Static and dynamic rectangular finite elements for plate vibration analysis

Transcription

1 UDC 34.: ]:9.6 Originl scientific pper Received: Sttic nd dynmic rectngulr finite elements for plte virtion nlysis Ivo Senjnović, Mrko Tomić, Neven Hdžić, Nikol Vldimir University of Zgre, Fculty of Mechnicl Engineering nd Nvl Architecture, Zgre, CROATIA e-mil: SUMMARY Nowdys dynmic finite elements with frequency dependent unified stiffness mtrix re considered in order to reduce finite element mesh density, nd, consequently, system of lgeric equtions in eigenvlue formultion. In this pper, n ttempt to develop 4-noded rectngulr finite element for thin plte ending is presented. Since there is no simple generl nlyticl solution of plte virtions for ritrry oundry conditions, dynmic shpe functions re ssumed s products of em shpe functions in longitudinl nd trnsverse direction. Dynmic finite element with frequency dependent stiffness nd mss mtrix is derived y energy pproch. Convergence of results is checked nd it is found tht such finite element formultion is not suitle for wider use. Therefore, strip finite element for virtion nlysis of rectngulr plte with simply supported two opposite edges is developed, which gives exct results. Appliction of oth finite elements is illustrted y severl numericl exmples. KEY WORDS: thin plte, dynmic finite elements, rectngulr element, strip element, dynmic stiffness nd mss mtrix.. INTRODUCTION The finite element method (FEM) is numericl method in which physicl properties from the element re re condensed into its nodes. In tht wy, the governing differentil equtions descriing the ehviour of n element re trnsformed into the system of lgeric equtions sed on energy equivlence []. FEM is widely used for structurl nlysis of engineering structures due to the possiility of detiled modelling of complex structure topology nd reltively esy computing. Depending on the type of structure, different finite elements hve een developed for strength nlysis, virtion nlysis, stility nlysis, etc. []. At the very eginning of computer ppliction, mtrix methods hve een developed s sis for lter development of nowdys generlly ccepted FEM [, 3]. Two FEM formultions hve een developed, i.e. the displcement finite element method nd force finite element method. In the former nd ltter cse, nodl ENGINEERING MODELLING 9 (06) -4, -

2 displcements nd nodl forces re unknown, respectively. Nowdys, the displcement FEM is mostly used s more prcticl solution []. This pper dels with thin plte virtion nlysis [4]. For this purpose, two types of 4-noded rectngulr finite elements re used; one with deflection comptiility (non-conforming) nd the other with deflection nd slope comptiility (conforming) []. In oth cses, shpe functions re presented in polynomil form. Sttic stiffness nd mss mtrix re developed. Numericl solution converges to the exct one y incresing finite element mesh density. Nowdys, there is tendency to develop dynmic finite elements with dynmic stiffness. For this purpose, nlyticl solution of differentil eqution of motion is used. However, tht is reltively esy tsk if D prolems s rs nd ems re considered, or if D prolem cn e reduced to D prolem, s in the cse of plte with simply supported two opposite edges []. An dvntge of dynmic finite elements is the pplicility of corser finite element mesh nd high ccurcy. In this pper, n ttempt to develop dynmic 4-noded rectngulr finite element with frequency dependent stiffness nd mss mtrix is descried. Shpe functions re ssumed in the form of products of the nlyticl em shpe functions. Since such plte shpe functions re n pproximtion of the rel ones, the usul energy pproch is pplied. Also, dynmic strip finite element for virtion nlysis of pltes with simply supported two opposite edges is presented. Its ppliction is illustrted in severl numericl exmples. Depending on oundry conditions, oth displcement nd force finite element methods re used in order to void the ill-conditioned formultion of eigenvlue prolem nd rther complicted procedure for its solving [6].. STATE-OF-THE ART. GENERAL There re two sic 4-noded rectngulr plte ending finite elements. One stisfies only deflection comptiility conditions long the edges, nd the other oth deflection nd slope comptiility conditions. The former is clled non-conforming element nd my yield over or under-stiff results. The ltter is clled conforming element nd demonstrtes monotonic convergence from the over-stiff side to the exct solution [, ]. Shpe functions of the non-conforming element re expressed in polynomil terms sed on sttic plte deformtion. Shpe functions of conforming element re defined s product of em shpe functions in longitudinl nd trnsversl directions. The element properties, i.e. stiffness mtrix, mss mtrix, geometric stiffness mtrix nd lod vector for sttic, dynmic nd stility nlyses, re sed on the vrition principle or energy pproch, nd re minly presented in symolic form. ENGINEERING MODELLING 9 (06) -4, -

3 . FINITE ELEMENT WITH DEFLECTION COMPATIBILITY The 4-noded rectngulr finite element is shown in Figure in Crtesin (0, x, y) nd nturl (0, ξ, η) coordinte systems. Fig. Rectngulr plte ending finite element with origin in node Nodl displcements, i.e. nodl deflection nd rottion ngles re indicted. Deflection function is ssumed in the polynomil form involving terms equl to the numer of degrees of freedom (DOF): w = + ξ + η + ξ + ξη + η + ξ + ξ η + ξη + η + ξ η + ξη, () where ξ = x /, η = y / nd nd re finite element length nd redth, respectively. In Eq. () ll polynomil terms of the third order nd lower re included. Among terms of the 3 3 fourth order only two mixed terms i.e., ξ η nd ξη re tken into ccount s the est choice from the ccurcy point of view. Rottion ngles re specified s: ϕ w w w w = =, ψ = =. () y η x ξ Shpe functions, i.e. deflection functions due to unit vlue of one nodl displcement (deflection or rottion) nd zero vlue of the others, re otined in the following form: ENGINEERING MODELLING 9 (06) -4, - 3

4 { Φ} ξη (3 ξ )ξ ( η) ( ξ )(3 η)η ( ξ )η( η) ξ( ξ ) ( η) (3 ξ )ξ ( η) + ξη( η)( η) ξη( η) ( ξ )ξ ( η) = (3 ξ )ξ η ξη( η)( η) ξ( η)η ( ξ )ξ η ( ξ )(3 η)η + ξ( ξ )( ξ )η ( ξ )( η)η ξ( ξ ) η Shpe functions due to unit nodl displcements w i re of order for unit nodl rottion ngle ϕ nd ψ, i.e. 3 ξ η nd (3) 3 ξη, s well s those 3 ξη ndξ 3 η, respectively. The ove shpe functions nd corresponding stiffness mtrix in symolic form re shown in Refs. [, 7]. Moreover, stiffness nd mss mtrix in symolic form re presented in Refs. [4, 8, 9]..3 FINITE ELEMENT WITH DEFLECTION AND SLOPE COMPATIBILITY Finite element shown in Figure is considered. Shpe functions Φ j(ξ,η), j =,..., re specified s products of em shpe functions in ξ nd η directions, X i(ξ ) nd Y i(η), i =,, 3, 4 []: Φ = X Y = (+ ξ )( ξ ) (+ η)( η) Φ = X Y = (+ ξ )( ξ ) η( η) Φ = X Y = ξ( ξ ) (+ η)( η) Φ = X Y = (3 ξ )ξ (+ η)( η) Φ = X Y = (3 ξ )ξ η( η) Φ = X Y = ( ξ )ξ (+ η)( η) Φ = X Y = (3 ξ )ξ (3 η)η Φ = X Y In this cse, ll shpe functions re of = (3 ξ )ξ ( η)η Φ = X Y = ( ξ )ξ (3 η)η Φ = X Y = (+ ξ )( ξ )(3 η)η Φ = X Y = (+ ξ )( ξ ) ( η)η Φ = X Y = ξ( ξ ) (3 η)η. 3 3 ξ η order. Stiffness mtrix is shown in the symolic form in Ref. [], while oth stiffness nd mss mtrices re specified in Ref. [9]. (4) 4 ENGINEERING MODELLING 9 (06) -4, -

5 3. FINITE ELEMENT BASED ON FREE PLATE NATURAL MODES Finite element of virting plte cn e considered s free structurl element in spce. Therefore, it is resonle to ssume its deflection function s set of free plte nturl modes. Only nturl virtions of pltes with simply supported two opposite edges nd ritrry oundry conditions on the remining two edges cn e determined nlyticlly in reltively simple wy. Strictly speking, some other oundry vlue prolems re lso solved nlyticlly ut in very complicted mnner [0, ]. Therefore, for ritrry oundry conditions, numericl methods like Ryleigh-Ritz method, finite difference method nd finite element method (FEM) re preferred. However, nturl virtions of free plte cn lso e nlysed pproximtely in simple nlyticl wy y the Ryleigh quotient, if the mode shpes re ssumed in the form of products of free em nturl modes, W(ξ,η) = X(ξ )Y(η) []. Bem nturl modes re divided into two groups, i.e. symmetric nd ntisymmetric. Therefore, the origin of the coordinte system is locted in the middle of the em length, l =. Free em nturl modes re expressed y the nturl coordinte ξ = x /, s follows []: symmetric: ntisymmetric: chγiξ cosγiξ X0 =, X i = +, i =,4... γ =.360, γ4 =.4978,... chγi cos γi shγiξ sinγiξ X = ξ, X i = +, i = 3,... γ3 = 3.966, γ = ,... shγi sinγi The sme expressions re vlid for em nturl modes in the cross-direction η = y /, denoted s Y i(η). The first three plte nturl modes re due to rigid ody motion with zero nturl frequencies since there is no restoring stiffness. Finite element virtion nlysis of free squre plte shows tht elstic nturl modes re of ordinry form []: W (ξ,η) X (ξ )Y (η) ij = i j, (7) () (6) or extrordinry form: W ± (ξ,η) = X (ξ )Y (η) ± X (ξ )Y (η). (8) mn kl m n k l The first elstic nturl modes re shown in Figure nd the corresponding formule re specified in Tle. Shpes of the nd nd 3 rd modes re hyperolic nd n ellipticl proloid, respectively []. Dimensionless frequency prmeters determined y oth FEM nlysis nd the Ryleigh quotient (RQ) re listed in Tle. Their ccurcy is estimted with respect to the relile results determined y the Ryleigh-Ritz method (RRM) in Ref. [3]. ENGINEERING MODELLING 9 (06) -4, -

6 Tle Frequency prmeter ( ) Ω = ω π ρh D of free squre plte elstic modes, Figure 3 Mode no. Mode type FEM Discrepncy Discrepncy RRM, RQ % % Liew [3] XY XY0 X0Y XY0 + X0Y XY XY X3Y0 X0Y X3Y0 + X0Y XY X3Y XY X3Y + XY XY (.0)* X3Y (.0)* * With respect to FEM Fig. Free squre plte nturl modes, FEM 6 ENGINEERING MODELLING 9 (06) -4, -

7 Fig. 3 Rectngulr plte ending finite element in Crtesin nd nturl coordinte systems It is necessry to point out tht extrordinry modes for rectngulr plte move to higher frequencies y incresing vlue of the plte spect rtio /. Therefore, deflection of rectngulr finite element shown in Figure 3 is ssumed s set of three rigid ody modes nd 9 squre plte elstic modes. Ordinry modes nd constitutive prts of extrordinry modes re tken into ccount. Symmetric mode X Y is excluded. Hence, one cn write: where { } is vector of unknown coefficients nd: { } W (ξ,η) = P(ξ,η), (9) P(ξ,η) = ξ η ξη X Y X3 Xη ξy Y3 X3η ξy3. (0) Structure of P(ξ,η) is similr to the polynomil terms in the cse of finite element with deflection comptiility conditions, Eq. (). For rottion ngles, the following is otined: w P ϕ = = η η { } w P ψ = = { }, ξ ξ () ENGINEERING MODELLING 9 (06) -4, - 7

8 P η P ξ = = ' ' ' ' ξ 0 Y 0 X ξy Y X ξy ' ' ' ' ' η X 0 X X η Y 0 X η Y. () Displcement field is specified s: w ψ { f } = ϕ = [ Z(ξ,η)]{ } [ Z(ξ,η)] Displcement vector of the i-th node ccording to Eq. (3) reds: { } ϕ [ ] { } i nd the finite element displcement vector:, (3) P P =. (4) η P ξ wi δ = i i = Z, i =,,3,4 i ψ { } { } { } [ ]{ } i () δ = δ = Z, (6) [ Z] [ Z] [ Z] [ Z] [ Z] =. (7) 3 4 By sustituting {} from Eq. (6) into Eq. (9) one cn express deflection function y the nodl displcement vector: is vector of shpe functions. { δ} w(ξ,η) = Φ, (8) Φ [ ] Deformtion (curvture) vector is otined in the form: = P(ξ,η) Z (9) 8 ENGINEERING MODELLING 9 (06) -4, -

9 w x w y w x y { κ} = = [ B]{ δ} [ B] [ A][ Z], (0) = () [ A] P ξ p P = q = η P r ξ η " " " " 3 3 p = X 0 X X η 0 0 X η 0 " " " " 3 3 q = Y 0 0 ξy Y 0 ξy () (3) ' ' ' ' 3 3 r = X Y 0 X Y. According to the definition of stiffness mtrix []: T [ ] [ ] [ ][ ] K = B D B d A, (4) A 3 ν 0 Eh D = D d, D =, d = ν 0 ( ν ) ν 0 0 [ ] [ ] [ ] () is mtrix of flexurl rigidity, the following is otined y employing Eq. (): [ ] [ ] T ( ) [ ][ ] =, (6) K D Z k Z T [ ] [ ] [ ][ ] k A d A dξ dη =. (7) Tking into ccount Eq. (), the element of mtrix [k] cn e written in index nottion: ENGINEERING MODELLING 9 (06) -4, - 9

10 i j = i j + j + i j + j + i j, (8) k p p νq q q νp ( ν)r r where p i, q i nd r i re defined y Eq. (3). Mss mtrix is expressed with shpe functions: [ ] { Φ} M = m Φ d A, (9) A where m = ρh is mss per unit plte re. By sustituting Eq. (9) into Eq. (9), the following is otined: The finite element eqution reds: [ ] [ ] T ( ) [ ][ ] =, (30) M m Z J Z [ J] { } = P P dξ dη. (3) { F} ([ K ] ω [ M] ){ δ} =, (3) where { F } nd { δ } re vectors of nodl forces nd nodl displcements, respectively. The first two free em nturl modes, Eqs. () nd (6), re shown in Figure 4. Virtion nlysis of free squre plte modelled y the presented finite element shows tht ll nturl frequencies determined y only one finite element gree very well with the exct solution, wheres for dense finite element mesh results converge to slightly higher vlues of the exct ones, Tle. Therefore, em nturl modes, Eqs. () nd (6), re pproximted y polynomils: = ( ), X3 ξ ( ξ 3) X 3ξ = (33) nd compred in Figure 4. Discrepncies of nturl frequencies of free squre plte determined y one finite element re very lrge for higher modes. However, the results rpidly converge to the exct solution y incresing the finite element mesh density, Tle. 0 ENGINEERING MODELLING 9 (06) -4, -

11 Tle Frequency prmeter Ω = ω( / π ) ρh/ D of free squre plte, Figure 3 Free plte Polynomil, mesh Mode No. Liew [3] modes, mesh x x x 4x4 6x6 8x8 0x Fig. 4 The first two free em elstic modes nd their polynomil pproximtion Bsed on the foresid numericl test, the polynomilly pproximted em nturl modes, Eq. (33), re used for the derivtion of finite element properties. Tht lso enles their symolicl presenttion. Mtrix of the polynomil coefficients nd its inversion red: ENGINEERING MODELLING 9 (06) -4, -

12 [ Z] [ Z ] = = Shpe functions, y using Eq. (9), cn e presented in index nottion for ech node i =,, 3, 4: Φi = + ξiξ(3 ξ ) + ηiη(3 η ) + ξiηiξη(4 ξ η ) 8 Φ i = (+ ξiξ + ηiη + ξiηiξη)(η ) 8 Φ i3 = (+ ξiξ + ηiη + ξiηiξη)( ξ ). 8 Shpe functions of the finite element re numered s follows: where k =,.... k i j (34) (3) (36) Φ = Φ, i =,,3,4, j =,,3, (37) Stiffness mtrix [K] is determined ccording to Eqs. (6), (7) nd (8), nd mss mtrix [M] ccording to Eqs. (30) nd (3). Mtrix [J] in [M] is digonl since polynomil terms in Eq. (0), where X (ξ ) nd X 3(ξ ) re specified y Eq. (33), nd Y (η) nd Y 3(η) in n nlogous wy, re orthogonl, i.e. ENGINEERING MODELLING 9 (06) -4, -

13 digonl [ ] J = (38) Stiffness nd mss mtrix cn e derived in symolic form. They re the sme s those presented in Refs. [4, 9], since polynomil Eq. (0) with functions Eq. (33) cn e rerrnged into the form Eq. (). The dvntge of the present pproch is the physicl explntion of the 3 polynomil terms s pproximte free plte nturl modes. As result, polynomil terms ξ η nd 3 ξη re chosen insted of 4 ξ nd 4 η due to lower nturl frequencies, i.e. strin energy, Tle. Another dvntge of the present formultion is the specifiction of the shpe functions in index nottion, which fcilittes the progrmming of the finite element. 4. FINITE ELEMENT BASED ON BEAM NATURAL MODES Finite element with the origin of the coordinte system plced in node is considered, Figure. Its shpe functions cn e specified s the product of em shpe functions X i(ξ ) nd Y i(η), i =,, 3, 4, s follows: Plte deflection is presented in the form: Node Node Node 3 Node4 Φ = X Y Φ = X Y Φ = X Y Φ = X Y Φ = X Y Φ = X Y Φ = X Y Φ = X Y Φ = X Y Φ = X Y Φ = X Y Φ = X Y { }. (39) w(ξ,η) = Φ δ, (40) where Φ is the vector of shpe functions nd { δ } is the vector of nodl displcements. Curvture vector is defined s: w x w y w x y { κ} = = [ B]{ δ} [ B] Φ ξ p Φ = q = η Φ r ξ η, (4) (4) ENGINEERING MODELLING 9 (06) -4, - 3

14 " " " " " " " " " " " " p = X Y X Y X Y X Y X Y X Y X Y X Y X Y X Y X Y X Y " " " " " " " " " " " " q = X Y X Y X Y X Y X Y X Y X Y X Y X Y X Y X Y X Y ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' r = X Y X Y X Y X Y X Y X Y X Y X Y X Y X Y X Y X Y. (43) Stiffness mtrix is defined ccording to Eq. (4): where referring to Eq. (8):, (44) 0 0 T [ K ] = [ B] [ D][ B] dξ dη = D[ k] i j = i j + i j + ( i j + i j ) + i j. (4) 0 0 k p p q q ν p q q p ( ν )r r dξdη Prmeters p i, q i nd r i re specified y Eq. (43). For mss mtrix Eq. (9), one cn write: [ ] { Φ} M m Φ dξ dη =, (46) 0 0 In order to determine em shpe functions, the deflection of nturlly virting em with length l nd nturl coordinte ξ = x / l, 0 ξ, is expressed y Krylov functions U i(ξ ) due to simplicity: X = AU + A U + A3 U3 + A4 U4, (47) U = ( chγξ + cosγξ ) U = ( shγξ + sinγξ ) U3 = ( chγξ cosγξ ) U4 = ( shγξ sinγξ ). An dvntge of Krylov functions is their chrcteristic vlues t ξ = 0 : U (0) =, 3 4 nd reltion etween derivtives nd functions: (48) U (0) = U (0) = U (0) = 0 (49) ' ' ' ' = 4 = 3 = 4 = 3 " " " " = 3 = 4 3 = 4 = U γu, U γu, U γu, U γu U γ U, U γ U, U γ U, U γ U,etc. Hence, derivtives of Krylov functions cn e expressed y the sme functions with the chnge of indexes in the nticlockwise direction. Furthermore, one otins for the ngle of rottion: (0) ϕ = dx γ ( A U A U A U A U ). dx = () 4 ENGINEERING MODELLING 9 (06) -4, -

15 The integrtion constnts A i cn e expressed in terms of nodl displcements. For the st node ξ = 0 nd one finds from Eqs. (47) nd (0) A = X(0) = W, nd A = ϕ(0 ) = φ γ γ. Constnts A 3 nd A 4 re expressed y A nd A from the sme equtions setting for the nd node X() = W nd ϕ () = φ. Hence, deflection function Eq. (47) cn e written in the form: X(ξ ) = X (ξ )W + X (ξ )φ + X 3(ξ )W + X 4(ξ )φ, () where X i(ξ ) refers to shpe functions with unit vlue of deflection nd rottion ngle t ξ = 0 nd ξ =, respectively: with the following coefficients: X (ξ ) = U (ξ ) + U (ξ ) + U (ξ ) 3 4 X (ξ ) = [ U (ξ ) + U 3(ξ ) + U 4(ξ )] γ X (ξ ) = U (ξ ) + U (ξ ) X 4(ξ ) = [ 4U 3(ξ ) + 4U 4(ξ )] γ shγ sinγ = U ()U 3() U 4 () =, C chγcosγ chγ sinγ shγcosγ = [ U ()U 4() U ()U 3() ] =, C chγcosγ chγ cos γ shγ sinγ 3 = U 3() =, 4 = U 4() =, C chγcosγ C chγcosγ shγcosγ + chγ sinγ = [ U 3()U 4() U ()U () ] =, C chγcosγ shγ sinγ = U ()U 3() U () =, C chγcosγ shγ + sinγ chγ cos γ 3 = U () =, 4 = U 3() =, C chγcosγ C chγcosγ C = U ()U 4() U 3 () = ( chγcosγ ). The chrcteristic feture of the first shpe function is tht sher force: tkes zero vlue t ξ 0 3 d X D 3 Q( ξ ) = D = γ U ( ξ ) + U ( ξ ) + U ( ξ ), 3 3 dx =, i.e. ( ) 3 3 Q 0 Dγ 0 (3) (4) 4 () = =. This condition is relted to ccording to Eq. (4), one otins eigenvlue eqution thγ + tgγ = 0, with roots Eq. (). In similr wy, ending moment of the second shpe function: tkes zero vlue t ξ 0 d X D M ( ξ ) = D = γ U ( ξ ) + U ( ξ ) + U ( ξ ), dx =, i.e. ( ) M 0 Dγ 0 = 0 nd 4 (6) = =. Tking into ccount tht eigenvlue function thγ tgγ = 0 is otined with eigenvlues Eq. (6). = 0, Eq. (4), ENGINEERING MODELLING 9 (06) -4, -

16 Finite element is constructed with em shpe functions X ( ξ ) nd X3 ( ) =, nd X ( ξ ) nd ( ) γ γ.360 ξ with rgument X4 ξ with rgument 3 =. Similr expressions to Eq. = with corresponding (3) cn e written for em functions in η direction Yi ( η ), i,,3,4 Krylov function Vj ( η ), j,,3,4 =. For derivtives of Krylov functions, reltions in Eq. (0) re ville. Stiffness nd mss mtrix re specified y Eqs. (44) nd (46), respectively. The first two em shpe functions expressed y Krylov functions re shown in Figures nd 6 nd compred with ordinry em shpe functions known s Hermite polynomils []. The " difference etween X i(ξ ) vlues is smll, wheres the one etween X i (ξ ) vlues is lrger. Polynomil " i X (ξ ) function is stright line, nd tht expressed y Krylov functions is of " prolic shpe. Curvture functions X i (ξ ) influence the stiffness mtrix nd deflection functions X i(ξ ) influence mss mtrix. If shpe function X (ξ ) is defined with γ, it is very close to the polynomil shpe function, Figure 6. For γ 0., shpe functions in terms of Krylov functions give the sme numericl vlues s Hermite polynomils, Figures nd 6. If trigonometric nd hyperolic functions in Krylov functions Eq. (48) nd coefficients Eq. (4) re expnded into power series nd setting γ 0, shpe functions Eq. (3) re reduced to polynomil em shpe functions in Eq. (4). Fig. Comprison of the first em shpe function determined y Krylov functions nd polynomil 6 ENGINEERING MODELLING 9 (06) -4, -

17 Fig. 6 Comprison of the second em shpe function determined y Krylov functions nd polynomil In order to investigte the ccurcy of different finite element formultions, virtion nlysis of clmped squre plte is performed. Plte is modelled y different mesh density, nd corresponding frctions of prmeters γ nd γ 3. The otined results for the first 8 nturl modes re shown in Tle 3, nd compred with the Liew vlues [3]. Mesh x with γ nd γ 3 gives only the first three nturl frequencies due to only one centrl free node with 3 d.o.f., which re highly ccurte. For mesh 4x4 discrepncies re incresed, ut vlues of frequency prmeter converge to the ccurte vlues, Tle 3. Tle 3 Frequency prmeter Ω = ω( / π ) functions, γ =.360, γ3 = ρh/ D of clmped squre plte, dynmic shpe Mesh Mode Mode Mode 3 Mode 4 Mode Mode 6 Mode 7 Mode 8 x γ, γγ (0.34%) (0.47%) (0.47%) 4x4 γ/, γγ3/ x6 γ/3, γγ3/ x8 γ/4, γγ3/ x0 γ/, γγ3/ x0 γ/0, γγ3/ x30 γ/, γγ3/ x40 γ/0, γγ3/ Liew [3] ENGINEERING MODELLING 9 (06) -4, - 7

18 If γ = 0. is tken into ccount, vlues of frequency prmeter re equl to those determined y the sttic finite element with polynomil shpe functions, Eqs. (4), Tle 4. Vlues of frequency prmeter converge to the exct solution y incresing finite element mesh density. Polynomil shpe functions cn e presented in simpler form, if the coordinte system of the finite element is plced in the middle of the element, Figure 3. In tht cse one otins: where ξ nd η. Xi = + ξiξ(3 ξ ), i =,3 4 Xi = ξ i(4 + 3ξiξ )(ξ ), i =,4 6 Yi = + ηiη(3 η ), i =,3 4 Yi = η i(4 + 3ηiη)(η ), i =,4, 6 Stiffness nd mss mtrix, Eqs. (44) nd (46), cn e generted in symolic form. They re the sme s those in Ref. [], since polynomils Eq. (7) cn e trnsformed into form Eq. (4) y chnging the coordinte system. Tle 4 Frequency prmeter Ω = ω( / π ) ρh/ D of clmped squre plte, shpe functions in terms of Krylov functions ( γ = 0.) nd Hermite polynomils (7) Mesh Mode Mode Mode 3 Mode 4 Mode Mode 6 Mode 7 Mode 8 x x x x x x x x Liew [3] STRIP FINITE ELEMENT FOR SIMPLY SUPPORTED PLATE AT TWO OPPOSITE EDGES. THEORETICAL CONSIDERATION Differentil eqution of plte virtion reds [4]: W W W m + + ω W = x x y y D In cse of simply supported longitudinl edges one cn ssume deflection in the form: (8) W ( ξ,η) = X ( ξ ) sinγη, γ = nπ, n =,, K (9) 8 ENGINEERING MODELLING 9 (06) -4, -

19 By sustituting Eq. (9) into Eq. (8), ordinry differentil eqution of Lévy form is otined [4]: m X γ X + γ ω X = 0. D Solution of Eq. (60) is ssumed in the exponentil form X = e nd four roots re otined, i.e. λ, = ± α, λ3,4 = ± iβ, λξ (60) m m α = ω + γ, β ω γ. D = D (6) Hence, deflection function Eq. (9) nd rottion ngle Eq. () red: W W ( ξ,η) = X ( ξ ) sinγη, ψ( ξ,η) = = Ψ ( ξ ) sinγη, (6) x ( ) X ξ = A shγη + A chγη + A sinβξ + A cosβξ, 3 4 (63) Ψ ( ξ ) = ( A αchαξ + A αshαξ + A3 βcosβξ A4 β sinβξ ) sinγη. Bending moment nd totl shering force, ccording to Ref. [4], red: W W 0 Mx ( ξ,η) = D + ν = Mx ( ξ ) sinγη, x y (64) 3 3 W W 0 Qx ( ξ,η) = D + ( ν) Qx ( ξ ) sinγη, 3 = x x y nd: ( ) = ( + ) ( ) = ( + + ) 0 x 3 4 M ξ D A c shαξ A c chαξ A c sinβξ A c cosβξ, 0 x 3 4 Q ξ D A d chαξ A d shαξ A d cosβξ A d sinβξ, α γ β γ c = ν, c ν, = + α γ α β γ β d = ( ν ), d ( ν ). = + Nodl displcements X ( 0) = X, Ψ ( 0) = Ψ, X ( ) = X nd Ψ ( ) Ψ (6) (66) = cn e written in the mtrix nottion: 0 0 X α β A 0 0 Ψ A =. (67) X shα chα sinβ cos β A 3 Ψ α α β β A 4 chα shα cosβ sinβ ENGINEERING MODELLING 9 (06) -4, - 9

20 On the other side nodl forces Q ( 0) = Q, M ( 0) = M, Q ( ) = Q nd ( ) lso presented in the mtrix nottion: x x x Q d 0 d 0 A M 0 c 0 c A = D. Q dchα dshα d cosβ d sinβ A3 M cshα cchα csinβ ccosβ A 4 M = M re x Equtions (67) nd (68) cn e written in condensed mtrix form { δ} [ B]{ A} { F} = [ C]{ A}, nd y eliminting vector { A }, the strip finite element eqution is otined: { } ( ) { } (68) = nd F = K ω δ, (69) ( ) [ ][ ] K ω = C B, (70) is 4x4 symmetric dynmic stiffness mtrix with 6 independent terms shown in the Appendix. Eqution (69) is derived y the displcement finite element method. In some cses, it is more convenient to use the finite element eqution otined y the force finite element pproch, i.e. { } ( ) { } δ = L ω F, (7) ( ) = ( ) = [ ][ ] L ω K ω B C, (7) is dynmic flexiility mtrix []. Its terms re lso given in the Appendix. A plte with uniform stiffness nd mss distriution is modelled y only one or two strip elements, depending on oundry conditions. For ech vlue of γ = nπ, n =,, K infinite numer of exct nturl frequencies nd modes cn e theoreticlly determined. If nturl virtions of plte with vrile properties in longitudinl direction is nlysed, plte is modelled y set of strip finite elements. Nturl frequencies re determined from condition Det Ks ( ω) = 0.. ILLUSTRATIVE EXAMPLES Simply supported squre plte, SSSS Only one strip finite element is sufficient to solve this tsk with comined geometric nd physicl oundry conditions. By tking into ccount oundry conditions X = X = 0, the finite element eqution is reduced to two d.o.f. system: M k k = M k k Ψ. Ψ Since k44 = k nd k4 = k4 one cn write for the determinnt of Eq. (73): 4 4 (73) Det = k + k k k = 0. (74) Ech of the ove expressions in the rckets cn e equl to zero. By pplying formule (A) from the Appendix, the following is otined: 0 ENGINEERING MODELLING 9 (06) -4, -

21 k ± k = ( c + c ) α sinβ( chαm ) βshα ( m cos β) = 0. (7) 4 Dk Deflection of simply supported plte is descried only y trigonometric functions, nd excluding hyperolic functions from Eq. (7), one finds sinβ = 0, i.e. β = mπ. Finlly, y employing the second formul of Eq. (6) for β, the well-known formul for nturl frequencies of simply supported plte is otined: Dimensionless frequency prmeter: tkes the simple form: mπ nπ D ω = +. (76) m Ω = ω π Ω = m + n m, (77) D. (78) Vlues of Ω for squre plte re compred with those determined numericlly y Ryleigh- Ritz method in Ref. [3], Tle. Tle Frequency prmeter ( ) Ω = ω π ρh D of squre plte simply supported t longitudinl edges Mode no. SSSS CSCS FSFS Liew, m x n Exct [3] n Liew, Exct [3] n Liew, Exct [3] x x x x x x x x x x Clmped squre plte t trnsverse edges, CSCS This prolem with geometric oundry conditions t trnsverse edges cn e solved with t lest two strip finite elements. By tking into ccount oundry conditions, the reduced ssemly eqution reds: ( ) ( ) ( ) ( ) Q k33 + k k34 + k X =, M ( ) ( ) ( ) ( ) k 43 k k44 k Ψ + + (79) ENGINEERING MODELLING 9 (06) -4, -

22 where ( p) k, p =, re relted to strip elements, nd Q, M, X nd Ψ re relted to the ij middle node of the ssemly. Since two finite elements re equl, one cn write ccording to Eqs. (A): Determinnt of Eq. (79) reds: ( ) ( ) ( ) ( ) ( ) ( ) k = k = k, k = k = k, k = k. (80) Det = 4k k = d α + d β c + c DK αshα cos β + βchα sinβ αchα sinβ βshα cos β = 0. Two trnscendentl frequency equtions re otined from Eq. (8), i.e. (8) αthα + βtgβ=0, βthα αtgβ=0. (8) The former nd the ltter re relted to symmetric nd ntisymmetric modes, respectively. Tking into ccount Eqs. (6), vlues of frequency prmeter Ω, Eq. (77), re determined for squre plte nd listed in Tle. Tht exct solutions re compred with numericl vlues clculted y the Ryleigh-Ritz method in Ref. [3]. Free squre plte t trnsverse edges, FSFS Due to physicl oundry conditions t the trnsverse edges, it is more convenient to solve this tsk y using the forced finite element formultion nd t lest two strip elements. Tking oundry conditions into ccount, the reduced ssemly eqution tkes the form: ( ) ( ) ( ) ( ) X l33 + l l34 + l Q =, Ψ ( ) ( ) ( ) ( ) l M 43 l l44 l + + ( p) where l, p =, re elements of the dynmic flexiility mtrix Eq. (79) for two elements. In ij similrity to the previous exmple, ccording to Eqs. (A): nd the determinnt of Eq. (84) reds: (83) ( ) ( ) ( ) ( ) ( ) ( ) l = l = l, l = l = l, l = l, (84) Det = 4l l = d α + d β c + c DL c d chα sinβ c d shαcosβ c d chα sinβ + c d shαcosβ = 0. Two trnscendentl frequency equtions re otined from Eq. (8), i.e. (8) cdthα cdtgβ=0, cdthα + cdtgβ=0. (86) The former nd ltter re relted to ntisymmetric nd symmetric modes, respectively. By employing reltions (6), vlues of the frequency prmeter Ω, Eq. (77), re determined for the squre plte nd included in Tle. They re compred with numericlly determined vlues from Ref. [3]. In the literture, the ove tsk is solved y employing ordinry used displcement finite element method nd only one finite element. However, in tht cse one is fced with the prolem of nturl frequencies determintion since Det K ( ω) 0. This prolem is overcome ENGINEERING MODELLING 9 (06) -4, -

23 y specil nd rther complex Wittrick-Willims lgorithm [, 6]. However, y using the force finite element formultion for similr prolems, the solution is otined stright wy nd unnecessry complictions re voided. 6. CONCLUSION Nowdys, dynmic finite elements with frequency dependent properties re more nd more used, due to course finite element mesh nd high ccurcy. If nlyticl solution of the considered issue is pplied, unique dynmic stiffness mtrix is constructed nd exct results re otined. If free em nturl modes re used for descriing plte deflection, the energy pproch is employed nd sttic stiffness nd mss mtrix re derived. The ltter pproch is used for the construction of non-conforming 4-noded rectngulr plte finite element (Section 3). Course mesh gives pproximte results for the spectrum of nturl frequencies quite close to the exct ones. In cse of conforming finite element with shpe functions s product of dynmic em shpe functions, only the first few nturl frequencies re close to the exct vlues (Section 4). By incresing finite element mesh density nd decresing vlues of rgument of trigonometric nd hyperolic functions ccordingly, the otined results converge to the exct ones. If very smll vlue of tht rgument is used, dynmic shpe functions ecome equl to sttic shpe functions, nd sttic finite element is otined with ordinry convergence of numericl results. In cse of plte simply supported t two opposite edges, D prolem is reduced to D, i.e. strip element, with closed-form nlyticl solution. Dynmic stiffness mtrix sed on tht solution gives exct vlues of nturl frequencies for ny comintion of oundry condition t the remining two plte edges. The developed 4-noded dynmic finite elements re not relile for wider prcticl use. On the contrry, ppliction of the -noded strip dynmic finite element is very effective. In future reserch, the sme pproch cn e used for the development of -noded finite element for thick pltes, y employing the modified Mindlin theory [4]. 7. ACKNOWLOGEMENT This work ws supported y the Ntionl Reserch Foundtion of Kore (NRF) grnt funded y the Koren Government (MSIP) through GCRC-SOP (Grnt No ). ENGINEERING MODELLING 9 (06) -4, - 3

24 8. APPENDIX: DYNAMIC STIFFNESS AND FLEXIBILITY MATRIX OF THE PLATE SIMPLY SUPPORTED AT TWO OPPOSITE EDGES Elements of dynmic stiffness mtrix ccording to Eqs. (67), (68) nd (70): k = k = d α + d β αshαcosβ + βchα sinβ, 33 DK k = k = ( d α d β)( chαcosβ ) + ( d α + d β) shα sinβ, 34 DK 3 DK 4 3 DK 44 DK K k = d α + d β αshα + βsinβ, k = k = d α + d β chα cosβ, k = k = c + c αchα sinβ βshα cos β, D k = = c + c βshα α sinβ, 4 DK ( ) ( ) = αβ chα cos β + α β shα sinβ. Elements of dynmic flexiility mtrix ccording to Eqs. (67), (68) nd (7): l = l = c + c c d chα sinβ c d shαcosβ, 33 DL l = l = d d c c chαcosβ + c d + c d shα sinβ, ( ) 34 D L l = c + c c d shα c d sinβ, 3 DL l = l = d d c + c chα cosβ, 4 3 DL l = l = d α + d β c d chα sinβ + c d shαcosβ, 44 DL l = = d α + d β c d shα + c d sinβ, 4 DL ( ) ( ) D = c c d d chαcosβ + c d c d shα sinβ. L (A) (A) 9. REFERENCES [] O.C. Zienkiewicz, The Finite Element Method in Engineering Science, McGrw-Hill, New York, 97. [] J.S. Przemieniecki, Theory of Mtrix Structurl Anlysis, McGrw-Hill, New York, 968. [3] D.J. Dwe, Mtrix nd Finite Elements Displcement Anlysis of Structures, Clrendon Press, Oxford, 984. [4] R. Szilrd, Theory nd Applictions of Plte Anlysis, John Wiley & Sons, Hooken, ENGINEERING MODELLING 9 (06) -4, -

25 [] M. Boscolo, J.R. Bnerjee, Dynmic stiffness formultion for pltes using first order sher deformtion theory, st AIAA/ASME/ASCE/AHS/ASC Structures, Structurl Dynmics, nd Mterils Conference <BR> 8 th - April 00, Orlndo, Florid. [6] F.W. Willims, S. Yun, K. Ye, D. Kennedy, M.S. Djoudi, Towrds deep nd simple understnding of trnscendentl eigenprolem of structurl virtions, Journl of Sound nd Virtion, Vol. 6, No. 4, pp , 00. [7] J. Sorić, The Finite Element Method, Golden mrketing Tehničk knjig, Zgre, 003. (in Crotin). [8] I. Holnd, K. Bell, Finite Element Method in Stress Anlysis, Tpir-Trykk, 970. [9] I. Senjnović, The Finite Element Method in Ship Structure Anlysis, University of Zgre, Zgre, 998. (In Crotin). [0] Y. Xing, B. Liu, Closed form solutions for free virtions of rectngulr Mindlin pltes, Act Mechnic Sinic, Vol., No., pp , 009. [] I. Senjnović, M. Tomić, N. Vldimir, D.S. Cho, Anlyticl solution for free virtion of modertely thick rectngulr plte, Hindwi Pulishing Corportion, Mthemticl Prolems in Engineering, Volume 03, Article ID [] I. Senjnović, M. Tomić, N. Vldimir, An pproximte nlyticl procedure for nturl virtion nlysis of free rectngulr pltes, Thin-Wlled Structures, Vol. 9, pp. 0-4, 0. [3] K.M. Liew, Y. Xing, S. Kitipornchi, Trnsverse Virtion of Thick Rectngulr Pltes I. Comprehensive Set of Boundry Conditions, Computers & Structures, Vol. 49, No., pp. - 9, 993. [4] I. Senjnović, N. Vldimir, M. Tomić, An dvnced theory of modertely thick plte virtions, Journl of Sound nd Virtions, Vol. 33, pp , 03. ENGINEERING MODELLING 9 (06) -4, -