams for the system responses

where the values for 1198961and 119896

2are chosen as 40 and 10

respectively Then the responses for the system of (1) can beobserved as following subsection

61 System Responses With the initial conditions imposed as

119902119894(0) = 0 119902

119894(0) = 0 for 119894 = 1 2 (32)

we integrate (7) by employing Runge-Kutta schemes to seekthe system responses of (7) Here the mass is normalized asunit and the coefficients of 119888

1and 1198882are set to be 4 and 2

respectivelyThe excitation frequency120596 is taken as 4with Γ1=

1 and Γ2= 2 The calculated responses of displacement and

velocity of the system are depicted in Figure 1 and their phasediagrams in Figure 2

62 Recovering Solutions by Zero-Crossing Time As men-tioned in Section 5 even if we derived the closed formsolutions for the present identification they depend on theunknown two zero-crossing times namely (119905ze

119869)119894and (119905

zeacc)119894

in (8) and (13) respectivelyAs a first case we attempt to find the zero-crossing

times from the steady-state region By a proper numerical

(a) (119905ze119869)1 and (119905

zeacc)1

(b) (119905ze119869)2 and (119905

zeacc)2

Figure 3 Illustration of zero-crossing times in the region of steadystate

Table 1 Identified coefficients compared with the exact onessteady-state region

1198881

1198882

Γ1

Γ2

Exact 4 2 1 2Identified 39761 20195 09988 20003Error () 06 10 01 002

equation solver the results are calculated as (119905ze119869)1= 8401 sec

(119905ze119869)2= 8408 sec (119905zeacc)1 = 9182 sec and (119905

zeacc)2 = 9186 sec

each of which is indicated in Figure 3 We then substitute thezero-crossing times into (25) (28) and (29) for realizing theclosed form solutions of identification Table 1 presents thenumerical values for the solutions by the proposed methodwhich is compared with the exact ones In addition time-simulated oscillatory motions using the identified excitationsand damping by the proposed method are depicted inFigure 6 where the real (exact) time-simulated oscillatorymotions also appear for comparison

In a similar manner as a second case Figure 4 showsthe zero-crossing times selected from the transient regionwhich are substituted again into (25) (28) and (29) for theidentification The results compared with the exact ones are

6 Journal of Applied Mathematics

(a) (119905ze119869)1 and (119905

zeacc)1

(b) (119905ze119869)2 and (119905

zeacc)2

Figure 4 Illustration of zero-crossing times from the transientregion

Table 2 Identified coefficients compared with the exact onestransient region

1198881

1198882

Γ1

Γ2

Exact 4 2 1 2Identified 39120 24773 09984 20227Error () 22 2387 016 114

listed in Table 2 Finally (third case) we choose (119905ze119869)119894from

the transient region and (119905zeacc)119894 from the steady-state region

for each 119894 = 1 2 as shown in Figure 5 Their correspondingresults are listed in Table 3 For all three cases the values forthe condition numbers of Α in (18) are of 119874(10

2) meaning

that our system equation (17) is well conditioned

7 Concluding Remarks

As discussed in Section 1 of introduction many of engineershave doubted if it is possible to identify not only the externalforcing but also nonlinear damping characteristics by onlyusing system responses This is a basic idea underlying thisstudyWe have introduced the concept of zero-crossings with

(a) (119905ze119869)1 and (119905

zeacc)1

(b) (119905ze119869)2 and (119905

zeacc)2

Figure 5 Illustration of zero-crossing times (119905ze119869)119894from the transient

region and (119905zeacc)119894 from the steady-state region (119894 = 1 2)

Table 3 Identified coefficients compared with the exact ones(119905

ze119869)119894mdashthe transient region (119905zeacc)119894mdashthe steady-state region (119894 = 1 2)

1198881

1198882

Γ1

Γ2

Exact 4 2 1 2Identified 39702 20948 09988 20003Error () 075 474 01 002

regard to the system responses which becomes a crucial keyto answer this question successfully Thereby we present amathematical analysis for a systematic process of identifica-tion as well as a general method for the nonlinear simul-taneous identification Even though the numerical exampleprovided herein is a highly nonlinear system the proposedmethod yields a good approximation of the exact expressionin identifying the two nonlinear physical quantities in a stableand accurate manner

Acknowledgments

This research is supported byBasic ScienceResearchProgramthrough the National Research Foundation of Korea (NRF)

Journal of Applied Mathematics 7

(a)

ExactIdentified

0 2 4 6 8 10

0

005

01

minus01

minus005

1 3 5 7 9

119905 (s)

1199022(m

)

(b)

Figure 6 The recovered (or identified) responses compared withthe exact ones

funded by theMinistry of Education Science andTechnology(Grant no 2011-0010090) In addition this research was alsosupported by Basic Science Research Program through theNational Research Foundation of Korea (NRF) funded bythe Ministry of Education Science and Technology (Grantno K20903002030-11E0100-04610) Finally the author wouldlike to extend his special thanks to the students Miss Hye-ree Bae andMr Jinsoo Park in Naval Architecture and OceanEngineeringDepartment PusanNational University for theirhelp in the paper

References

[1] M Kulisiewicz ldquoA nonparametric method of identification ofvibration damping in non-linear dynamic systemsrdquo Interna-tional Journal of Solids and Structures vol 19 pp 601ndash609 1983

[2] D V Iourtchenko andM F Dimentberg ldquoIn-service Identifica-tion of non-linear damping from measured random vibrationrdquoJournal of Sound and Vibration vol 255 pp 549ndash554 2002

[3] T S Jang H S Choi and S L Han ldquoA new method fordetecting non-linear damping and restoring forces in non-linear oscillation systems from transient datardquo InternationalJournal of Non-Linear Mechanics vol 44 pp 801ndash808 2009

[4] T S Jang ldquoNon-parametric simultaneous identification ofboth the nonlinear damping and restoring characteristics ofnonlinear systems whose dampings depend on velocity alonerdquoMechanical Systems and Signal Processing vol 25 pp 1159ndash11732011

[5] S F Masri and T K Caughey ldquoA nonparametric identificationtechnique for nonlinear dynamic problemsrdquo ASME Journal ofApplied Mechanics vol 46 pp 433ndash447 1979

[6] J Doyle ldquoA wavelet deconvolution method for impact forceidentificationrdquo Experimental Mechanics vol 37 pp 403ndash4081997

[7] Q H Zen M Chapin and D B Bogy ldquoA force identificationmethod for sliderdisk contact forcemeasurementrdquo IEEETrans-actions on Magnetics vol 36 pp 2667ndash2670 2000

[8] T S Jang H S Baek H S Choi and S G Lee ldquoA newmethod for measuring nonharmonic periodic excitation forcesin nonlinear damped systemsrdquo Mechanical Systems and SignalProcessing vol 256 pp 2219ndash2228 2011

[9] T S Jang H S Back S L Han and T Kinoshita ldquoIndirectmeasurement of the impulsive load to a nonlinear system fromdynamic responses inverse problem formulationrdquo MechanicalSystems and Signal Processing vol 24 pp 1665ndash1681 2010

[10] T S Jang S L Han and T Kinoshita ldquoAn inversemeasurementof the sudden underwater movement of the sea-floor by usingthe time-history record of the water-wave elevationrdquo WaveMotion vol 47 no 3 pp 146ndash155 2010

[11] M Taylan ldquoThe effect of nonlinear damping and restoring inship rollingrdquo Ocean Engineering vol 27 pp 921ndash932 2000

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Journal of Applied Mathematics 5

(a) 1199021 and 1199021

0

002

004

006

008

0

01

02

03

minus008

minus006

minus004

minus002

minus02

minus01

1199022 (m)

1198891199022119889119905(m

s)

(b) 1199022 and 1199022

Figure 2 Phase diagrams for the system responses

where the values for 1198961and 119896

2are chosen as 40 and 10

respectively Then the responses for the system of (1) can beobserved as following subsection

61 System Responses With the initial conditions imposed as

119902119894(0) = 0 119902

119894(0) = 0 for 119894 = 1 2 (32)

we integrate (7) by employing Runge-Kutta schemes to seekthe system responses of (7) Here the mass is normalized asunit and the coefficients of 119888

1and 1198882are set to be 4 and 2

respectivelyThe excitation frequency120596 is taken as 4with Γ1=

1 and Γ2= 2 The calculated responses of displacement and

velocity of the system are depicted in Figure 1 and their phasediagrams in Figure 2

62 Recovering Solutions by Zero-Crossing Time As men-tioned in Section 5 even if we derived the closed formsolutions for the present identification they depend on theunknown two zero-crossing times namely (119905ze

119869)119894and (119905

zeacc)119894

in (8) and (13) respectivelyAs a first case we attempt to find the zero-crossing

times from the steady-state region By a proper numerical

(a) (119905ze119869)1 and (119905

zeacc)1

(b) (119905ze119869)2 and (119905

zeacc)2

Figure 3 Illustration of zero-crossing times in the region of steadystate

Table 1 Identified coefficients compared with the exact onessteady-state region

1198881

1198882

Γ1

Γ2

Exact 4 2 1 2Identified 39761 20195 09988 20003Error () 06 10 01 002

equation solver the results are calculated as (119905ze119869)1= 8401 sec

(119905ze119869)2= 8408 sec (119905zeacc)1 = 9182 sec and (119905

zeacc)2 = 9186 sec

each of which is indicated in Figure 3 We then substitute thezero-crossing times into (25) (28) and (29) for realizing theclosed form solutions of identification Table 1 presents thenumerical values for the solutions by the proposed methodwhich is compared with the exact ones In addition time-simulated oscillatory motions using the identified excitationsand damping by the proposed method are depicted inFigure 6 where the real (exact) time-simulated oscillatorymotions also appear for comparison

In a similar manner as a second case Figure 4 showsthe zero-crossing times selected from the transient regionwhich are substituted again into (25) (28) and (29) for theidentification The results compared with the exact ones are

6 Journal of Applied Mathematics

(a) (119905ze119869)1 and (119905

zeacc)1

(b) (119905ze119869)2 and (119905

zeacc)2

Figure 4 Illustration of zero-crossing times from the transientregion

Table 2 Identified coefficients compared with the exact onestransient region

1198881

1198882

Γ1

Γ2

Exact 4 2 1 2Identified 39120 24773 09984 20227Error () 22 2387 016 114

listed in Table 2 Finally (third case) we choose (119905ze119869)119894from

the transient region and (119905zeacc)119894 from the steady-state region

for each 119894 = 1 2 as shown in Figure 5 Their correspondingresults are listed in Table 3 For all three cases the values forthe condition numbers of Α in (18) are of 119874(10

2) meaning

that our system equation (17) is well conditioned

7 Concluding Remarks

As discussed in Section 1 of introduction many of engineershave doubted if it is possible to identify not only the externalforcing but also nonlinear damping characteristics by onlyusing system responses This is a basic idea underlying thisstudyWe have introduced the concept of zero-crossings with

(a) (119905ze119869)1 and (119905

zeacc)1

(b) (119905ze119869)2 and (119905

zeacc)2

Figure 5 Illustration of zero-crossing times (119905ze119869)119894from the transient

region and (119905zeacc)119894 from the steady-state region (119894 = 1 2)

Table 3 Identified coefficients compared with the exact ones(119905

ze119869)119894mdashthe transient region (119905zeacc)119894mdashthe steady-state region (119894 = 1 2)

1198881

1198882

Γ1

Γ2

Exact 4 2 1 2Identified 39702 20948 09988 20003Error () 075 474 01 002

regard to the system responses which becomes a crucial keyto answer this question successfully Thereby we present amathematical analysis for a systematic process of identifica-tion as well as a general method for the nonlinear simul-taneous identification Even though the numerical exampleprovided herein is a highly nonlinear system the proposedmethod yields a good approximation of the exact expressionin identifying the two nonlinear physical quantities in a stableand accurate manner

Acknowledgments

This research is supported byBasic ScienceResearchProgramthrough the National Research Foundation of Korea (NRF)

Journal of Applied Mathematics 7

(a)

ExactIdentified

0 2 4 6 8 10

0

005

01

minus01

minus005

1 3 5 7 9

119905 (s)

1199022(m

)

(b)

Figure 6 The recovered (or identified) responses compared withthe exact ones

funded by theMinistry of Education Science andTechnology(Grant no 2011-0010090) In addition this research was alsosupported by Basic Science Research Program through theNational Research Foundation of Korea (NRF) funded bythe Ministry of Education Science and Technology (Grantno K20903002030-11E0100-04610) Finally the author wouldlike to extend his special thanks to the students Miss Hye-ree Bae andMr Jinsoo Park in Naval Architecture and OceanEngineeringDepartment PusanNational University for theirhelp in the paper

References

[1] M Kulisiewicz ldquoA nonparametric method of identification ofvibration damping in non-linear dynamic systemsrdquo Interna-tional Journal of Solids and Structures vol 19 pp 601ndash609 1983

[2] D V Iourtchenko andM F Dimentberg ldquoIn-service Identifica-tion of non-linear damping from measured random vibrationrdquoJournal of Sound and Vibration vol 255 pp 549ndash554 2002

[3] T S Jang H S Choi and S L Han ldquoA new method fordetecting non-linear damping and restoring forces in non-linear oscillation systems from transient datardquo InternationalJournal of Non-Linear Mechanics vol 44 pp 801ndash808 2009

[4] T S Jang ldquoNon-parametric simultaneous identification ofboth the nonlinear damping and restoring characteristics ofnonlinear systems whose dampings depend on velocity alonerdquoMechanical Systems and Signal Processing vol 25 pp 1159ndash11732011

[5] S F Masri and T K Caughey ldquoA nonparametric identificationtechnique for nonlinear dynamic problemsrdquo ASME Journal ofApplied Mechanics vol 46 pp 433ndash447 1979

[6] J Doyle ldquoA wavelet deconvolution method for impact forceidentificationrdquo Experimental Mechanics vol 37 pp 403ndash4081997

[7] Q H Zen M Chapin and D B Bogy ldquoA force identificationmethod for sliderdisk contact forcemeasurementrdquo IEEETrans-actions on Magnetics vol 36 pp 2667ndash2670 2000

[8] T S Jang H S Baek H S Choi and S G Lee ldquoA newmethod for measuring nonharmonic periodic excitation forcesin nonlinear damped systemsrdquo Mechanical Systems and SignalProcessing vol 256 pp 2219ndash2228 2011

[9] T S Jang H S Back S L Han and T Kinoshita ldquoIndirectmeasurement of the impulsive load to a nonlinear system fromdynamic responses inverse problem formulationrdquo MechanicalSystems and Signal Processing vol 24 pp 1665ndash1681 2010

[10] T S Jang S L Han and T Kinoshita ldquoAn inversemeasurementof the sudden underwater movement of the sea-floor by usingthe time-history record of the water-wave elevationrdquo WaveMotion vol 47 no 3 pp 146ndash155 2010

[11] M Taylan ldquoThe effect of nonlinear damping and restoring inship rollingrdquo Ocean Engineering vol 27 pp 921ndash932 2000

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Journal of Applied Mathematics

(a) (119905ze119869)1 and (119905

zeacc)1

(b) (119905ze119869)2 and (119905

zeacc)2

Figure 4 Illustration of zero-crossing times from the transientregion

Table 2 Identified coefficients compared with the exact onestransient region

1198881

1198882

Γ1

Γ2

Exact 4 2 1 2Identified 39120 24773 09984 20227Error () 22 2387 016 114

listed in Table 2 Finally (third case) we choose (119905ze119869)119894from

the transient region and (119905zeacc)119894 from the steady-state region

for each 119894 = 1 2 as shown in Figure 5 Their correspondingresults are listed in Table 3 For all three cases the values forthe condition numbers of Α in (18) are of 119874(10

2) meaning

that our system equation (17) is well conditioned

7 Concluding Remarks

As discussed in Section 1 of introduction many of engineershave doubted if it is possible to identify not only the externalforcing but also nonlinear damping characteristics by onlyusing system responses This is a basic idea underlying thisstudyWe have introduced the concept of zero-crossings with

(a) (119905ze119869)1 and (119905

zeacc)1

(b) (119905ze119869)2 and (119905

zeacc)2

Figure 5 Illustration of zero-crossing times (119905ze119869)119894from the transient

region and (119905zeacc)119894 from the steady-state region (119894 = 1 2)

Table 3 Identified coefficients compared with the exact ones(119905

ze119869)119894mdashthe transient region (119905zeacc)119894mdashthe steady-state region (119894 = 1 2)

1198881

1198882

Γ1

Γ2

Exact 4 2 1 2Identified 39702 20948 09988 20003Error () 075 474 01 002

regard to the system responses which becomes a crucial keyto answer this question successfully Thereby we present amathematical analysis for a systematic process of identifica-tion as well as a general method for the nonlinear simul-taneous identification Even though the numerical exampleprovided herein is a highly nonlinear system the proposedmethod yields a good approximation of the exact expressionin identifying the two nonlinear physical quantities in a stableand accurate manner

Acknowledgments

This research is supported byBasic ScienceResearchProgramthrough the National Research Foundation of Korea (NRF)

Journal of Applied Mathematics 7

(a)

ExactIdentified

0 2 4 6 8 10

0

005

01

minus01

minus005

1 3 5 7 9

119905 (s)

1199022(m

)

(b)

Figure 6 The recovered (or identified) responses compared withthe exact ones

funded by theMinistry of Education Science andTechnology(Grant no 2011-0010090) In addition this research was alsosupported by Basic Science Research Program through theNational Research Foundation of Korea (NRF) funded bythe Ministry of Education Science and Technology (Grantno K20903002030-11E0100-04610) Finally the author wouldlike to extend his special thanks to the students Miss Hye-ree Bae andMr Jinsoo Park in Naval Architecture and OceanEngineeringDepartment PusanNational University for theirhelp in the paper

References

[1] M Kulisiewicz ldquoA nonparametric method of identification ofvibration damping in non-linear dynamic systemsrdquo Interna-tional Journal of Solids and Structures vol 19 pp 601ndash609 1983

[2] D V Iourtchenko andM F Dimentberg ldquoIn-service Identifica-tion of non-linear damping from measured random vibrationrdquoJournal of Sound and Vibration vol 255 pp 549ndash554 2002

[3] T S Jang H S Choi and S L Han ldquoA new method fordetecting non-linear damping and restoring forces in non-linear oscillation systems from transient datardquo InternationalJournal of Non-Linear Mechanics vol 44 pp 801ndash808 2009

[4] T S Jang ldquoNon-parametric simultaneous identification ofboth the nonlinear damping and restoring characteristics ofnonlinear systems whose dampings depend on velocity alonerdquoMechanical Systems and Signal Processing vol 25 pp 1159ndash11732011

[5] S F Masri and T K Caughey ldquoA nonparametric identificationtechnique for nonlinear dynamic problemsrdquo ASME Journal ofApplied Mechanics vol 46 pp 433ndash447 1979

[6] J Doyle ldquoA wavelet deconvolution method for impact forceidentificationrdquo Experimental Mechanics vol 37 pp 403ndash4081997

[7] Q H Zen M Chapin and D B Bogy ldquoA force identificationmethod for sliderdisk contact forcemeasurementrdquo IEEETrans-actions on Magnetics vol 36 pp 2667ndash2670 2000

[8] T S Jang H S Baek H S Choi and S G Lee ldquoA newmethod for measuring nonharmonic periodic excitation forcesin nonlinear damped systemsrdquo Mechanical Systems and SignalProcessing vol 256 pp 2219ndash2228 2011

[9] T S Jang H S Back S L Han and T Kinoshita ldquoIndirectmeasurement of the impulsive load to a nonlinear system fromdynamic responses inverse problem formulationrdquo MechanicalSystems and Signal Processing vol 24 pp 1665ndash1681 2010

[10] T S Jang S L Han and T Kinoshita ldquoAn inversemeasurementof the sudden underwater movement of the sea-floor by usingthe time-history record of the water-wave elevationrdquo WaveMotion vol 47 no 3 pp 146ndash155 2010

[11] M Taylan ldquoThe effect of nonlinear damping and restoring inship rollingrdquo Ocean Engineering vol 27 pp 921ndash932 2000

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Journal of Applied Mathematics 7

(a)

ExactIdentified

0 2 4 6 8 10

0

005

01

minus01

minus005

1 3 5 7 9

119905 (s)

1199022(m

)

(b)

Figure 6 The recovered (or identified) responses compared withthe exact ones

funded by theMinistry of Education Science andTechnology(Grant no 2011-0010090) In addition this research was alsosupported by Basic Science Research Program through theNational Research Foundation of Korea (NRF) funded bythe Ministry of Education Science and Technology (Grantno K20903002030-11E0100-04610) Finally the author wouldlike to extend his special thanks to the students Miss Hye-ree Bae andMr Jinsoo Park in Naval Architecture and OceanEngineeringDepartment PusanNational University for theirhelp in the paper

References

[1] M Kulisiewicz ldquoA nonparametric method of identification ofvibration damping in non-linear dynamic systemsrdquo Interna-tional Journal of Solids and Structures vol 19 pp 601ndash609 1983

[2] D V Iourtchenko andM F Dimentberg ldquoIn-service Identifica-tion of non-linear damping from measured random vibrationrdquoJournal of Sound and Vibration vol 255 pp 549ndash554 2002

[3] T S Jang H S Choi and S L Han ldquoA new method fordetecting non-linear damping and restoring forces in non-linear oscillation systems from transient datardquo InternationalJournal of Non-Linear Mechanics vol 44 pp 801ndash808 2009

[4] T S Jang ldquoNon-parametric simultaneous identification ofboth the nonlinear damping and restoring characteristics ofnonlinear systems whose dampings depend on velocity alonerdquoMechanical Systems and Signal Processing vol 25 pp 1159ndash11732011

[5] S F Masri and T K Caughey ldquoA nonparametric identificationtechnique for nonlinear dynamic problemsrdquo ASME Journal ofApplied Mechanics vol 46 pp 433ndash447 1979

[6] J Doyle ldquoA wavelet deconvolution method for impact forceidentificationrdquo Experimental Mechanics vol 37 pp 403ndash4081997

[7] Q H Zen M Chapin and D B Bogy ldquoA force identificationmethod for sliderdisk contact forcemeasurementrdquo IEEETrans-actions on Magnetics vol 36 pp 2667ndash2670 2000

[8] T S Jang H S Baek H S Choi and S G Lee ldquoA newmethod for measuring nonharmonic periodic excitation forcesin nonlinear damped systemsrdquo Mechanical Systems and SignalProcessing vol 256 pp 2219ndash2228 2011

[9] T S Jang H S Back S L Han and T Kinoshita ldquoIndirectmeasurement of the impulsive load to a nonlinear system fromdynamic responses inverse problem formulationrdquo MechanicalSystems and Signal Processing vol 24 pp 1665ndash1681 2010

[10] T S Jang S L Han and T Kinoshita ldquoAn inversemeasurementof the sudden underwater movement of the sea-floor by usingthe time-history record of the water-wave elevationrdquo WaveMotion vol 47 no 3 pp 146ndash155 2010

[11] M Taylan ldquoThe effect of nonlinear damping and restoring inship rollingrdquo Ocean Engineering vol 27 pp 921ndash932 2000

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of