Modal Identification of Non-Linear Structures and the Use of Modal Model in Structural Dynamic Analysis

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1 Modal Identfcaton of Non-Lnear Structures and the Use of Modal Model n Structural Dynamc Analyss Özge Arslan and H. Nevzat Özgüven Department of Mechancal Engneerng Mddle East Techncal Unversty Ankara 06531, TURKEY Emal: arslan@me.metu.edu.tr, ozguven@metu.edu.tr NOMENCLATURE [D] Dynamc structural modfcaton matrx {f} Generalzed external forcng vector {F} Ampltude vector of harmonc external forcng Unt magnary number [H] Lnear structural dampng matrx [I ] Identty matrx [K] Lnear stffness matrx [M] Lnear mass matrx {N} Internal non-lnear forcng vector n Number of degrees of freedom v Descrbng functon {x} Dsplacement vector {X} Complex ampltude vector of steady state harmonc dsplacements [α] Receptance matrx [Δ] Non-lnearty matrx [ΔC] Vscous dampng matrx of modfyng system [ΔK] Stffness matrx of modfyng system [ΔM] Mass matrx of modfyng system [γ] Receptance matrx of modfed system ω Exctaton frequency ABSTRACT One of the major problems n structural dynamcs s to dentfy nonlnearty, whch s usually local n large structural systems, and to conduct dynamc analyss of the non-lnear system. In ths work, a new approach s suggested for modal dentfcaton of a non-lnear system. Modal parameters obtaned through modal dentfcaton are used n harmonc response predcton at dfferent forcng levels. The response at only the fundamental harmonc s consdered. The model can also be used to predct the response of the non-lnear system coupled wth a lnear system and/or subjected to structural modfcaton. An teratve soluton method s used n structural dynamc analyses. The dentfcaton method proposed s for systems where nonlnearty s between a sngle coordnate and the ground. Response dependent modal parameters of the non-lnear system are obtaned va modal testng at dfferent response levels. The method presented s verfed through case studes. In the case studes presented, a subsystem wth cubc stffness type non-lnearty s consdered and the smulated FRFs of the subsystem generated for varous response levels are used as pseudo expermental values.

2 1. INTRODUCTION Nonlnearty s a generally encountered phenomenon n mechancal structures. Although structures n general have some amount of nonlnearty, response ampltudes are so small that non-lnear forces are not excted n most cases. Therefore a lnear model wll be suffcent for dynamc analyss. However n some cases, especally when there are jonts ntroducng hgh nonlnearty or when the response level s so hgh that nonlnear forces are comparable to lnear forces, a lnear model wll not suffce and a non-lnear analyss wll be requred. There exst a varety of studes n non-lnear structural dynamcs area, most of whch concentrate on detecton, localzaton, and dentfcaton of nonlnearty [1-8]. In recent years studes on modelng and dentfcaton of nonlnearty have also ncreased. However, constructon of relable models for non-lnear structures s stll an uncertan ssue n ths feld. Former studes [9-11] show that lnear dentfcaton of nonlnear systems causes msleadng results. In [9] Özgüven and İmregün appled lnear modal analyss on classcally damped non-lnear systems. The results showed hghly complex modes that ndcate non-lnear behavor snce the ntal dampng was proportonal. Ths ndcated that lnear dentfcaton methods can reveal nonlnearty n a structure but they fal to provde a relable modal model. As lnear modal analyss tools are not compatble wth the non-lnear theory, majorty of the studes n ths feld [11-13] focus on development of non-lnear modal analyss technques. Chong and İmregün [14-15] suggested a non-lnear modal analyss for a mult-degree-of-freedom system by frst dentfyng modal parameters from measured response. For verfcaton they used smulated non-lnear response data. They were able to predct the nonlnear response of the system for other exctaton levels by usng the modal parameter varatons wth respect to modal dsplacement. The approach proposed n ths study bases on the fact that non-lnear structures exhbt lnear behavor under certan condtons, whch makes the use of lnear modal dentfcaton methods possble. Modal parameters dentfed under these condtons are used to construct a modal model for the non-lnear system to be analyzed. In that respect, ths study follows a smlar approach wth that of Chong and İmregün [14-15]. However, n the present study physcal dsplacements are used, unlke n references [14] and [15] n whch modal dsplacements were employed. Furthermore, the present work uses a sem analytcal approach for the modal model, and also extends the use of the modal model n structural modfcaton problems n addton to response predcton and dynamc couplng analyss.. THEORY When a constant ampltude harmonc force s appled over a frequency range, the non-lnear elements n a structure wll act lke equvalent dampngs and/or stffnesses wth dfferent values at each frequency. However, when the response level s kept constant n a experment, non-lnear elements wll behave as equvalent lnear elements, and the structure wll behave lnearly for that response level as dscussed n [] and expermentally shown n [16]. Then f an FRF of a non-lnear system s measured by keepng the response ampltude constant (wth dsplacement controlled experments), a lnear dentfcaton can be carred out, and a set of modal parameters for each response level can be obtaned. Then, the modal parameters dentfed at several response levels can easly be employed n harmonc response analyses..1 Modal Identfcaton of Non-lnear Structures from Response Level Controlled FRFs Modal Model of a Non-lnear Structure Consder the equaton of moton of a non-lnear MDOF system: [ M ]{ x} + [ K]{ x} + [ H]{ x} + { N( x, x )} = { f} (1) where matrces [M], [H] and [K] represent the mass, structural dampng and stffness matrces, respectvely. Vectors {x} and {f} stand for the response and external force appled on the system, respectvely. The vector {N} corresponds to the non-lnear nternal forces n the system. Ths force vector s usually a functon of dsplacement and/or velocty response, dependng on the nonlnearty present n the system. In ths study the nonlnearty between a sngle coordnate and the ground s consdered, therefore {N} ncludes only one nonzero element. Consderng a snusodal exctaton at a frequency ω and assumng that the response s also harmonc at the same frequency, the forcng and response vectors can be wrtten as;

3 and { f } { F} e ωt = () { x} = { X} e ωt (3) respectvely. The vector {X} conssts of complex values to accommodate phase nformaton. Wth ths assumpton, the non-lnear forces can be expressed n a matrx form as frst suggested by Budak and Özgüven [17-18] and then extended by Tanrıkulu et. al. [19]. Let us consder the non-lnear force n n the system. Here subscrpt represents the coordnate where the non-lnear element s connected as shown n Fgure 1. X m k +1 k k k 3 m k m 1 k 1 Fgure 1. Non-lnear system consdered n the study As the response {x} s assumed to be snusodal at the same frequency, then the non-lnear nternal force (, ) n x x can be wrtten as (, ) (, ) n x x = v x x x (4) such that v provdes the best average of the true restorng force. In other words, t s the descrbng functon for the non-lnear element present n the system. Dervaton of v for varous nonlnearty types can be found n reference [0]. Usng Eq. (4), the vector {N} can now be wrtten as Here [ ( x, x) ] { (, )} = [ Δ(, )]{ } t N x x x x X e ω (5) Δ s the response dependent nonlnearty matrx and the elements of t are gven n terms of descrbng functons v rj as follows [19]: n Δ rr = vrr + v rj r = 1,,..., n j= 1 j r Δ rj = v rj r j r = 1,,..., n (7) (6)

4 As the nonlnearty consdered n ths study s between a sngle coordnate and the ground, {N} ncludes only one nonzero element. Consequently, the nonlnearty matrx for the system conssts of only one nonzero dagonal element v as can be seen from Eq. (6). Substtutng Eqs. (), (3) and (5) nto Eq. (1), yelds ( ωt ωt ω [ M ] + [ K] + [ H] + Δ ( x, x ) ){ X} e = { F} e (8) Then the receptance matrx of the system can be wrtten as: (, X) = ω [ M] + [ K] + [ H] + Δ( x, x) α ω 1 (9) From Eq. (6) t s seen that the nonlnearty can be consdered as an added equvalent stffness matrx whch s a functon of the response ampltude, provded that the descrbng functon for the nonlnearty s a functon of only the response ampltude. Then, t can be concluded that response controlled measurements provde lnear FRFs, each correspondng to a dfferent response level. Response controlled measurements are to be performed by keepng the response ampltude of the non-lnear coordnate constant. Here the term non-lnear coordnate refers to the coordnate to whch the non-lnear element s attached. Identfcaton by usng these FRFs results n a set of modal parameters for the system, each set correspondng to a specfc response ampltude value. The above dervaton s for a system wth structural dampng, and t can easly be extended to a vscously damped system. As the dentfed modal parameters vary wth respect to the response ampltude X, dentfed modal parameters can be expressed as a functon of X as follows ( X ) ( X ) ( ) ωr = ωr (10) ηr = ηr (11) r A kl = r A kl X (1) Ths model then can be used n harmonc response predcton of the non-lnear system, as well as n couplng and modfcaton analyses, whch wll be dscussed n detal below.. Harmonc Response Predcton n Non-lnear Structures by Modal Superposton Once the varaton of the non-lnear modal parameters wth respect to the response ampltude s known, the harmonc response ampltude X can be wrtten as ( ω, ) X = αj X F j (13) Here, F j s the ampltude of the harmonc force appled at j th coordnate and α j s the response level dependent receptance value. Note that, α j should actually be called pseudo receptance as t s not possble to talk about receptance for a non-lnear system. The pseudo receptance expresson n Eq. (13) can be wrtten as a modal summaton n terms of modal parameters dentfed above, as follows αj ( ω, X ) ( X ) n r A j = r = 1 + ( ωr ( X )) ω ( ωr ( X )) ηr ( X ) (14) Then, the soluton of Eq. (13) requres teraton. Once a convergent soluton s obtaned for X, then the rest of the response ampltudes X j (j = 1,,.., n, j ) can be calculated drectly by usng the modal data correspondng to the convergent response value, X. Eq. (13) s an mplct equaton n X so t can be solved teratvely wth a proper numercal soluton method for a gven ω. In ths study the fxed pont teraton method s used for the numercal soluton, and a weghtng

5 coeffcent s employed n order to provde convergence at frequences near resonance. Note that ths soluton wll be vald only for the case where there s a harmonc force wth ampltude F j appled at the j th coordnate..3 Structural Couplng by Usng Modal Model The modal model obtaned above can also be employed n dynamc analyses of the non-lnear system coupled wth lnear ones by usng the receptance couplng method. Receptance values for the lnear system can be found expermentally or computatonally by lnear modal analyss tools, whereas, the pseudo receptance values of the non-lnear system are calculated from the modal model as explaned above. Let us consder a non-lnear system coupled rgdly wth a lnear system as shown n Fgure. x x m f x k f k Nonlnear MDOF substructure Lnear MDOF substructure Coupled nonlnear MDOF structure x, x c f c Fgure. Non-lnear system coupled rgdly wth a lnear system The receptance matrces of the non-lnear and lnear substructures can be parttoned as follows αmm [ αmn ] [ α ] [ α ] α NL = nm nn (15)

6 α kk α kl α L = α lk α ll (16) Here subscrpts m and k refer to the coordnates to be coupled, and subscrpts n and l represent the rest of the coordnates n the non-lnear and lnear systems, respectvely. By usng classcal receptance couplng technques, the receptance matrx of the coupled system can be obtaned as where C C α C nn α nl α = C C α ln α ll C α nl = C α nn = αnn [ α ] nm α kl αmm + α kk [ α ][ α ] nm mn αmm + α kk (17) (18) (19) C α ln = α lk [ α ] mn αmm + α kk (0) α C lk α kl α ll = α ll αmm + α kk (1) As the elements of C n α α NL are response level dependent pseudo receptances, so are the receptance values : C C α = α ( ω, ) X C Then α from Eq. () can be substtuted n Eq. (13) to fnd the response of the system to a harmonc forcng j appled at the j th coordnate. The soluton, agan requres teraton. When rgd couplng s made as above, the formulaton does not permt to take the coordnate where the nonlnear element s attached as the couplng node; because, n rgd couplng the receptance matrx of the coupled system does not nclude the receptances related to the couplng node, whereas the ampltude of the dsplacement of the non-lnear coordnate s requred n the analyss. Yet, flexble couplng can easly overcome ths dffculty. ().4 Structural Modfcaton of a Non-lnear Structure by Usng Modal Model The basc methodology presented n ths study can also be mplemented n structural modfcaton problems. Consder a MDOF lnear system for whch the receptance matrx [α] at frequency ω s known. When ths system s modfed wthout ncreasng the total degrees of freedom of the system, the receptance matrx [γ] of the modfed structure can be wrtten as [1]

7 where 1 (3) [ γ ] = [ I] + [ α][ D] [ α] [ D] [ K] ω [ M] [ H] = Δ Δ + Δ (4) Here, [ΔK], [ΔM] and [ΔH] represent the stffness, mass and dampng matrces of the modfyng structure, respectvely. If the modfcaton s local, the above equatons can be wrtten n terms of parttoned matrces [1], whch reduces the computaton tme consderably. Now f we apply the same modfcaton method to a non-lnear system, then [α] n Eq. (3) wll be the pseudo receptance matrx of the non-lnear system whch can be obtaned by modal synthess by usng dentfed response dependent modal parameters dscussed above. Thus the response dependent pseudo receptance of the modfed system wll be gven by 1 γ ( ω, X ) = [] I + α( ω, X ) [ D] α( ω, X ) (5) whch can be used to fnd X. Agan, an teratve soluton s requred. For large systems, the formulaton gven n [1] can be employed to reduce the computatonal effort. 3. CASE STUDIES The applcaton and the valdaton of the methods proposed are demonstrated by usng smple dscrete systems for the sake of smplcty. In each case, smulated FRFs generated at dfferent response levels are used as pseudo expermental data. In practce, sne sweep vbraton test s to be used. Modal parameters obtaned through the dentfcaton of pseudo expermental FRFs are expressed as a functon of response ampltude. The modal model of the non-lnear system constructed s then employed n response computatons. Identfcaton of modal parameters s performed by usng MODENT. The responses obtaned from modal model are compared wth the frequency response of the system calculated by harmonc balance approach. 3.1 Case Study 1 The system used n [9] s consdered n ths case study. As shown n Fgure 3, t s a -DOF system wth cubc stffness. M X 1 K C M 1 K 1+ K* C 1 Fgure 3. The system consdered n Case Study 1 The propertes of the system are as follows: M 1 = 0 kg, M = kg, * 10 K = 10 X 1 [ N / m], 6 5 K 1 = 8 10 N / m, K = 8 10 N / m, C 1 = Ns/ m, C = 1.5 Ns/ m

8 By usng the pseudo expermental FRF curves obtaned n the frequency range of Hz, for several X 1 values rangng between.4 mm and 8.6 mm, X 1 dependent modal parameters are obtaned as gven n Fgures 4 to 6. ω 1 [Hz] ω [Hz] a) X [m] 1 x 10-3 b) X [m] 1 x 10-3 Fgure 4. Natural frequency varatons wth respect to X 1. (a) mode 1 (b) mode η 1 (%) η (%) a) X [m] 1 x 10-3 b) X [m] 1 x 10-3 Fgure 5. Dampng rato varatons wth respect to X 1. (a) mode 1 (b) mode A 11 [1/kg] A 11 [1/kg] a) X [m] 1 x 10-3 b) X [m] 1 x 10-3 Fgure 6. Modal constant varatons wth respect to X 1. (a) mode 1 (b) mode As can be seen from the fgures, modal parameters of the non-lnear system follow a trend and therefore they can be expressed n terms of proper mathematcal functons that ft to the correspondng data ponts. The scattered values observed n the graphs of dampng rato and modal constants are beleved to be due to the dentfcaton algorthm used n MODENT. In practcal applcatons even more scattered ponts are expected due to measurement errors. In order to see the effect of measurement errors, case studes wth polluted smulated FRF values were also carred out. Snce analytcal expressons are ft to modal data ponts, t was observed that havng scattered data does not affect the results sgnfcantly.

9 Frequency response of the system at forcng levels of 000N and 4000N are calculated by usng the modal model and presented n Fgures 7-9 n decbel scale. The force s appled at the frst mass to whch the nonlnear element s connected. The excellent match of the responses obtaned from the modal model wth those calculated by harmonc balance method demonstrates the valdty of the modal model and the approach suggested. Note that both approaches are vald f the basc assumpton (harmonc exctaton results n harmonc response at the same frequency) holds true. Fgures 8 and 9 show solutons for both ncreasng and decreasng. Very slght dfferences are observed between the results of ths study and the HBM only around jump, and t s beleved that they are due to usng dfferent numercal computatonal algorthms n both approaches. -90 α α 1 HBM smulatons -110 pseudo receptance(db) frequency(hz) Fgure 7. Pseudo FRFs of the system at F 1 = 000N ncreasng decreasng HBM smulaton (ncreasng sweep) HBM smulaton (decreasng sweep) α 1 [db] frequency [Hz] Fgure 8. Pseudo receptance, α 1 of the system at F 1 = 4000N

10 ncreasng decreasng HBM smulaton (ncreasng sweep) HBM smulaton (decreasng sweep) -15 α 11 [db] frequency [Hz] Fgure 9. Pseudo receptance, α 11 of the system at F 1 = 4000N 3. Case Study In ths case study, the -DOF system consdered n the frst case study s coupled wth an undamped lnear subsystem, and the coupled system s analyzed by usng the approach proposed. System matrces of the lnear subsystem are as follows: 1 0 M L = kg 0 1, 5 5 K L = / 5 5 N m The coupled system s obtaned by rgdly connectng the frst mass of the lnear system to the second mass, M, of the non-lnear system shown n Fgure 3. Pseudo receptance matrx of the non-lnear system s calculated by usng the modal model obtaned n the frst case study, and the pseudo receptance matrx of the coupled 3-DOF non-lnear system s calculated by usng the receptance couplng approach proposed. The frequency response of the coupled system at a forcng level of 3000N appled at the non-lnear coordnate s shown n Fgure 10 n decbel scale. Agan a very good agreement s obtaned between the results of ths study and HBM solutons, except around jump frequency at the thrd mode, most probably due to the reason explaned n Case Study Case Study 3 In ths case study, agan the system n the frst case study s used as the orgnal system, and the followng mass and stffness modfcatons are made: 1 0 Δ M = kg 0 1, 5 5 K Δ = / 5 5 N m Frequency response of the modfed system at forcng level of 000N s calculated from the modal model of the orgnal non-lnear system and the modfyng mass and stffness matrces. The result s shown n Fgure 11

11 n decbel scale. Force s appled at the second mass. In ths fgure, solutons for both ncreasng and decreasng are shown. It s agan possble to observe the jump around the frst resonance of the system, where the results devate from those of HBM pseudo receptance [db] HBM smulaton (ncreasng sweep) HBM smulaton (decreasng sweep) ncreasng decreasng frequency [Hz] Fgure 10. Pseudo receptance, α 11 of the coupled system when there s a force of F = 3000N at the nonlnear coordnate pseudo receptance [db] orgnal system -150 HBM smulaton (ncreasng sweep) HBM smulaton (ncreasng sweep) -160 ncreasng decreasng frequency [Hz] Fgure 11. Pseudo receptance α 11 of the modfed system when there s a force of F = 000N at the nonlnear coordnate

12 4. CONCLUSIONS A method s proposed for modal dentfcaton and modal analyss of non-lnear systems. The modal model of the system s obtaned by dentfyng modal parameters from the lnear FRF drawn for constant dsplacement ampltude of the non-lnear coordnate (the coordnate where non-lnear element s connected). Repeatng the dentfcaton for dfferent response ampltudes, dentfed modal parameters can be expressed as functons of the response ampltude of the non-lnear coordnate. It s demonstrated wth case studes that the modal model can successfully be employed n the response predcton of the system at any forcng level. The uses of the same modal model n dynamc couplng of the dentfed non-lnear system wth a lnear system, and n the dynamc modfcaton analyss of the non-lnear system are also formulated and valdated by case studes. As the modal parameters are represented as functons of the response tself, an teratve soluton s requred n each analyss. Fxed pont teraton s used as the numercal soluton, and a weghtng coeffcent s employed n order to provde convergence at frequences near resonance. The agreement observed between the frequency responses obtaned by usng the method proposed n ths study wth those of the harmonc balance method demonstrates the valdty of the modal model and the methods suggested, provded that the basc assumpton harmonc exctaton results n harmonc response at the same frequency holds true, as both approaches base on ths assumpton. However, a slght msmatch s observed between the solutons of ths study and those of HBM only around frequences where jump occurs. It s beleved that ths s due to usng dfferent numercal soluton algorthms n both approaches. The current work nvolves n employng better soluton algorthms n both approaches. The methods suggested are applcable to non-lnear systems where nonlnearty s between a sngle coordnate and the ground. Furthermore the descrbng functon for the nonlnearty should be frequency ndependent. Extenson of the study to generalze the method for systems where nonlnearty s at an arbtrary locaton s the subject of the current work. 5. REFERENCES [1] Mertens, M., Van der Auweraer, H., Vanherck, P., Snoeys, R., Detecton of Nonlnear Dynamc Behavour of Mechancal Structures, 4 th Internatonal Modal Analyss Conference, Los Angeles, Calforna, USA, , [] He, J., Ewns, D. J., "A Smple Method of Interpretaton for the Modal Analyss of Non-lnear Systems," 5 th Internatonal Modal Analyss Conference, London, England, , [3] Bohlen, S., Gaul, L., Vbratons of Structures Coupled by Nonlnear Transfer Behavour of Jonts; A Combned Computatonal and Expermental Approach, 5 th Internatonal Modal Analyss Conference, London, England, 86-91, [4] Ln, R. M., Ewns, D. J., On the Locaton of Structural Nonlnearty from Modal Testng - A Feasblty Study, 8 th Internatonal Modal Analyss Conference, , [5] Vakaks, A. F., Ewns, D. J., Effects of Weak Nonlneartes on Modal Analyss, 10 th Internatonal Modal Analyss Conference, San Dego, Calforna, USA, 73-78, 199. [6] Song, H. W., Wang, W. L., Non-Lnear System Identfcaton Usng Frequency Doman Measurement Data, 16 th Internatonal Modal Analyss Conference, Santa Barbara, Calforna, USA, , [7] Özer, B., Özgüven, H. N., A New Method for Localzaton and Identfcaton of Nonlneartes n Structures, 6 th Bennal Conference on Engneerng Systems Desgn and Analyss, İstanbul, Turkey, 00. [8] Goge, D., Snapus, M,, Fullekrug, U., Lnk, M., Detecton and Descrpton of Non-Lnear Phenomena n Expermental Modal Analyss va Lnearty Plots, Internatonal Journal of Non-Lnear Mechancs, Volume 40, Issue 1, Non-lnear Flud Mechancs, 7-48, 005. [9] Özgüven, H. N., İmregün, M., Complex Modes Arsng from Lnear Identfcaton of Nonlnear Systems, The Internatonal Journal of Analytcal and Expermental Modal Analyss, Volume 8, Issue, , 1993.

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