Avoidance of numerical singularities in free vibration analysis of Euler-Bernoulli beams using Green functions

Transcription

1 WSEAS TRASACTIOS o APPLIED ad THEORETICAL MECHAICS Goraka Štimac Ročević, Braimir Ročević, Ate Skoblar, Saji Braut Avoidace of umerical sigularities i free vibratio aalysis of Euler-Beroulli beams usig Gree fuctios GORAKA ŠTIMAC ROČEVIĆ Departmet of Egieerig Mechaics Faculty of Egieerig, Uiversity of Rijeka Vukovarska 58, Rijeka, CROATIA gstimac@riteh.hr ATE SKOBLAR Departmet of Egieerig Mechaics Faculty of Egieerig, Uiversity of Rijeka Vukovarska 58, Rijeka, CROATIA askoblar@riteh.hr BRAIMIR ROČEVIĆ Desig departmet Vulka ova, d.o.o. Spičićeva, Rijeka, CROATIA brocev@yahoo.com SAJI BRAUT Departmet of Egieerig Mechaics Faculty of Egieerig, Uiversity of Rijeka Vukovarska 58, Rijeka, CROATIA sbraut@riteh.hr Abstract: This paper ivestigates the reliability of a algorithm that implemets the Gree fuctio method i free vibratio aalysis of Euler-Beroulli beams. The ivestigatio is cocered with the robustess of the algorithm with respect to the occurrece of umerical sigularities i the calculatio procedure of mode shapes. The problem is studied for beams supported with a arbitrary umber of itermediate traslatioal sprigs, which ca be uderstood as a geeralizatio of the cases whe the beam is without elastic supports ad whe the beam rests o itermediate rigid supports. The problem of umerical sigularities arises from the fact that the elemets of the modal vector have to be expressed i terms of a "arbitrarily" chose referetial elemet of that vector, whose value ca vaish if it coicides closely eough with a ode of the sought mode shape. The problem is geerally tackled here with the itroductio of a fictitious sprig of a vaishigly small stiffess, ad the robustess of the algorithm depeds crucially o the appropriate placemet of that sprig. This paper presets several useful guidelies for the implemetatio of computer code based o these priciples ad its reliability is demostrated through examples. Key-Words: umerical sigularity, Gree fuctios, free vibratios, Euler-Beroulli beam, sprig support Itroductio The methods of derivig closed form expressios for the frequecy equatio ad the mode shape equatio of vibratig beams are a well-studied field of research. For Euler-Beroulli beam with various boudary coditios these aalytical solutios ca be foud i stadard textbooks [], ad their computatioal implemetatio geerally does ot etail ay complicatios regardig accuracy or sesitivity to the iput parameters. However, i more complex cases whe the beam carries ay kid of itermediate elemets ad restraits (sprigs, masses, dampers etc.), the mathematical complexity rises progressively with each elemet added ito the model. This mathematical complexity is, i tur, ofte accompaied by algorithmic complexity which is required to implemet such mathematical models. This paper is cocered with the method of Gree fuctios, whose primary weakess is the iability to produce some mode shapes whe iappropriately implemeted. We shall demostrate that this problem ca reliably be overcome with the itroductio of a coveietly placed fictitious sprig of egligible stiffess. By doig so, the full potetial of the Gree fuctio method ca be exploited practically without restrictios; with the two most promiet advatages of the Gree fuctio method beig straightforward extesio from free vibratio to forced vibratio aalysis ad the relative ease of icorporatig poit applied loads ad/or restrictios ito the model. For the purpose of this study, the beam is assumed to be supported with a arbitrary umber of traslatioal sprigs, which is a good approximatio i a umber of real structural problems. A good geeral isight ito the applicatio of Gree fuctios i beam vibratio aalysis ca, for example, be foud i [-7], ad this paper is a cotiuatio of the work preseted by the authors i [8]. This paper is structured as follows. Chapter presets the mathematical model of the stated problem, ad the aalysis of algorithmic robustess of the preseted approach ad problems ecoutered i some characteristic cases is give i Chapter 3. Chapter summarizes the results with cocludig remarks. E-ISS: Volume 3, 8

2 WSEAS TRASACTIOS o APPLIED ad THEORETICAL MECHAICS Goraka Štimac Ročević, Braimir Ročević, Ate Skoblar, Saji Braut Mathematical model The subject uder ivestigatio is a beam (Fig. ) supported with a arbitrary umber of traslatioal sprigs. I this case, the trasverse displacemet ca be obtaied from the differetial equatio: wxt (, ) wxt (, ) EI A k( x) w( x, t), () x t where w(x,t) is the deflectio, EI is flexural rigidity, ρ is the desity ad A is the cross-sectioal area of the beam. The support stiffess at positio x is desigated as k(x). If the beam is supported with liear sprigs (at locatios x, = ), it ca be writte k( x) k ( xx ), () where k is the stiffess of the -th sprig, ad δ is the Dirac delta fuctio. z x k Fig.. A beam pied at both eds supported with traslatioal sprigs. Gree fuctios The atural frequecies ω ad the mode shapes W(x) of the aalysed beam are determied from the homogeeous solutio of the differetial equatio (). Usig the method of separatio of variables, the solutio is assumed i the followig form: wxt (, ) W( xe ) it. (3) After substitutig (3) ito () ad itroducig the odimesioal variables with respect to the beam legth L: x x, ad L L W W, () L the left had side of equatio () is rewritte as: d W ( ) W( ) W ( ) K ( ). (5) d Here, ε is the odimesioal atural frequecy ad K is the odimesioal stiffess, defied as: x A L EI, (6) k x 3 K k L EI. (7) Moreover, by usig the stadard feature of the Dirac fuctio: W ( ) ( ), W ( ) ( ) (8),, the differetial equatio ca be writte as: d W ( ) W( ) W( ) ( ) K. (9) d The Gree fuctio method is applied here to solve the differetial equatio i (9). I the first step, the Gree fuctio G(ξ, ξ ) must be itroduced, which by defiitio satisfies the differetial equatio whe the forcig term is equal to the Dirac delta fuctio, that is: d G(, ) d G(, ) ( ). () Secodly, the Gree fuctio for a particular beam is obtaied by solvig equatio () i cojuctio with the set boudary coditios. The Gree fuctio for a pied-pied boudary coditio ca be show to be: a G(, ) 3 3sih( ) b si( ) H ( ), with a si( )si( )sih( ) sih( )sih( )si( ), b si( )sih( ). () Here, H( ) is the Heaviside fuctio. For more details the reader is referred to [8].. Frequecy equatio The solutio of (9) ca ow be writte as: W( ) W ( ) K G(, ). () I case whe the beam is supported with oe sprig equatio () is evaluated at positio ξ as: (, ) W ( ). (3) I case whe the beam is supported with two sprigs at positios ξ ad ξ, equatio () is evaluated at ξ ad ξ, yieldig: E-ISS: Volume 3, 8

3 WSEAS TRASACTIOS o APPLIED ad THEORETICAL MECHAICS Goraka Štimac Ročević, Braimir Ročević, Ate Skoblar, Saji Braut W( ) W( ) (, ) W( ) (, ), () W( ) W( ) K G(, ) W( ) K G(, ) or i matrix form: (, ) (, ) W ( ). (5) (, ) (, ) W ( ) Geerally speakig, by evaluatig equatio () for sprig positios (ξ = ξ m, m = ) a system of liear equatios is obtaied, which ca be assembled i matrix form as KW, (6) where the mode shape vector W is defied as: W ( ) W ( ) W. (7) W ( ) The ecessary coditio for a otrivial solutio of the vector W is: K. (8) This equatio is kow as the frequecy equatio. Its solutios are odimesioal frequecies ε, from which the atural frequecies are obtaied as EI AL. (9) Comparig expressios (3) ad (5) with (6) ad (8), the frequecy equatios for a beam supported with oe ad two sprigs are obtaied as: (, ), () (, ) (, ). () (, ) (, ).3 Mode shape equatio For each of the calculated odimesioal atural frequecies ε, the mode shape ca ow be obtaied from (). The first step is to choose oe elemet of the mode shape vector, for example W ( ), as the referetial oe ad to express the remaiig elemets relative to this referece. From () this ow yields: W ( ) W ( ) (, ) (, ). () W( ) ( ) W Afterwards, this equatio is evaluated for all sprig positios (ξ = ξ m ) ad a system of liear equatios is obtaied, which is assembled i matrix form as: KW R, (3) where: W ( ) (, ) W ( 3) W (, ), R W ( ). () W ( ) (, ) The matrix K is obtaied by removig the first colum ad the last row from the matrix K. The solutios of (3) are all members of the vector W, expressed relative to the chose oe, i this case W ( ). For example, for =, from equatio (3) it follows: (, ) ( ) W ( ). (5) (, ) W The mode shape equatios for a beam supported with oe ad two sprigs are ow obtaied as: W ( ) (, ) W ( ), (6) W ( ) (, ) (, ) G(, ). (7) W ( ) (, ) G 3 Model implemetatio, examples To illustrate the preseted theory ad the problems associated with its implemetatio a pied-pied beam is studied, keepig i mid that the preseted priciples equally apply to other cases of boudary coditios. All examples preseted i this paper are implemeted i Matlab. If the beam is supported with oly oe sprig, the atural frequecies are obtaied by solvig the frequecy equatio (), where the Gree fuctio G for a pied-pied beam is defied i (). Followig this, for ξ < ξ, it is obtaied: a K. (8) 3 b I order to speed up umerical calculatios, this equatio is rewritte i the umerator/deomiator form ad the zeros are ow obtaied oly for the umerator 3 f( ) K a b, (9) which correspod to the zeros of equatio (8). For example, if we take K = ad ξ =. the atural E-ISS: Volume 3, 8

4 996 WSEAS TRASACTIOS o APPLIED ad THEORETICAL MECHAICS Goraka Štimac Ročević, Braimir Ročević, Ate Skoblar, Saji Braut frequecies are obtaied from (8) as preseted i Fig. ad for K = ad ξ =. ad ξ =.5 from (9) as give i Fig. 3. becomes the ew first mode. The same pheomeo ca be see i Fig. at ξ =.5, where for K > K cr the curves related to the first mode shape touch the curves related to the secod mode shape. 5 f() Fig.. Zeros of the frequecy equatio as defied i (8) for K = ad ξ =..5 f() f() x x Fig. 3. Zeros of the frequecy equatio as defied i (9) for K = ad a) ξ =., b) ξ =.5 A more iformative diagram is obtaied if Eq. (9) is solved iteratively for various sprig positios ad stiffess values, Fig.. Here, each plotted curve shows the chage of the o-dimesioal atural frequecy with respect to the positio of the sprig for a costat value of K. Additioal isight is obtaied after plottig the odimesioal frequecies as fuctios of the odimesioal stiffess while the positio of the sprig is held costat, as show i Fig. 5. At ξ =.5 the critical value of K is foud umerically as K cr = 996. For K > K cr, o further icrease i ε is gaied with icreased stiffess. Hece, the origial secod mode Fig.. First three values of ε for a pied-pied beam for differet values of K ad ξ ; the results are plotted for five values of K ;, 5, 5, 996, K K K K Fig. 5. Depedece of odimesioal frequecies ε, ε ad ε 3 o K for pied-pied beam supported with a sigle sprig at a) ξ =., b) ξ =.5 (Δ FEM results) The mode shapes ca be obtaied from equatio (6) relative to the deflectio at the poit of applicatio of the first sprig. However, the problem arises if this deflectio approaches zero ad the modal deflectios calculated relative to such referetial value become ifiite. This ca geerally happe i two cases: () whe the sprig stiffess is too high, ad K E-ISS: -39 Volume 3, 8

5 WSEAS TRASACTIOS o APPLIED ad THEORETICAL MECHAICS Goraka Štimac Ročević, Braimir Ročević, Ate Skoblar, Saji Braut () if the mode shape has a zero at the poit where the sprig is located (regardless of the sprig stiffess). The problem of zero deflectio at the locatio of the sprig ca be solved with the itroductio of a fictitious sprig of egligible stiffess (K ) placed at a coveiet positio to the left of the positio of the sprig (i.e. ξ < ξ =.5). The chose positio must ot be i the immediate viciity of the expected modal zeros of the aalysed mode shapes. The system is ow modelled as a system with two sprigs, where the first sprig is fictitious. Fig. 6 presets mode shapes with sprig stiffess K = at ξ =.5 ad for two cases: with o fictitious ad with a fictitious sprig at ξ = mode mode mode mode - mode mode Fig. 6. Mode shapes for K = at ξ =.5; a) with o fictitious sprig the algorithm failed to produce the st ad the th mode shape, b) with a fictitious sprig at ξ =. It ca be observed that the algorithm failed to produce the st ad the th mode shape, whose zeroes coicide with the positio of the sprig (ξ =.5). I cotrast, with the itroductio of the fictitious sprig all desired mode shapes are preseted correctly. At the positio of the real sprig i Fig. 6a (ξ =.5) as well as at the positio of the fictitious sprig i Fig. 6b (ξ =.) the modal amplitudes equal uity i all mode shapes. This happes because the modes are always scaled with respect to the displacemets of the referetial sprig ad this is i accordace with the defiitio of mode shapes i Eq. (). The mode shapes of a beam with sprig stiffess K = at ξ =.5 are show i Fig. 7 for two cases: with o fictitious sprig ad with a fictitious sprig at ξ =.. 8 x 37 6 mode mode mode mode - mode mode Fig. 7. Mode shapes with K = at ξ =.5; a) with o fictitious sprig the algorithm failed to produce the st ad the 3 rd mode shape, b) with a fictitious sprig at ξ =. It must be oted that the preseted approach does ot rely o fidig some uiversally applicable locatio of the fictitious sprig, because that is impossible. A appropriate positio ca be deduced oly i the cotext of the structural layout of the problem at had ad the umber of modes that the aalyst wishes to obtai. As the structural complexity rises, so does the possibility of upredictable locatios of odes at higher order mode shapes, which might coicide with ξ. A more coveiet scalig of the mode shape amplitudes ca be obtaied with mass ormalizatio, which eables the compariso with the fiite elemet results, as show i Fig. 8 for two positios ξ =. ad ξ =.5, ad both with K =. As expected these results do ot deped o the positio of the fictitious sprig ad a complete agreemet with the results of the fiite elemet aalysis is observed. For that purpose all the required quatities had to be kow, therefore a solid rectagular beam with the followig geometrical ad material properties was aalysed: width.3 m, height.5 m, legth m, desity ρ = 785 kg/m 3 ad the modulus of elasticity E =. /m. If we assume that the beam is supported with two sprigs at positios ξ ad ξ, the frequecies of the beam ca be obtaied from the frequecy equatio (), with the Gree fuctio G for a pied-pied beam defied i (). E-ISS: -39 Volume 3, 8

6 WSEAS TRASACTIOS o APPLIED ad THEORETICAL MECHAICS Goraka Štimac Ročević, Braimir Ročević, Ate Skoblar, Saji Braut mode - mode mode - mode Fig. 8. Mass ormalized mode shapes with sprig stiffess K = at positio a) ξ =., b) ξ =.5 Fig. 9 presets the depedece of ε o sprig stiffesses K ad K whe the sprigs are located at ξ =/3 ad ξ =/3 (at zero poits of the third mode shape). The obtaied surface has a plateau where a icrease i either K ad/or K o loger produces a icrease i ε. This meas that the origial third mode becomes the ew first mode. K 9.8 Fig. 9. Depedece of odimesioal frequecy ε o K ad K for a pied-pied beam supported with two sprigs at ξ = /3 ad ξ = /3 K itroductio of a fictitious sprig of a egligible stiffess which must always be positioed so as to avoid modal odes. Some useful geeral guidelies o how to resolve these issues whe implemetig this method are demostrated trough preseted examples. Refereces: [] Karovsky, I. A.; Lebed, O. I., Formulas for Structural Dyamics: Tables, Graphs ad Solutios, McGraw-Hill,. [] Kukla, S., The Gree fuctio method i frequecy aalysis of a beam with itermediate elastic supports, Joural of Soud ad Vibratio Vol. 9, o., 99, pp [3] Kukla, S., Free vibratio of a beam supported o a stepped elastic foudatio, Joural of Soud ad Vibratio, Vol. 9, o., 99, pp [] Kukla, S.; Posiadala, B., Free vibratios of beams with elastically mouted masses, Joural of Soud ad Vibratio, Vol. 75, o., 99, pp [5] Kukla, S., Applicatio of Gree fuctios i frequecy aalysis of Timosheko beams with oscillators, Joural of Soud ad Vibratio, Vol. 5, 997, pp [6] Mohamad, A., Tables of Gree s fuctios for the theory of beam vibratios with geeral itermediate appedages, Iteratioal Joural of Solids ad Structures, Vol. 3, o., 99, pp [7] Abu-Hilal, M., Forced vibratio of Euler- Beroulli beams by meas of dyamic Gree fuctios, Joural of Soud ad Vibratio, Vol. 67, o., 3, pp [8] G. Štimac Ročević, B. Ročević, A. Skoblar, S. Braut, A comparative evaluatio of some solutio methods i free vibratio aalysis of elastically supported beams, Joural of the Polytechics of Rijeka, Vol.6, o., 8, pp Coclusio I this paper the robustess of the algorithm for free vibratio aalysis based o Gree fuctio method is ivestigated. The preseted problem is especially cocered with umerical sigularities which ievitably appear i the calculatio procedure of mode shapes if the equatios are formulated oly based o the real structure. The problem is overcome with the E-ISS: -39 Volume 3, 8