Dynamical stability and parametrical vibrations of the laminated plates with complex shape

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1 (3) Dyamical stability ad parametrical vibratios of the lamiated plates with comple shape Abstract The problem of oliear vibratios ad stability aalysis for the symmetric lamiated plates with comple shape loaded by static or periodic load i-plae is cosidered. I geeral case research of stability ad parametric vibratios is coected with may mathematical difficulties. For this reaso we propose approach based o applicatio of R-fuctios theory ad variatioal methods (RFM).The developed method takes ito accout pre-buckle stress state of the plate. The proposed approach is demostrated o testig problems ad applied to lamiated plates with cutouts. The effects of geometrical parameters load boudary coditios o stability regios ad oliear vibratios are ivestigated. Keywords idiya Kurpa Olga Mazur Victoria Tkacheko idiya Kurpa The Kharkov State Techical Uiversity Departmet of Applied Mathematics UKRAIE; Kurpa@kpi.kharkov.ua Olga Mazur The Kharkov State Techical Uiversity Departmet of Applied Mathematics UKRAIE; OlgaMazur@mail.ru Victoria Tkacheko The Kharkov State Techical Uiversity Departmet of Applied Mathematics UKRAIE; victoria_tkacheko@mail.ruuiversity of Tehra College of Egieerig Egieerig Optimizatio Research Group Tehra Ira Correspodig Author; eorg@ut.ac.ir; Tel.: Fa: ITRODUCTIO amiated composite plates are frequetly used i various egieerig applicatios i the aviatio aerospace marie mechaical ad others idustries. The use of composite materials requires comple aalytical ad umerical methods i order to predict accurately their respose to eteral loadig. A recet review of curret developmets i o-liear vibratio aalysis ad dyamic stability of the lamiated plates ad shells has bee preseted i refereces [6934] ets. From preseted review it follows that bucklig problems with o-uiform pre-buckle stress state are of special iterest. There are few works i which vibratio bucklig ad parametric istability behavior of a lamiated plate with iteral cutouts were studied []ets. It should be oted that the fiite elemet method (FEM) remais the oly way for dealig with comple structures ad the most versatile. I this work we propose alterative to FEM approach based o usig variatioal methods ad R-fuctios theory (RFM). The developed method takes ito accout pre-buckle stress-state ad allows ivestigatig the lamiated plates of a arbitrary form i particular case with iteral free simply supported ad clamped cutouts. Formerly this approach was developed i refereces [3] for isotropic ad orthotropic plates. The aim of this paper is etedig the theoretical model of this ati America Joural of Solids ad Structures (3) 75 88

2 76 idiya Kurpa Olga Mazur ad Victoria Tkacheko / Dyamical stability ad parametrical vibratios of the lamiated plates with comple shape method i order to carry out the oliear aalysis of lamiated plates with a arbitrary shape ad differet boudary coditios. The dyamic istability ad bucklig characteristics of the lamiated plates with cutouts are discussed. PROBEM FORMUATIO Geometrically oliear vibratio of simmetricaly lamiated composite plates subjected to iplae compressive periodic edge loadig p p pt cos t is studied. It is assumed that the delamiatio of the layers is abset. The mathematical formulatio of the problem is made i the framework of the classical lamiated plate theory. eglectig rotatory iertia the equatio of equilibrium [] may be writte as: () y () y M M M w w w mw (3) y yy y yy tt where ad M M M are stress resultats: C C C 6 C C C 6 C C C M D D D w 6 M M D D D w 6 yy M D D D w y. Deformatios T are epressed as [] u w v y y w v u y w w y where uvw are displacemets of the plate i directios O Oy ad Oz respectively. For coveiece we itroduce the followig otatio where T T ati America Joural of Solids ad Structures (3) 75 88

3 idiya Kurpa Olga Mazur ad Victoria Tkacheko / Dyamical stability ad parametrical vibratios of the lamiated plates with comple shape 77 u v y v u y w w y w w. y The stress resultats ad may be preseted as follows T T ( ) ( ) C ( ) ( ) C. as[]: The compoets C D ij 6666 of the stiffess matrices C ad D are defied ij ij hs s ij ij ij s h C D B z dz s where B () s ij are mechaical characteristics of the s-layer. The system of equatios ()-(3) is supplemeted by correspodig boudary coditios. 3. METHOD OF SOUTIO The proposed method is based o solvig a umber of auiliary problems.. Solvig problem about pre-buckle stress state of the lamiated plate. I order to determie the pre-buckle stress state of the plate let us cosider the followig system y y (4) supplemeted o the loaded part of the border by the followig boudary coditios u v T u v y (5) ( ) ( ) Operators T i (5) is defied by formulas l m lm T l m lm ati America Joural of Solids ad Structures (3) 75 88

4 78 idiya Kurpa Olga Mazur ad Victoria Tkacheko / Dyamical stability ad parametrical vibratios of the lamiated plates with comple shape where l cos( O) m cos Oy is ormal vector to border of the domai. Boudary coditios o the remai part deped o way of fiig. Solutio u y v y of this system (4)-(5) may be foud by RFM (R-fuctios method). Variatioal formulatio of the problem (4)-(5) is reduced to fidig miimum of the fuctioal I( u v ) ( ) d u l v m d ( ) ( ) ( ) ( ) (6) We seek the miimum of the fuctioal (6) o the set of basis fuctios costructed by R- fuctios theory [78]. Forces () T are defied after fidig solutios u ad v () C u v u v where () () () () T u v ( ) () u ( ) y v ( ) u v. y. Fidig bucklig load. et us fid the critical load provided that compressive load is varied proportioally to some parameter that is from the followig equatio 33w l3 u v w. (7) is foud by eergy approach. et us write the appropri- The critical value of the parameter ate fuctioal: I( w) [( M M M ) ( ( w ) y y ( w ) w w )] ddy. As before miimizatio of the fuctioal is performed o the set of basis fuctios costructed usig the RFM. As a result of the Ritz s method this problem is reduced to the eigevalue problem. 3. Solutio liear vibratio problem of the lamiated plate subjected to i-plae compressive static edge loadig p. Above metioed problem is solved by RFM too. The correspodig fuctioal is ati America Joural of Solids ad Structures (3) 75 88

5 idiya Kurpa Olga Mazur ad Victoria Tkacheko / Dyamical stability ad parametrical vibratios of the lamiated plates with comple shape 79 I U V (8) where U is total potetial eergy takig ito accout the ifluece of forces i the media plae: U( w) M M M y y p ( u v )( w ) ( u v )( w ) ( u v ) w w d ad V is kietic eergy: m V w d. et us deote eigefuctio correspodig to the first mode of free liear vibratios of loaded plate by w is relative atural frequecy. 4. Solutios of the auiliary problem like elasticity problem. et us cosider the followig system of the equatios: y l w y l w (9) The right side of the system (9) has the followig kid y l w l w y This system is supplemeted by the boudary coditios: ( ) ( ) ( ) ( ) w T T w () where l m lm T l m lm. The solutio of system (9)-() is foud by RFM. Deote this solutio by u y v y. Variatioal formulatio of the problem is reduced to fidig miimum of the followig fuctioal ati America Joural of Solids ad Structures (3) 75 88

6 8 idiya Kurpa Olga Mazur ad Victoria Tkacheko / Dyamical stability ad parametrical vibratios of the lamiated plates with comple shape I( u v) ( ( l( w) u l( w) v)) d ( ) ( ) ( )( ) ( )( ) w u l v m T w u m v l d. 5. Solutio of o-liear vibratio problem. et us represet ukow fuctios u v w i the followig way w y t y t w y u y t pu y y t u y v y t pv y y t v y. () Substitutig epressios () ito equatio ()-(3) ad usig the Bubov-Galerki method we receive ordiary differetial equatio: i case of static load i case of periodic load 3 y ( t) y( t) y ( t ) () 3 y ( t) ( k cos( t)) y( t) y ( t ). (3) Here k ad is defied by the followig epressios: k p t p ( u v w ) w ( u v w ) w ( u v w ) w d y yy m w. To solve the equatio () let us preset ukow fuctio yt () i the form y( t) Acos t. Usig the Bubov-Galerki method we receive the depedece betwee amplitude A ad frequecy ratio : 3. 4 A To idetify areas dyamic istability istead of (3) we use the equatio Mathieu: y t k cos t y t. t ati America Joural of Solids ad Structures (3) 75 88

7 idiya Kurpa Olga Mazur ad Victoria Tkacheko / Dyamical stability ad parametrical vibratios of the lamiated plates with comple shape 8 The mai area of dyamic istability (ear ) is limited by values ad [4]: k k. For the aalysis of oliear vibratios after the loss of stability we use oliear equatio (3). Depedece betwee the frequecy ratio / ad amplitude A are determied as follows [4]: A 3 4 k. 4. UMERICA RESUTS To illustrate our approach let us cosider symmetrically lamiated plate with cetral cutout (Fig. ). I this case it is eeded to determie the pre-buckle stress state of the lamiated plate. Suppose the plate is loaded logitudially alog the edges parallel to ais OY ( ). a Figure Form of plate with cutout. Boudary coditios of two types are cosidered:. The plate is assumed simply supported o the outer border but cutout is free (SS-F): w M p y y : a ; w M y u y y : b y ; (4) M Q T y 3 3 \.. The plate is simply supported o the all border (SS-SS): ati America Joural of Solids ad Structures (3) 75 88

8 8 idiya Kurpa Olga Mazur ad Victoria Tkacheko / Dyamical stability ad parametrical vibratios of the lamiated plates with comple shape w M p y y : a ; w M T y 4 4 \. (5) umerical results are obtaied for material with the followig relatios for the elasticity coefficiets: E=4. Gpa E=9.3 Gpa G=G3=5.95 Gpa G3=.96 Gpa ν=.33. The structures of solutio [78] for above metioed boudary value problems (4)-(5) (7) (8) (9)-() are take i the form:. boudary coditios SS-F (4): w P ; u i P i v i P i i ;. boudary coditios SS-SS (5): w P ; i P i i P i i where P Pj j 4 are idefiite compoets of the structure preseted as a epasio i a series of some complete system (power polyomials trigoometric polyomials splies etc.) y y y are equatios of the boudary domai ad its parts ad. For costructios of fuctios y y y theory of the R-fuctios is used: y f f f f y f f y f (6) 3 4 where are R-operatios [78] represeted below y y y y y y. Fuctios fi i..4 i (6) are determied as f a a f b b y f 3 c c f 4 c y c To validate the proposed approach we ivestigate cross-ply four layers lamiated plate with free cutout boudary coditios (4) ad for a=b=.5m h=.5m. This problem had bee solved by Dash S. ad others i [5]. The accuracy ad the efficiecy of the preset method are established through compariso of o-dimesioal bucklig load 3 b Eh with [5] fig.. ati America Joural of Solids ad Structures (3) 75 88

9 idiya Kurpa Olga Mazur ad Victoria Tkacheko / Dyamical stability ad parametrical vibratios of the lamiated plates with comple shape 83 Figure o-dimesioal bucklig load 3 b Eh with differet ratios of cutout. Further calculatios are carried out for geometrical parameters a/b= h=.5m ad boudary coditios (5). The ifluece of cutout size o o-dimesioal frequecies a / E h for various values of static compoet of load p is preseted i the table. Table. Effect of cutout size o atural frequecy a / E h c / b / 9 / / 9 / 9 / p / p / 9 / / 9 / Effect of parameter of cutout size ( c / b ) o istability regios ( k k ) are studied for p / p =.5.75 fig. 3. Obtaied results demostrate the same behavior i both of the cases: for a larger value of c / a the loss of istability occurs at lager values of ecitatio frequecy θ ad respose zoes occupy greater area. It should also be oted that istability zoes for p p =.5 correspod to greater value of ecitatio frequecy θ. / ati America Joural of Solids ad Structures (3) 75 88

10 84 idiya Kurpa Olga Mazur ad Victoria Tkacheko / Dyamical stability ad parametrical vibratios of the lamiated plates with comple shape Figure 3. Istability zoes for differet size of cutout (SS-SS p / p =.5.75). Istability aalysis is fulfilled for 345 layers plates. (fig.4). Results are obtaied for p / p =.5 ad c / b..4. More sigificatly the umber of layers effects o istability domais for plates with large cutouts. ati America Joural of Solids ad Structures (3) 75 88

11 idiya Kurpa Olga Mazur ad Victoria Tkacheko / Dyamical stability ad parametrical vibratios of the lamiated plates with comple shape 85 Figure 4. Istability zoes for differet umber of layers (SS-SS p / p =.5 c / b..4 ). oliear parametric vibratios are ivestigated for 3-layers plate. Effect of load parameter p t o respose curves is show o fig.5. Calculatios are performed for c / b.. p / p =.5. Decreasig of parameter p t leads to a covergece of curves. ati America Joural of Solids ad Structures (3) 75 88

12 86 idiya Kurpa Olga Mazur ad Victoria Tkacheko / Dyamical stability ad parametrical vibratios of the lamiated plates with comple shape Figure 5. Respose curve for differet values of parameter p t (SS-SS p / p =.5 c / b.. ). ati America Joural of Solids ad Structures (3) 75 88

13 idiya Kurpa Olga Mazur ad Victoria Tkacheko / Dyamical stability ad parametrical vibratios of the lamiated plates with comple shape 87 Figure 6. Respose curve for differet size of cutout (SS-SS p / p =.5.75). The effect of size of cutout o the respose curves is aalyzed for p / p =.5.75 pt / p =.5. The etesio of cutout leads to icrease of the vibratio amplitude. Chage of the static compoet of the load p affects the slope of respose curves. 5 COCUSIOS A effective method to ivestigate dyamical stability ad oliear vibratios of symmetrically lamiated plates with a comple form is developed. The proposed method is based o the origial meshless discretizatio procedure i the time ad variatioal methods combied with R-fuctios theory. Due to applicatio of the developed approach the iitial oliear problem is reduced to sequece of auiliary liear problems ad oliear ordiary differetial equatios (ODEs) with respect to time. The preset approach has advatage of beig suitable for cosiderig of differet types of the boudary coditios i domais of arbitrary shape. The proposed method has bee ati America Joural of Solids ad Structures (3) 75 88

14 88 idiya Kurpa Olga Mazur ad Victoria Tkacheko / Dyamical stability ad parametrical vibratios of the lamiated plates with comple shape applied to study dyamical stability ad oliear vibratios of simply supported rectagular plate with both free ad simply supported cetral square cutout. The umerical results for bucklig loads istability regios ad amplitude-frequecy respose curves are obtaied for various size of cutout umber of layers parameters of load. ote that due to icrease of cutout size of simply supported plate istability regios ted to shift to higher ecitatio frequecy with etesio of istability regios showig destabilizig effect of cutout o behavior of plate. The effect of layers umber o istability zoes is ot sigificat for plates with small cutout but icrease of cutout leads to essetial ifluece of layers umber. The amplitude-frequecy respose curves chage the slope ad locatio accordig to variatio of values of the load compoets. Refereces. S. A. Ambartsumya Theory Of Aisotropic Plates: Stregth Stability & Vibratios. // Taylor & Fracis; editio (99). P.36.. Awrejcewicz J. Research of stability ad oliear vibratio of plates by R-Fuctios Method / J. Awrejcewicz. Kurpa O. Mazur // Modelig Simulatio ad cotrol of oliear Egieerig Dyamical Systems. UK Spriger 9. P Awrejcewicz J. Kurpa. Mazur O. O the parametric vibratios ad meshless discretizatio of orthotropic plates with comple shape. Iteratioal Joural of oliear Scieces ad umerical Simulatio. V. Is. 5 P Boloti V.V. The dyamic stability of elastic systems // Holde-Day 964 P Dash S. Stability of lamiated composite curved paels with cutout usig fiite elemet method / Dash S. Asha A. V. Sahu S K. Proceedigs of the Iteratioal Coferece o Theoretical Applied Computatioal ad Eperimetal Mechaics (ICTACEM 4) December IIT Kharagpur. 6. Datta P. K. ad Biswas S. Research Advaces o Tesio Bucklig Behavior of Aerospace Structures: A Review. Iteratioal Joural of Aeroautical ad Space Scieces vol. issue P Rvachev V.. (98). The R-Fuctios theory ad its Some Applicatio aukova Dumka Kiev 98. (i Russia) 8. Kurpa.V. R-fuctios Method for Solvig iear Problems of Bedig ad Vibratios of Shallow Shells. Kharkiv TU Press Kharkiv 9 i Russia. 9. Qatu M. S. Sulliva R. W. Wag W. Recet research advaces o the dyamic aalysis of composite shells: 9. Composite Structures Volume 93 Issue December P Sahu S. K. Datta P. K. Dyamic Stability of amiated Composite Curved Paels with Cutouts. Joural of Egieerig Mechaics Vol. 9 o. ovember 3 pp Sahu S. K. Research Advaces i the dyamic stability behavior of plates ad shells: Part: coservative system / S. K. Sahu P. K. Datta // Applied mechaics reviews P Sahu S. K. Datta P. K. Dyamic stability of curved paels with cutouts. Joural of Soud ad Vibratio. V. 5 Issue 4 P Simitses G. J. Istability of dyamically loaded structures / G. J. Simitses // Applied mechaics reviews P Zhag Y.X. Yag C.H. Recet developmets i fiite elemet aalysis for lamiated composite plates. Composite Structures Vol. 88 o.. (March 9) P ati America Joural of Solids ad Structures (3) 75 88