Residual Modes on Non-linear Resonant Decay Method

Transcription

1 Residual Modes on Non-linea Resonant Decay Method D Mehdi Samast, Pofesso Jan R. Wight ABSRAC Non-linea Resonant Decay method (NL-RDM) addesses the identification of multi-degee of feedom non-linea systems. his method offes a pactical appoach to the identification of lumped paamete and continuous systems by poducing a non-linea extension of the classical linea modal model. he method is intoduced and its potential as a pactical identification appoach explained. his pape is concened with the inclusion of esidual modes, above and below any egion of inteest. Any stuctue that is not suppoted to eath will have igid body modes, being modes that have a natual fequency at zeo Hetz; a good example is an aicaft. hese modes ae sometimes ignoed but fo a complete mathematical model obtained fom NL-RDM the effects of igid body modes must be analysed. NL-RDM elies on a method of cuve fitting to geneate modal chaacteistics, in tems of foce, displacement, velocity and acceleation in modal space. he lowe esidual egion was obseved to contain igid body modes, and these wee obseved to affect significantly the esults in the egion of inteest. Monitoing this effect was shown to be too difficult cuently fo the NL-RDM, given technological estictions. A Mass Substitution method was geneated to model the system esponse moe accuately. Its accuacy was demonstated though case studies. Most systems also contain an indefinite numbe of esidual modes, these being modes that occu at fequencies geate than the fequency ange of inteest. he section will attempt to detemine the effects of excluding esidual modes on the accuacy of NL-RDM. he effect of the uppe esidual aea on the modelling of the system within the egion of inteest was obseved though seveal case studies. Simple validation methods have been consideed to peseve the accuacy of the model, taking into account both hamonic and coupled esidual modes and thei effects. Keywods: NL-RDM, Residual Modes, Rigid Body Mode, Out of Range modes. Manuscipt eceived on Mach, 3,. D. Mehdi Samast Autho is with Islamic Azad Univesity, East ehan Banch, ehan, Ian, phone: , Membe of IAENG, msamast@yahoo.com Pofesso Jan R. Wight is with he Univesity of Mancheste, Mancheste, M3 9PL, UK, jan.wight@mancheste.ac.uk I. MEHODOLOGY FOR NL-RDM he method is a development of the Resonant Decay Method, the Restoing Foce Suface method based in modal space and the Foce Appopiation methodology [, ]. It makes use of an appopiated excitation vecto applied in a bust to educe the numbe of modes equied in the cuve fit. In this section, the mathematical fom of the extended modal model fo non-linea systems will be descibed. Conside a lumped paamete system o a continuous one that has been discedited as N degees of feedom (DOF). he equation of motion in physical space fo a dynamic system including stiffness non-lineaities is M x C x K x K x x F(t) () L whee [M] is the mass matix, [C] is the damping matix, [K L ] is the linea stiffness matix, [K NL ] is the non-linea stiffness matix, {F(t)} is the vecto of applied nodal foces, and {x(t)} is the vecto of physical displacements. he tansfomation between physical and modal space is defined by N NL { x( t)} { } p [ ]{ p( t)} whee {p(t)} is a vecto of modal amplitudes, and [] is the modal matix of the N modes {}, =, N of the undelying linea system, which may be obtained by solving the classical eigenvalue poblem fo un-damped fee vibation. he coesponding un-damped natual fequencies ae ( =, N). In pactice, the numbe of degees of feedom equied to model the system with easonable accuacy can be educed by consideing only those modes NR << N having natual fequencies in the fequency ange of inteest, o that ae consideed impotant in the esponse. In this case, the modal matix [ ] is educed to dimension N by NR and the vecto of modal amplitudes {p(t)} to dimension NR by. Substituting the modal expansion into the system equations of motion and pe-multiplying by [] yields: M p( t) C p K K p( t) F L NL Using the othogonality of the modes, this equation of motion in modal space becomes () (3) ISSN: (Pint); ISSN: (Online)

2 m p c p [ k ] p( t) f f (4) L whee the modal mass matix [m] and linea modal stiffness matix [k L ] ae diagonal and the modal damping matix [c] is diagonal fo popotionally damped systems. {f} is the applied modal foce vecto and {f NL } is the vecto of nonlinea estoing foces in modal space. Clealy, in ode to be able to pefom this tansfomation, the modal matix [] is assumed to be known fom low foce level esults. Fo a paticula mode of a popotionally damped non-linea system, equation (4) is in the fom of a single degee of feedom system, namely m p c p k p fnl f,... NR (5) whee p is the th modal displacement, m, c and k ae the th mode modal mass, damping and stiffness, and f is the coesponding applied modal foce. he modal mass and stiffness fo the linea system ae elated by the un-damped natual fequency fom the equation k m [3]. Nonpopotional damping would lead to the pesence of modal damping coupling tems [4]. he tem f NL efes to the th mode non-linea modal estoing foce and in geneal includes othe modal co-odinates to allow fo non-linea coss coupling f ( p, p,.., p NR 3 s i A i ( s, s ) p NR i s ) NL NR NR s ts i 3i j NL A i j s, t p p whee the A tem is the amplitude of the non-linea tem in the th mode equation that is associated with the espective modal displacements p s and p t (fo modes identified by the indices s and t), aised to the powes defined by i and j. Fo example, the diect cubic non-linea stiffness tem associated with the th mode would be A p 3 (s=, t=, i=3, j=) and a typical coss-coupling tem between modes and q might be A p p q (s=, t=q, i=, j=). Note, in this case, that the set of possible basis functions is esticted to tems of 3 d ode involving no moe than two modes. Also, the basis functions in equation (6) ae polynomials wheeas non-polynomial tems may be used to cate fo diffeent types of non-lineaity (e.g. discontinuous) [5]. hee is no eason in pinciple why non-linea modal velocity tems cannot also be included in the extended modal model to cope with damping non-lineaity. A. SIMPLE EXAMPLE Non-lineaity is not easy to define in a geneal manne and so the NL-RDM cannot be applied to a geneal case. hus, it is ecommended that the consideation of the method stat with a simple example such as a two degee of feedom system with some kind of non-lineaity, which could be the common cubic stiffness, to take advantage of a elatively simple case. Some othe type of non-lineaity such i s j t (6) as quadatic damping, fiction and backlash will not be investigated in this pape. II. RIGID BODY AND OU OF RANGE MODES he pevious pape [6] attempted to detemine the effects the use of impefect data has on NL-RDM, assuming use of the coect model. his pape will assume the use of pefect data and test pocess, but will analyse some of the effects of using an incoect model. he analysis will include the effects of igid body modes, of poblems associated with hamonics, and of the esidual pat of the data. he effects will be analysed in tems of deived system paametes and the FRF. Any stuctue that is not suppoted to eath will have igid body modes, being modes that have a natual fequency at zeo Hetz; a good example is an aicaft. hese modes ae sometimes ignoed but fo a complete mathematical model obtained fom NL-RDM the effects of igid body modes must be analysed. NL-RDM elies on a method of cuve fitting to geneate modal chaacteistics, in tems of foce, displacement, velocity and acceleation in modal space. Most systems also contain an indefinite numbe of esidual modes, these being modes that occu at fequencies geate than the fequency ange of inteest. his chapte will attempt to detemine the effects of excluding esidual modes on the accuacy of NL-RDM. Finally, hamonic effects will be analysed. Specifically a system with a cubic stiffness non-lineaity will be obseved, when one mode natual fequency occus at pecisely 3 times the natual fequency of the othe mode. A. IDENIFYING RIGID BODY MODES Rigid body motion occus in many systems and affects the dynamic behaviou, so should be investigated. A igid body mode is a mode with a natual fequency of zeo Hetz and mode shapes which nomally consist of pue tanslation o otation; igid body modes occu in a pefect fee-fee stuctue. Expeimentally, igid body motion is obseved to cause no stain in any elastic element in the system. Rigid body behaviou can include six types of motion, which ae tanslation and otation in 3 dimensions. he way in which the NL-RDM would appoach the analysis of igid body modes needs to be discussed, as a complete model is equied in ode fo the dynamic behaviou of the system to be investigated. he poblems encounteed by this appoach give ise to thee pactical methods of analysing stuctues with igid body modes. Fistly, the mode is identified using the NL-RDM in the same way as fo any othe mode. Secondly, the igid body mode can simply be ignoed. he effects of this on accuate system epesentation ae analysed by compaison between pactical esults fo models with and without igid body modes. hidly, the igid body mode is simulated by a substituted modal mass element without the need fo any specific identification. ISSN: (Pint); ISSN: (Online)

3 MMIF Eigenvalue Phase FRF FRF A.. NON-LINEAR MDOF MODELS WIH RIGID MODY MODES In this section, two systems ae intoduced fo the evaluation of the pactical method, i.e. to detemine the effectiveness of appoximating a fee-fee system by a lightly spung fixed-fixed system. he thee degee of feedom lumped paamete system with stiffness nonlineaity between the second and thid masses (fee-fee stuctue) in Figue, is compaed to anothe stuctue, identical except fo being suppoted at both ends on soft spings (pactical stuctue) in Figue. stuctues intoduced in the pevious section, ae compaed in Figues 3 and FRFs phases in Figue 4. he FRFs show stong ageement, indicating the accuacy of the igid body mode appoximation and justifying the analysis of the suppoted stuctue fo igid body mode identification. Figue 5 shows the MMIF [7] fo the 3 degee of feedom pactical stuctue, whee the igid body mode on soft suppots is included. Clealy, appopiated foce pattens could be obtained fom such an analysis. 3 5 Pactical Pefect 5 5 Figue : Schematic Diagam of 3 Degee of Feedom Feefee System with Stiffness Non-lineaity (pefect fee-fee stuctue) 5 5 Fequency Figue 3: H FRFs fo the Pefect fee-fee Stuctue and the Pactical System 5 Pactical Pefect Figue : Schematic Diagam of 3 Degee of Feedom Fee- Fee System with Stiffness Non-lineaity but suppoted on Soft Spings (pactical stuctue) he FRF esults fo the pefect fee-fee and pactical systems ae shown in Figues 3 and 4 espectively. Compaison between the fist and second modes, i.e. the igid body mode and the fist flexible mode, gives an indication of the suitability of the pactical system fo appoximating the fee-fee system. At a atio of / the fist mode is consideed close enough to zeo Hetz to not have any significant effect on the flexible mode; in this example, the igid body mode natual fequency is.4 Hz and the fist flexible mode is 5.6 Hz. Fo bette compaison between the pefect and pactical system, the mode shapes o modal matix fo the two cases ae as follows.5 ModalMati x (7) Pefect.5 and.9967 ModalMatix P actical Clealy, the suppot has little effect on the modal chaacteistics. he FRFs of the pefect and pactical (8) Fequency Figue 4: H FRFs Phases fo the Pefect fee-fee Stuctue and the Pactical System Fequency (Hz) Figue 5: MMIF fo the 3DOF Pactical Fee-fee System A. IDENIFICAION OF A RIGID BODY MODE USING NL-RDM Having confimed that it is acceptable to conside a fee-fee stuctue to be suppoted on soft spings, one possibility fo identifying the igid body behaviou is to simply apply the NL-RDM [8] appoach just as fo a flexible mode. he NL-RDM is pefectly suited to identify a fee-fee stuctue by simulation, but poblems aise when ISSN: (Pint); ISSN: (Online)

4 H (m/s /N) H (m/s /N) seeking to identify a stuctue in pactice, i.e. expeimentally. Seveal poblems hinde the analysis of igid body modes pactically as the igid body mode may need to be investigated with vey low fequencies. his may ceate a significant poblem with noise, as the level of noise is close to the low acceleation levels expeienced at low fequencies. he vey low fequencies ceate futhe poblems expeimentally when geneating foce and monitoing esponse. Piezo-electic tansduces ae not accuate at low fequency levels as the chaacteistics olloff at low fequencies. Howeve, it would be possible to use acceleometes that opeate down to DC (e.g. piezo-esistive acceleometes) o even displacement tansduces. Commonly, shakes would be used to geneate foce and acceleometes used to monito the motion, so the ability to geneate acceptably low levels of foce and monito the low levels of motion, and hence obtain accuate esults, is limited by the capability of these pieces of equipment. It is well known that it is difficult to use an electodynamic shake to geneate a sinusoidal foce at low fequency. Futhe poblems ae encounteed when using the NL- RDM to identify the igid body mode. Poblems may aise when pepaing fo foce appopiation. A poo MMIF, specifically a pooly identified igid body mode causes an inaccuate foce vecto and poo tuning of the system. hese poblems ae citical with a lowe flexible mode fequency. Fo example, with a fist flexible mode aound 5 Hz, the igid mode fequency would be aound.5 Hz and the pocedue of detecting and identifying the mode becomes much moe difficult than when a stuctue s fist flexible mode fequency appeas aound Hz and the igid body fequency could be moe like Hz. Fo such a stiffe stuctue, thee would be no paticula poblem in using the NL-RDM to identify the igid body mode linea chaacteistics; no non-linea behaviou would be expected. ypically, the instumentation poblems mentioned above become wose below about 4 Hz, coesponding to a flexible mode at 4 Hz if the igid / flexible mode fequency atio is %. heoetically, using impoved foce geneation / ecoding, with bette shakes and acceleometes could poduce bette, moe accuate esults at vey low fequencies. echnologies ae aleady available o being developed that could impove the accuacy of these devices (e.g. lase based measuement systems). Clealy, impoved low-level accuacy of components would have a significant effect on the difficulties associated with igid body mode expeimentation. With accuate enough components, the igid body mode could theoetically be analysed by the NL- RDM in the same way as any othe mode. Howeve, if diect identification of vey low fequency igid body modes is too difficult to pefom accuately, maybe thee ae othe appoaches? Pehaps the igid body mode could simply be ignoed o a theoetically based estimate included? A.3 NEGLECING HE RIGID BODY MODE IN HE Non-IDEALIZED MODEL Anothe method that could be employed to analyse the system esponse in a non-linea system with a igid body mode is to entiely ignoe the igid body mode itself. Fo example, in the flutte analysis of an aicaft, often only the flexible modes ae included. he non-linea system is then analysed like any othe system, in tems of its flexible modes. he ease of this method is clealy its geatest advantage, but what may be a significant mode is missed out. Case studies ae caied out in this section to demonstate the accuacy of this appoach as compaed to identifying the igid body mode fom measued expeimental esults. he same system as intoduced in Figue is chosen fo analysis. Figues 6 and 7 show the calculated FRFs fo the system when the igid body mode is ignoed (Geen colou), and compaes them to esults fom a complete analysis, with the igid body mode on soft suppots included in the pactical system (Blue colou). While the geneal shape of the FRFs match aound the two flexible mode peaks, it is clea that esponse levels ae all somewhat inaccuate when the igid body mode is ignoed. he effect occus not only whee the igid body peak is not modelled, but at othe fequencies too, because the influence of the igid body mode extends acoss the entie fequency ange of the system. hus, while this method can be used as an appoximation to the tue system, its accuacy fo geneating system esponse and thus its application may be seiously limited. Figue 8 implements a flowchat, which indicates the way the compaison between systems is done Pactical neglected 5 5 Fequency (Hz) Figue 6: Compaisons of FRF (H ) Pactical neglected 5 5 Fequency (Hz) Figue 7: Compaisons of FRF (H ) ISSN: (Pint); ISSN: (Online)

5 a modal equation to allow fo the igid body mode contibution. he mass epesentation method involves analysing the igid body mode in tems of an added modal mass, that mass being deived fom a mass summation ove the stuctue, assuming a igid body mode shape. his method has the advantage of a moe complete system analysis, without making any majo changes o additions to the oiginal method. he appoach will be illustated fo the simple case of the 3 DOF system consideed ealie. he extension to moe complex poblems will be outlined late. he igid body modal mass M, which is also in a sense a mass summation, could be calculated as M * M * (9) Physical Figue 8: Compaison between Systems Rigid Body Ignoed o Included A.4 REPRESENING HE RIGID BODY MODE WIHOU SPECIFIC IDENIFICAION Due to the inaccuacy of measuement of low fequency igid body modes, a complete identification using NL-RDM may be consideed too inaccuate to be wokable; cetainly this is the case fo an aicaft whee the flexible modes occu at low fequencies (typically -5 Hz). heefoe a method is equied that will eadily allow fo the igid body mode effects in the identified model; fo example, in cuve fitting an FRF, it is common to include a lowe esidual tem in the model to appoximate the igid body effect [9]. In this section, a method is poposed that will simply add in whee the igid body mode shape fo the 3 DOF system is () and M Physical is the physical mass matix of the system. hus the igid body modal mass is M M * M * M 3 () o M M () M o moe geneally M 3 M M M (3) i otal hus, if the total mass of the stuctue is known by diect measuement o calculation, and if the igid body tanslation mode shape is assumed to have unit values, the igid body modal mass is simply equal to the total mass of the stuctue. In ode to include the igid body mode behaviou as an additional equation to accompany the identified flexible mode equations, then the appopiate modal foce and possibly modal mass and damping values should be estimated. By analogy to the analysis fo mass pefomed above, the igid body modal stiffness K will equal the summation of the suppot stiffnesses (e.g. bungees) and the modal damping C will be the summation of the suppot damping coefficients, both tansfomed into modal space. he igid body modal foce may be detemined though the physical foces to be applied to the system, which means that: F * F (4) Physical F F * F (5) F3 F F (6) F F3 ISSN: (Pint); ISSN: (Online)

6 o moe geneally F Fi Fotal (7) hus the igid body modal foce is simply equal to the summation of the physical foces applied to the stuctue. he final modal equation fo the igid body mode (including suppots if desied) will then be M q Cq Kq F (8) whee q is the igid body modal coodinate. his modal equation is simply added to those identified fo the nonlinea flexible modes. It should be noted that no non-linea modal couplings will exist between igid body and flexible modes, povided the suppot stiffnesses ae linea, because no non-linea foces will be geneated by igid body motion and foces geneated in motion of a flexible mode will not do any wok in the igid body mode. Fo implementation in the simulation pogamme, a system with 3 degee of feedom with cubic stiffness nonlineaity and one igid body mode is consideed. Figue shows the schematic system of the pactical stuctue. Fistly, only the igid body modal mass is included (i.e. effect of suppot stiffness is ignoed) and then the suppot effects ae included though the modal stiffness K. he accuacy is clealy impoved in compaison with the Neglected method, with the FRFs being mostly coincident. he only vaiation is seen in the epesentation of the igid body mode itself, aound which point the methodically calculated FRF levels off, giving a lowe esponse level aound the igid body mode. hus this appoach is a simple and pactical one to including igid body mode effects in the identified nonlinea model without needing to cay out difficult low fequency measuements. he appoach epesents the igid body effects accuately, povided the estimated mass is easonably accuate; indeed, the epesentation of the igid body modes is consideed to be moe accuate than using the lowe esidual tem in a cuve fit. If suppot stiffnesses wee only included to make the test possible, and a model of the tue fee-fee system equied, then only the igid body modal mass and foce need be included in the final model, as the suppot stiffness and damping effects can be ignoed. A.5 PRACICALIIES FOR MORE COMPLEX SRUCURES In eality a stuctue could have as many as six igid body modes. his will affect the mass epesentation method. Fo the 3 tanslation modes, the modal masses will all equal the total mass of the stuctue and the mode shapes will be equal tanslation of all points in the same diection, i.e. simple extension of the above appoach. hus, fo a heave mode, Heave M M M (9) i otal and so on. he othe igid body modes will involve igid body otation about the cente of mass (e.g. oll, pitch and yaw). In this case, the substitution would be an inetia epesentation, as opposed to being mass based fo a tanslation type mode. hus fo a pitch mode, it may be shown that Pitch M Ii I () otal whee I i is the pitch moment of inetia of the i th component of the stuctue about the cente of mass and I otal is the total pitch moment of inetia of the stuctue about the cente of mass. he pitch mode modal foce will be the sum of applied moments of each of the physical foces about the cente of mass. he coesponding otation mode shape must be a adian otation if the modal mass is the moment of inetia. In pactice, if the suppot stiffness effects ae to be included in the igid body modal equations, the values may be found by diect measuement of the sping stiffnesses o by detemining the igid body natual fequencies by exciting o distubing the stuctue into a igid body motion, even though the data may be noisy, and then calculating the modal stiffness fom the modal mass. Flexible modes would be identified as nomal using NL-RDM. B. UPPER RESIDUAL EFFECS It is possible to divide the ange of fequencies into thee egions. Of cental concen is the paticula egion of inteest, which will often include seveal impotant flexible and maybe non-linea modes. If this aea is consideed in isolation, it may be modelled inaccuately, as the enegy fom modes is less than expected in that egion. he peceding section has consideed the low fequency egion below the egion of inteest, containing one o moe igid body modes, and obseved the effect of its pesence on the egion of inteest; appoaches fo allowing fo the igid body effects wee poposed. Additionally, the egion of fequencies highe than the egion of inteest should be consideed and its effects on the egion of inteest obseved. hese highe fequency modes occu at lowe levels of displacement, and so ae moe likely to be linea, and theefoe aguably may be less of a concen. he poblem with such out of ange high fequency modes is that thei effects will appea in the measued data, up to aound the Nyquist fequency [], but the model constucted by the NL-RDM appoach will only cove a limited numbe of modes, likely to be smalle than the numbe of modes in the measued ange. In cuve fitting of an FRF, it is common to include an uppe esidual tem to allow cudely fo the modes above the ange analysed. Howeve, thee appeas to be no obvious way of allowing fo out of ange modes above the ange of inteest; adding a epesentative mode would seem to be athe atificial. It seems the only way is to ensue that the fequency ange ISSN: (Pint); ISSN: (Online)

7 H(m/s /N) analysed is somewhat wide than that fo which the model is eally equied; this is moe impotant fo a non-linea stuctue than fo a linea one as will be shown late. B. EFFECS OF MISSED MODES In a continuous system, thee ae a geat many modes pesent. It would be impactical to model all of them, and so the effect of missed modes should be obseved. hese missed modes may be coupled to included modes and so the effect of those missed modes should not be neglected. o obseve the effect of missed modes on a simple example, a thee degee of feedom non-linea lumped paamete system is modelled, and foce, displacement, velocity and acceleation data deived. hen, in this study case, the thid and highest mode is omitted fom the fitted model (but not its effects fom the data) and the cuve fit would be un fo both systems (3 and DOF). he FRF esult is shown in Figue 9. Results have shown some loss of enegy and inaccuacy on the FRF. Howeve, the estimated paametes fo the two identified modes wee actually the same (i.e. not in eo); this is because thee was no non-linea coupling and 3 shakes wee used so the appopiation was pefect. Eos could be pesent if the appopiation was impefect and if non-linea couplings wee pesent; any eo would then depend upon how lage the missed mode contibution was and how close in fequency Fequency Figue 9: FRF, when hid mode is missed hee is no way to eliminate these eos; the only appoach is to ensue that any modes with a lage esponse ae included and that the fequency ange chosen is adequately lage so that the effect of highe fequency modes is small. B. HARMONIC EFFECS One paticula case of out of ange modes that is of inteest is when an out of ange mode occus at a hamonic of a mode within the ange of inteest. Hamonic effects need to be consideed fo a non-linea system. he hamonic behaviou depends on the ode of non-lineaity, which means that if a system has a quadatic o cubic non-lineaity, hamonic effects may be obseved at two o thee times any natual fequency espectively. he so called hamonic modes outside the egion of inteest may stoe a significant amount of enegy, and so shouldn t be neglected. he effects of these hamonic modes ae consideed though the modelling of a system with two natual fequencies, at 7. and.36 Hz, the second mode occuing at the thid hamonic of the fist mode. he simulation pogamme is un fo cases aound and at exactly thee times the atio between natual fequencies to indicate any effects on the esults. When the system was identified with both modes pesent, the identified paametes wee not affected because all the elevant tems wee included in the analysis; howeve, consideation of this hamonic issue is equied when the 3 times natual fequency is outside the egion of inteest. he esults showed the esponse of the highe fequency (hamonic) mode is clealy much lage in this case, especially fo the acceleation. Anothe compaison based on examining time domain signals showed that the modal acceleation fo mode when seeking to excite mode was doubled when the fequencies diffeed by a facto of 3. What is of moe inteest is that when mode was analysed alone using NL-RDM, with the pesence of mode completely ignoed, the modal paametes identified wee moe in eo when the two modes diffeed by exactly a facto of 3. B.3 PRACICALIIES Pactically, it may be impotant to validate the esponse model geneated within the egion of inteest. his would be best achieved though modelling of the system with one additional mode outside this egion, and obseving the effect of the inclusion of this mode on the oiginally identified paametes and fit to a validation excitation. If thee is no significant change, then the ange chosen is aguably adequate. If thee is a significant diffeence, then the ange needs to be extended. In ecent sections has been concened with the inclusion of esidual modes, above and below any egion of inteest. he lowe esidual egion was obseved to contain igid body modes, and these wee obseved to possibly affect significantly the esults in the egion of inteest. Obtaining igid body paametes expeimentally was shown to be too difficult cuently fo the NL-RDM, given technological estictions. A mass epesentation method was geneated to model the system esponse moe accuately by adding in a modal equation fo each igid body mode. Its accuacy was demonstated though case studies. he effect of the uppe esidual aea on the modelling of the system within the egion of inteest was obseved though seveal case studies. No obvious way of allowing fo highe fequency modes seems available and so it is impotant that an adequate ange of data is consideed. Modes whose fequency is at a hamonic facto of modes of inteest need to be consideed caefully as if they ae omitted, significant eos may occu. Simple validation methods have been consideed to peseve the accuacy of the model, taking into account both hamonic and coupled esidual modes and thei effects. ISSN: (Pint); ISSN: (Online)

8 III. CONCLUSIONS In this pape, poblems with esidual effects have been investigated, including igid body mode effects in the lowe esidual aea, and missed / coupled modes in the uppe esidual aea. Hamonic effects, whee one mode natual fequency was an intege multiple of anothe mode wee also investigated. Measuing any igid body mode is difficult because, even though the stuctue is suppoted on low stiffness mounts, the fequency will usually be vey low and excitation and measuement ae inaccuate. So, some methods wee intoduced in the simulation pogamme fo and 3 degee of feedom systems to detemine o maybe neglect a igid body mode. he fist option could be to neglect the igid body mode. he esults show some inaccuacy not just aound the igid mode that neglected, but also at othe fequencies even when the igid body mode was at a significantly lowe fequency than the flexible mode(s). Anothe option could be consideed when, instead of identifying the igid body mode, the modal mass may be epesented theoetically by a summation of physical mass and inetia paametes and a theoetical modal equation added to the oveall model; a modal stiffness tem may be estimated if the igid body fequencies on the suppots can be estimated. he esults show good ageement when compaed with neglecting the igid body mode. In a eal stuctue, estimating accuate mass and inetia paametes could become a poblem although the FE model could be used. In eal stuctues, because a significant numbe of modes ae pesent in the fequency domain, only egions of inteest ae consideed but then how could uppe esidual effects be allowed fo? Effects of this egion have been investigated. Missing o including a mode in a egion of inteest is compaed fo a 3 degee of feedom system. Unlike linea identification, thee seems no way of including an uppe esidual tem in the non-linea model to allow fo the effect of the out of ange modes. o avoid this poblem, it is ecommended that exta modes be added to the egion of inteest and, the effect of the inceased size model on the validation checked to see if the diffeence is significant; so maybe moe modes need to be included than fo a linea identification. Altenatively, the level of enegy in the out of ange modes could be assessed to see if they need to be included. One moe issue, is about the so-called hamonic effect whee an out of ange mode fequency is at an intege multiple of a mode in the egion of inteest. As explained, the hamonic effect should be consideed caefully in a nonlinea system and depends on the non-linea ode (i.e. is non-lineaity of second o thid ode). Fo example fo cubic stiffness non-lineaity, 3 times fequency should be examined and, depending on the level of enegy in that mode, could be consideed o left out of the egion of inteest. he oveall study showed that the NL-RDM could allow a non-linea modal model to be identified. he levels of inaccuacy pesent fo expected levels of eo was consideed to be acceptable; even though high levels of inaccuacy wee sometimes encounteed, the eos wee highe than might be expected fom moden instumentation and methodologies. his is the natue of non-lineaity fo which thee is no pefect solution to non-linea system identification up to now. he method needs to be modified, but at the end of the day, NL-RDM is still in elatively ealy stages of exploation and development should continue. ACKNOWLEDGMEN he authos ae gateful to D Michael F. Platten fo helpful discussions on the methodology pesented in this pape. REFERENCES [] M. Samast, Identification of Non-Linea Dynamic Systems using the Non-Linea Resonant Decay Method (NL-RDM), PhD hesis, he Univesity of Mancheste, 6. [] M.F. Platten, J.R. Wight, J.E. Coope, M. Samast, Identification of multi-degee-of-feedom non-linea simulated and expeimental systems, in: Poceedings of the Intenational Semina on Modal Analysis (ISMA), Leuven,, pp. 95. [3] J.R. Wight, M.F. Platten, J.E. Coope, M. Samast, Identification of multi-degee-of-feedom weakly nonlinea systems using a model based in modal space, in: Poceedings of the Intenational Confeence on Stuctual System Identification, Kassel,, pp [4] S. Naylo, Identification of Non-popotionally Damped Stuctues using Foce Appopiation, PhD hesis, he Univesity of Mancheste, 998. [5] D. J. Ewins, Modal esting: heoy and Pactice, Reseach Studies Pess 995. [6] Mehdi Samast and Jan R. Wight, Sensitivity of NL- RDM o Eos In Paametes Used Fom he Undelying Linea System, Poceedings of the ASME 9 Intenational Mechanical Engineeing Congess & Exposition, Novembe 3-9, Lake Buena Vista, Floida, [7] R. Williams, J. Cowley and H. Vold, he Multivaiate Mode Indicato Function in Modal Analysis, Poceeding of 4 th Intenational Modal Analysis Confeence, Los Angeles 986. [8] J. R. Wight, M. F. Platten, J. E. Coope and M. Samast, Expeimental Identification of Continuous Non-linea Systems using an Extension of Foce Appopiation, st Intenational Modal Analysis Confeence, Floida, 3. [9] M. I. McEwan, J. R. Wight, J. E. Coope and A.Y.. Leung, A Finite Element echnique fo Non-linea plate and Stiffness Panel Response Pediction, AIAA Confeence Pape 595,. [] K. Woden and G. R. omlinson, Nonlineaity in Stuctual Dynamics, Institute of Physics Publishing,. ISSN: (Pint); ISSN: (Online)