Large Amplitude Free Vibrations of Mindlin Square Plates A Novel Formulation

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1 Reserch Article Interntionl Journl o Current Engineering nd echnology E-ISSN , P-ISSN INPRESSCO, All Rights Reserved Aville t rge Amplitude Free Virtions o Mindlin Squre Pltes A Novel Formultion K.Meer She Ȧ * nd G.Venkteswr Ro Ḃ Ȧ eprtment o Mechnicl Engineering, University college o engineering(a,jnu Kkind , Andhr Prdesh, Indi Ḃ eprtment o Mechnicl Engineering, Vrdhmn College o engineering, Shmshd, Hyderd, 50 8, Andhr Prdesh, Indi. Accepted 0 Jnury 04, Aville online 0 Ferury 04, Specil Issue-, (Ferury 04 Astrct A novel ormultion to study the lrge mplitude ree virtion ehvior o uniorm modertely thick squre pltes with xilly immovle edges is presented. he ormultion is very simple s the lrge mplitude ree virtion ehvior is ccurtely evluted knowing the totlly unrelted quntities like the tensions developed in the plte due to lrge delections nd the corresponding uckling lod. he sis or this ormultion is the recent work o the uthors on predicting the ree virtion ehvior o initilly loded pltes. his novel ormultion involves three sic steps, through which the lrge mplitude ree virtion ehvior o squre pltes cn e very esily nd ccurtely evluted y ollowing the ssumed spce mode (ccurte single term dmissile unctions depending on the oundry conditions pproch used y mny reserches. Numericl results, with emphsis on the undmentl requency, or the uniorm modertely thick squre pltes, re presented in the digitl orm in this pper long with the stndrd solutions other reserchers, to conclusively demonstrte the eiccy o the strtling simplicity o the present novel ormultion. Keywords: Free virtion etc.. Introduction Pltes, whether thin or modertely thick, re the most widely used structurl memers in mny erospce structurl elements which re highly optimized. Evlution o the lrge mplitude ree virtions o such pltes is essentil or studying the dynmic ehvior o the structurl memers under the inluence o severe dynmic environment. he sic requirement is to ind the vrition o the requency o ree virtion o the reltively complex modertely thick squre pltes, or given mode, with respect to given mximum mplitude o virtion. Bsed on this inormtion urther investigtions on the dynmic ehvior o these structurl memers ollow. his pper presents novel ormultion to study the lrge mplitude ree virtions o modertely thick squre pltes, where the eects o sher deormtion nd rotry inerti hve to e considered, with ll edges simply supported. he clssic work on the topic o lrge mplitude ree virtions o hinged-hinged ems is initilly due to Woinowsky-Krieger who considered the nonlinerity involved in the strin-displcement reltion (von-krmn type tking into ccount the constnt xil tension developed in the em ecuse o the lrge delections with the xilly immovle end conditions. his tension term when included in the dynmic eqution o equilirium, mkes it nonliner (hrdening type nd the *Corresponding uthor: K.Meer She solution is otined in terms o the elliptic integrls, with n ssumed spce mode or the hinged-hinged em. It is the endevor o the uthors to develop strtlingly simple nd t the sme time ccurte nd relile solutions, to some complex prolems o prcticl interest in structurl mechnics, nd successully developed such ormultions or predicting the undmentl requency prmeter o initilly loded structurl memers. he uthor s hve een successully studied lrge mplitude ree virtion ehvior o uniorm sher lexile em nd thin pltes using simple novel ormultion. With the sme spirit nd ollowing the erlier work, the uthors hve successully developed novel ormultion or studying the lrge mplitude ree virtions o uniorm modertely thick squre pltes with immovle edges, where the nonlinerity exists in the strin-displcement reltion o von-krmn type. In the present pper single term spce mode pproch is used nd the eect o the nonlinerity on the undmentl requency, is studied, ollowing the work o mny reserchers on this topic. his ormultion is clled novel ecuse it is sed on the knowledge o two totlly unrelted quntities like xil tensile lods developed in the modertely thick squre pltes due to lrge delections nd the corresponding uckling lods due to the ixil compressive lods t the edges. he sic steps involved in the present novel ormultion re: i Presenting riely, or the ske o completeness, the derivtion o n lredy developed simple design ormul OI: Interntionl Conerence on Advnces in Mechnicl Sciences 04

2 K.Meer She et l Interntionl Journl o Current Engineering nd echnology, Specil Issue- (Fe 04 to otin the requency prmeters o initilly loded pltes, ii Applying the design ormul given in step (i to otin the rtios o the nonliner to liner rdin requencies in terms o the mximum mplitude o virtion o the squre pltes y using the tensions developed due to the lrge delections s the initil lod, where the nonliner rdin requency is otined y using the SHM ssumption nd iii Finlly, to otin the rtios o the nonliner to liner rdin requencies y eliminting the SHM ssumption using the hrmonic lnce method[ (HBM, tht provides correction ctor which corrects the error induced ecuse o the SHM ssumption. he present ormultion is minly sed on the physicl concepts o the prolem nd corresponding logicl deductions without much complex mthemticl tretment. Following these three sic steps, the numericl results o reltively complex prolem o the lrge mplitude ree virtions o isotropic modertely thick squre pltes with immovle edges re presented in the digitl orm. Solution or the cse o ll edges simply supported re demonstrted nd given long with other reserchers using complex mthemticl ormultions re used to ssess the ccurcy, reliility nd simplicity o the present novel ormultion. he mjor three steps, mentioned erlier, used in the proposed ormultion re riely presented in the ollowing sections. Initilly oded Free Virtions o Pltes he liner ree virtion ehvior o squre pltes due to externlly pplied concentrted xil lod cn e ccurtely evluted ollowing the nlysis methods like the Ryleigh-Ritz method, weighted residul methods, corresponding inite element ormultions etc. he inl governing eqution o equilirium is otined in the mtrix orm, ssuming SHM, s [K] {}- i [G] {}- [M] {}=0 ( It cn e seen rom Eq. ( tht or positive (compressive initil lod, the plte is less sti nd the corresponding nturl requencies decrese nd when the initil lod is negtive (tensile the structurl system gets stiened nd the corresponding nturl requencies increse. From Eq. (, the degenerte cse o the uckling eqution, y neglecting the mss mtrix, is given y [K] { } = [G] { } ( Eqution ( cn e written s [G] { }= (/ [K] { } (3 Similrly, the degenerte cse or the ree virtion prolem without the initil lod cn e otined rom Eq. (, y neglecting the geometric stiness mtrix, s [M] { } = (/ 0 [K] { } (4 Sustituting Eqs. (3 nd (4 in Eq. (, we get i [K] [ K]{ } [ K]{ } 0 (5 with resonle nd vlid ssumption, prticulrly or the irst uckling nd ree virtion modes, { } { } {} (6 Eqution (5 ecomes [K] i (7 [ K]{ } [ K]{ } 0 or [K] {}- [ K]{ }( / / 0 (8 Eqution (8 implies tht i i 3 Rtios o Nonliner to iner Rdin Frequencies with SHM Assumption H he prmeters nd in Eq. (9 cn e interpreted s 0, the nonliner requency prmeter nd ( m 0, the liner requency prmeter nd the initil lod prmeter s i (tensile rising due to lrge delections s mentioned in the previous section. With these interprettions, Eq. (9 cn e written s H H (0 4 Correction to Rtios o Nonliner to iner Rdin Frequencies Otined with SHM Assumption he procedure to evlute o H / is given in the previous section, with the ssumption o SHM, which is not relistic, s the lrge mplitude ree virtion ehvior, even though periodic, does not exhiit SHM. However, the liner ree virtions ollow the SHM. he ollowing mthemticl tretment, pplying the HBM[0], gives correction ctor o 3/4, which compenstes the error cused ecuse o the ssumption o SHM in otining the rtios o / H 5 Appliction o Novel Formultion to ll Edges Simply Supported Modertely hick Squre Plte he procedure to evlute o non dimensionl ension Prmeter is given s ollows he tension developed in the plte in x-direction s Eh dw( x x dx 0 dx ( x y w sin sin ( Sustituting Eq.( in Eq.( nd ter simpliiction Eh 4 (3 3 Eh Sustituting in the ove eqution (9 545 Interntionl Conerence on Advnces in Mechnicl Sciences 04

3 K.Meer She et l Interntionl Journl o Current Engineering nd echnology, Specil Issue- (Fe 04 le: Buckling lod prmeters( or ll edges simply supported modertely thick squre plte with dierent thickness rtios(h/ h/ le : / vlues o ll edges simply supported squre plte or undmentl mode. h/ PS FEM FEM[ PS [] ] PS FEM[] PS FEM[] PS FEM[] PS FEM[] le3: Buckling lod prmeters( or ll edges clmped modertely thick squre plte with dierent thickness rtios(h/ h/ le 4: / vlues o ll edges clmped squre plte or undmentl mode. h/ PS FEM[] PS FEM[] PS FEM[] PS FEM[] PS FEM[] PS FEM[] h 3 h (4 he tension developed in the plte does not contin ny sher lexile terms. he non-dimensionl tension prmeter rom the ove expression s x 3 (5 h he non dimensionl uckling lod prmeter when squre plte sujected to ixil compression long the edges ( the edges x=0, x= sujected N x per unit length nd the edges y=0, y= sujected N per unit length N x (6 h Numericl Results nd iscussion As the present ormultion to evlute the lrge mplitude ree virtion ehvior o structurl elements requires the xil tensile lods developed due to the lrge delections nd the uckling lod prmeter otined consistent with the tensile lods developed, i.e., edge compressive lods, which re evluted y suitly y 546 Interntionl Conerence on Advnces in Mechnicl Sciences 04

4 K.Meer She et l Interntionl Journl o Current Engineering nd echnology, Specil Issue- (Fe 04 selecting the single term dmissile unctions with one undetermined coeicient or the trnsverse displcement w. Further, or ormultion like the one presented here tht is minly sed on the physicl concepts nd susequent logicl conclusions, it is essentil to show how the vlues o H otined initilly nd computed using the correction ctor mtch with the FEM[ existing solution, ville in the open literture. Buckling lods which re required to evlute lrge mplitude ree virtion ehvior re given in lewith ll edges simply supported nd or dierent thickness rtios nd re given in le. he present results compre very well with those otined using the inite element ormultion nd re given in le or ll edges simply supported squre plte. It is worth mentioning here tht the present ormultion directly gives the rtios o / H nd the rtios / re otined y correcting these vlues using HBM. Similr uckling lods nd time period rtios re given in respectively in tle3 nd le4 or ll edges clmped squre plte nd ound tht the present results nd FEM results re very closely mtching. Conclusions he eiccy o the proposed novel ormultion or studying the lrge mplitude ree virtions o modertely thick squre pltes hs een conclusively demonstrted in this pper. he present ormultion is strtling simple ecuse it is primrily sed on the physicl concepts involved in the prolem nd susequent logicl deductions with very little mthemtics, wheres, the erlier studies on this topic re highly mthemticlly oriented. he uthors re o the opinion tht the present ormultion, which involves three simple sic steps, ttrcts oth the nlysts nd reserchers elonging to the sic engineering reserch. Most o the reserchers on this topic concentrted on the irst mode o virtion (undmentl requency while studying the lrge mplitude ree virtion phenomenon, ssuming simple ut ccurte single term dmissile unctions. Hence, the uthors re lso tken the single term dmissile unctions with one undetermined coeicient nd studied the lrge mplitude ree virtion ehvior o modertely thick squre pltes. he results otined y present simple ormultion very closely mtches with the FEM nd is given in le nd le4. Acknowledgements he uthor is thnkul to the uthorities o University College o Engineering(A, JNU-Kkind or sponsoring the necessry economicl support or presenting the pper under EQIP phse. Second uthor grteully cknowledges the support given y INAE during the course o this work. Reerences G.V.Ro,, nd N.R. Nidu, (00 Prediction o Fundmentl Frequencies o Stressed Spring-Hinged pered Bems, AIAA J.,39, pp G.V. Ro, (00, Simple Formul to Predict the Fundmentl Frequency o Initilly Stressed Squre Pltes, J. Sound Vi., 46, pp.85-\ G.V. Ro, ndk.k. Rju, (00 Simple Formul or Evluting the Fundmentl Frequency Prmeter o Initilly Stressed Uniorm Bems on Elstic Foundtion, Fire echnol., 4, pp.4 G.V.Ro, nd R.Neeth, (003 Prediction o Fundmentl Frequency o Initilly In plne oded hick Circulr Pltes, J. Sound Vi., 59, pp G.V.Ro, S. Renjit, nd G.Rjeev Nir, (004 esign Formul to Evlute the Fundmentl Frequency o hin Annulr Pltes Sujected to Uniorm Compressive od t the outer Edge, Advnces in virtion Engineering, 3, pp.5-60 G.V.Ro, nd S.Renjit, (005 Applicility o the esign Formul or Predicting the Fundmentl Frequency o Initilly Stressed Modertely hick Annulr Pltes, Journl o Structurl Engineering, 3, pp G. V. Ro, K.Meer She nd G. Rng Jnrdhn, (008 Simple ormul to Study the lrge Amplitude ree Virtions o Bems nd Pltes, J.Appl.Mech., 75, pp.-4. G.V.Ro, K. Meer She, ndg. Rng Jnrdhn, (007 rge Amplitude Free Virtions o Uniorm imoshenko Bems- A Novel Formultion, AIAA J., 45, pp azrr, nd R.G.White, (999 A Semi Anlyticl Approch to the Nonliner ynmic Response Prolems o S-S nd C-C Bems t rge Virtion Amplitudes, Prt I: Generl heory nd Appliction to the Single Mode Approch to Free nd Forced Virtion Anlysis, Journl o Sound nd Virtion, 4, pp K. Meer She, G.V. Ro, ndg. Rngjnrdhn, (009 Buckling nd Free Virtion o Modertely hick Squre Pltes Using Coupled isplcement Field Method, Journl o structurl Engineering,36,PP K.K. Rju, G.V. Ro, ndi.s. Rju, (978 Eect o Geometric Nonlinerity on he Free Flexurl Virtions o Modertely hick Rectngulr Pltes, computers nd structures, 9, pp Nomenclture = squre plte dimention m = mximum mplitude t the centre o the squre plte [G] = system geometric stiness mtrix I = re moment o inerti [K] = elstic stiness mtrix m = mss o the em per unit length [M] = system mss mtrix = non liner time period = liner time period X = tension developed in x direction due to lrge delections {} = eigenvector (virtion mode shpe with initil lod { } = eigenvector (uckling mode shpe { } = eigenvector (virtion mode shpe without initil lod = N uckling lod prmeter ( xcr = 4 requency prmeter with initil lod ( Woinowsky-Krieger, S.,(950 he Eect o n Axil Force on the Virtion o Hinged Brs, SME J.Appl.Mech., 7, pp = requency prmeter without initil lod ( Interntionl Conerence on Advnces in Mechnicl Sciences 04

5 K.Meer She et l Interntionl Journl o Current Engineering nd echnology, Specil Issue- (Fe 04 N i = Initil lod prmeter ( x = nonliner requency prmeter( 4 H = nonliner requency prmeter with SHM H ssumption ( 4 = xil tensile lod prmeter ( x = Poission s rtio (tken s 0.3 = liner requency prmeter with initil lod = nonliner rdin requency without SHM ssumption 0 = requency prmeter without initil lod = liner rdin requency = nonliner rdin requency with SHM ssumption H 548 Interntionl Conerence on Advnces in Mechnicl Sciences 04