Vibration response of sandwich plate under Low-velocity impact loading

Transcription

1 Vbraton response o sandwh plate under Low-veloty mpat loadng M. WALI, M. ABDNNADHR, T. FAKHFAKH, M. HADDAR Researh Unt o Dynams o Mehanal Systems (UDSM), Natonal shool o engneers o Sax BP Sax TUNISI mondherwal@yahoo.r Abstrat - The vbraton response o sandwh panels, under low-veloty mpat, s presented by usng nte element method (FM) and Hertz ontat low. The vbratons on the sandwh plate under the mpat are evaluated usng the Wgner-Vlle dstrbuton. As a result, by ormulatng a smple model nvolvng the moton o a rgd mpator ombned wth dynam equaton o sandwh plate, the eet o ntal mpat veloty and the geometr propertes o plate are dented. Ths numeral smulaton that determne the mpat ore and estmate the behavour under mpat would be very helpul durng the sandwh strutures desgn. Keyword: - Low-veloty/ Impat/ Hertz ontat law / Impat ore/ Wgner-Vlle dstrbuton/ Indentaton 1 Introduton Impat s dened as the proess nvolved n the ollson o two or more objets. Now, elast advaned plates have been wdely used n arrat, aerospae, automotve and marne. Durng mpat, knet energy s lost. The loss o knet energy s aused by wave propagaton, plast deormaton or vsoelast phenomena and depends on the shapes and materal propertes o the olldng bodes as well as on ther relatve velotes [1]. Some methods and results o the mpats on the lexble strutures are reaptulated by Goldsmth [1], Johnson [] and Stronge [3]. In low-veloty mpat, Fnte element analyss and a moded Hertz ontat law were employed to relate the mpator movement wth the dsplaement hstory o the sandwh panels, [4, 5]. An other approah based on a system havng threedegree o reedom, onsstng o sprng-massdamper-dashpot (SMDD) or sprng-mass-damper (SMD), used by Malekzadeh et al. [6] to model the nteraton between the mpator and the omposte sandwh panel. Also, n the modellng o the omposte strutures mpat, Abrate [7] used an energy-balane model and sprng-mass models to analyse dynam mpats. In ths paper, the low-veloty mpats between a rgd sphere and a lamped sandwh plate have been modelled. Impat response analyss s perormed usng FM and Hertz ontat law. The mpat problem ormulaton s presented, the method used to alulate mpat ores and then an algorthm or solvng elast mpat problems s developed. Numeral results, nludng ontat ore, veloty o mpator, ndentaton and dynam response o plate subjeted to low-veloty mpat s presented. Formulaton o the problem In ths seton mpat problem o a rgd sphere aganst a ully lamped sandwh plate s ormulated. The sandwh plate s modelled by usng the nte-element method (FM).The Hertz ontat law s used to smulate the pressure dstrbuton n the ontat area between the sphere and the sandwh plate..1 B-dmensonal sandwh plate model Three peretly glued layers orm the studed sandwh plate. The two external layers, alled skns, are supposed elast, sotrop and homogeneous, and the ore layer s assumed to be vsoelast. A partal ross seton o the plate n the xz-plane as shown n Fg. 1. ISSN: Issue 1, Volume 6, January 11

2 To model the sandwh plates a lnear longtudnal dsplaement eld hypothess s adopted n eah layer. The two sandwh plate skns and the ore ollow, respetvely, the assumpton o Love Krhho and the assumpton o Mndln. Takng nto aount the above-mentoned assumptons, the two skns and the ore longtudnal dsplaement an be expressed n terms o 7 undamental quanttes, the longtudnal dsplaements u m and v m, the relatve longtudnal dsplaements [u] and [v], the transversal dsplaement w and the skns rotatons β x and β y, (Fg. 1) [8]: Skns plates [ u] u = um + + ( z z ) βx [ v] v = vm + + ( z z ) β y w = w Vsoelast ore u = um zmβ x + zϕ x v = vm zmβ y + zϕ y w = w (=1,) (1) where, u1 + u v + v um = v v = v v (3) [ u] u1 u 1 m = () = [ ] 1 [ u] z1-z [ v] z1-z ϕx = + -1 βx h h z m z = + z 1 ϕ y = + -1 β y h h (4) (5) From dsplaement expresson, the stress-stran relatonshp at eah pont or th layer o the skn plate s: σ xx D11 D1 ε xx σ yy = D1 D εyy σ D γ xy 33 xy (=1,) (6) where, D =D = ; D = 1-ν 1- νν ; 11 ( ) 1 D 33 = (7) 1+ ν In the lnear vsoelast ore, the stress-stran relaton s obtaned by onsderng a relaxaton model, [9, 1]: t dεkl σj () t = C jkl(t-τ) dτ (8) dτ Ths representaton, used to model vsoelast materal behavour, leads to omplex relatonshp between the stress and stran n the ore, [1]. σ xx C xx 11 C1 ε σ yy C εyy 1 C σ xy = C γ 33 xy (9) σxz C44 γxz σ yz C 55 γ yz where, 11 C =C = ; C 1-ν ν 1 = 1- ν ; C 33 =C 44 =C 55 = G (1) The dynam equaton s obtaned by the use o FM. So a quadrlateral homogenzed sandwh nte element s used whh has our nodes and seven degrees o reedom per node (u m, v m,[u], [v],β x, β y, w). The equaton o the sandwh plate moton an be wrtten as: M W t + K W t = F t (11) ( ) ( ) ( ) In ase o proportonal dampng, there s relatonshp between hysteret and vsous dampng models [11], leadng to equvalent dynam system o the sandwh plate whh s gves (Appendx B): M W t + CW t + KW t = F t (1) ( ) ( ) ( ) ( ) ISSN: Issue 1, Volume 6, January 11

3 .3 Numeral models or the mpat proess In ths seton we study the problem o the rtonless normal low veloty mpat o a rgd sphere aganst an elast plate.the dynam equatons or both the sandwh plate and mpator (rgd sphere) an be wrtten as: Fg. 1. Sandwh plate dsplaements n the x-zplane.. Applaton o Hertz law n ball-plate ontat I the ndentaton due to the mpat load s muh smaller than the plate thkness, the ontat ore between the mpator and the sandwh plate durng the mpat s assumed to be governed by the nonlnear Hertzan ontat law o the orm: F() t = k () 3 δ t (13) where δ s the ndentaton o the ball-plate bodes at the ontat ponts, dened by: b b h1 δ = ZI W,, (14) where Z I denotes the mpator dsplaement and W s the transverse top ae plate dsplaement at the mpat loaton. And k s the Hertzan ontat stness, dened n reerene [4, 1] by: 4 RI k = (15) 1- ν I 1- ν P 3 ( + ) I P Thereore, or applyng equaton (15) n sandwh plates mpat analyss, P and ν P an be estmated usng the ollowng relatons, [13]: h P = ; h + h + h 1 1 h h h h h h 1 ν P= ν1 + ν + ν (16) ( ) ( ) ( ) ( ) W ( ) = W ( ) = W ( ) MW t + CW t + KW t = F t where, and M I Z I + F t = where, Z I = Z I = Z I = () ( ) ( ) ; ( ) V (17) (18) For the resoluton o the two dynam equatons (17) and (18) smultaneously, an teratve numeral method aordng to the proedure desrbed below was used. At the ntal moment t = (begnnng o the shok), the sphere punhes the plate wth an ntal veloty V. We assume an arbtrary ontat ore F t = F. value ( ) The dynam equatons (17) and (18) are solved usng the Newmark ntegraton sheme. That leaded to alulate the ndentaton δ. The ore F( t ) s orreted then by usng the Hertzan ontat law expressed by the relaton (13) ater havng determned the penetraton δ o the mpator n the plate, gven by the expresson (14). The value o F( t ), thus obtaned, s renjeted n the equatons (17) and (18), and to take agan the teratve proess untl onvergene o F( t ). Two ases are possble: - I δ >, the ontat between the sphere and the plate exst; the mpat ore s dened by the Hertzan ontat law and F( t ) wll be realulated usng a new value o δ, and s then njeted nto the dynam equatons to be solved agan. - I δ, the ontat s lost and the dynam responses are determnate. ISSN: Issue 1, Volume 6, January 11 3

4 3 Numeral results and dsusson 3.1 Valdaton In order to ensure the auray o the present model and the omputer program developed, mpated elast plate system s valdated by Lu et al. [14]. Lu et al onsdered a lamped 1 x 1 x. mm 3 steel plate mpated by a steel sphere at ts entre. The ntal ondton o zero dsplaement and zero veloty are assumed or the plate. The mehanal propertes o the plate are gven n Table1. Table 1. Mehanal eatures o elast plate Young Densty Posson modulus (GPa) (kg/m 3 ) oeent The radus o the sphere s 5 mm and the veloty just beore mpat s V =1 m/s. Fg. llustrated the omparson between the results o the entre plate deleton obtaned by the present study and by Lu et al. [14], who used the Lagrangan approah. The numeral results showed onorm reprodublty wth Lu s results at the mpat moment ([,.] perod tme). Ater. ms there was no ontat between the mpator and the plate, and these urves represented the plate ree vbraton ater mpat. Fg.. Comparson o dsplaement hstory or a steel plate. 3. Study o mpated sandwh plate In ths seton, a parametr study o a sandwh plate subjeted to mpat loadng s onduted, and the eets o varyng the derent parameters, suh as ntal veloty and thkness o the skns, on the mpat proess are nvestgated usng proposed model. The mpator s onsdered rgd, wth a mass 3.8 g and radus R I =1 mm, and t s assumed that ths spheral mpator hts the plate at ts enter wth an ntal veloty V The geometral and mehanal haratersts o the studed plates are mentoned on Tables and 3. Table. Geometral sandwh plate haratersts Sandwh plate Sde b (mm) Thkness (mm) P1 1 h 1 =.5;h =1; h =.5 P 1 h 1 =1 ; h =1 ; h =1 Table 3. Mehanal sandwh plate haratersts Young Modulus (GPa) Posson Coeent ν Densty ρ Materal propertes Skns =69 GPa.33 7 kg/m 3 Alumnum (MPa) (MPa) η (loss ator) G (MPa) G (MPa) η G (loss ator Densty ρ Materal propertes Core kg/m 3 HRX C7.13 [1] 3..1 Inluene o mpator ntal veloty In ths part the eet o the ntal veloty on the dynam behavour o sandwh plate P1 s studed. Fgs. 3 and 4 llustrate the mpat ore and the mpator veloty, aordng to the tme axes, or the sandwh plate P1 and wth derent ntal mpator veloty. These gures showed that the nrease n mpator veloty led to an nrease n mpat ore but nvolved a derease n the loadng tme. Fg. 5 llustrates the requeny responses o the sandwh plate P1 or varous ndental velotes ISSN: Issue 1, Volume 6, January 11 4

5 o the mpator, whereas Fg. 6 llustrates the temporal responses o the sandwh plate P1 or varous ndental veloty o the mpator. In the ase o an elasto-dynam mpat problem, the shok, on the sandwh plate P1, usng a steel sphere (mpator) extes the same requenes even we vared the ntal veloty o the mpator. we an also note that only the ampltude o aeleraton nreases wth the nrease o the ntal mpat veloty. Fg. 5. Frequeny responses o mpated sandwh plate or derent ntal veloty o mpator. Fg. 3. mpat ore on the sandwh plate P1 Fg. 6. Temporal responses o mpated sandwh plate or derent ntal veloty o mpator. Fg. 4. Veloty o the mpator durng the mpat proess. 3.. The skns thkness Inluene In order to analyze the nluene o the skns thkness on the dynam behavour o the sandwh ISSN: Issue 1, Volume 6, January 11 5

6 plate subjeted to mpat, two examples o sandwh plate are taken. Fg. 9. Dsplaements o the enter plates or an ntal mpat veloty V = 1m/s. Fg. 7. Impat ore o sandwh plate P1 and P or an ntal mpat veloty V = 1m/s. Fg. 7 represents the evoluton aordng to the tme o the mpat ore on the plates P1 and P or an ntal mpat veloty V = 1m/s. Ths gure shows that the nrease o the skns thkness o the plates generates an nrease n the mpat ore and a reduton n the shok duraton. Ths s due to the nrease n the rgdty o the plate wth the nrease o the skns thkness. Also the eet o vsoelast ore durng the mpat proess dereases wth the nrease o the layers thkness. Fg. 8 llustrates the ndentaton, aordng to the tme axes, or the tow plates. Fg. 9 llustrates the omparson between the results o the entre plate deleton o sandwh plates P1 and P Fg. 8. Indentaton o sandwh plate P1 and P or an ntal mpat veloty V = 1m/s. Comparson between two plates P1 and P showed that: The nreasng o skns thkness aused a derease n ndentaton and n dsplaement. Durng ollsons, energy s both absorbed by the bendng o the plate and the absorbng harater o the vsoelast ore. By observng Fgs 8 and 9, we note that the energy absorbed by lexng o the plate s larger n the ase o plate P1 (dsplaement o plate P1 s large than plate P), and the energy absorbed by transverse shear o the ore s larger or the plate P1 (ndentaton n the plate P1 s large than n the plate P). ISSN: Issue 1, Volume 6, January 11 6

7 ln Sale Aeleraton (m/s ) x Sgnal n tme WV, log sale nergy Spetral densty Frequene [khz] x Tme [ms] Fg. 1. Wgner-Vlle Dstrbuton, requeny response and temporal response o the Sandwh plate P1 under mpat o a sphere wth an ntal veloty V =1m/s. ln Sale Aeleraton (m/s ) x Sgnal n tme WV, log Sale nerge Spetral densty Frequene [khz] x Tme [ms] Fg. 11. Wgner-Vlle Dstrbuton, requeny response and temporal response o the Sandwh plate P under mpat o a sphere wth an ntal veloty V =1m/s. ISSN: Issue 1, Volume 6, January 11 7

8 Fgs. 1 and 11 llustrate a suesson o loal analyss o the sgnal observed through a wndow usng the energy dstrbuton o the sgnal n requeny-tme plan. For plates P1 and P there are three man omponents are separated by ther ampltude and duraton. It then dstngushes other seondary omponents or the two plates P1 and P. From Fgs. 1 and 11, we note that ater mpat the osllatons o sandwh plates P1 and P are thereore non-sustaned (not mantaned). We note also that the requenes begn wth a delay that depends on the duraton o shok, these delayed mounted, vsble on the Wgner-Vlle dstrbuton ders rom one struture to another, represent the natural requenes o sandwh plates P1 and P. Durng ree osllatons, the vbraton o the sandwh plate P1 dsspates very qukly. Ths aster dsspaton, n the plate P1 s due n our parametr study to the perentage o the vsoelast ore partpaton n the general behavour o the plate. We also note that the spetral densty o energy (vbraton level) s hgher durng an mpat or the plate P. By omparng the two dstrbutons o Wgner-Vlle, the requeny omponents o vbraton n the plate P (less damped) are mantaned at relatvely hgh levels ompared to the requeny omponents o the vbraton o the plate P1 ( qukly dsspated). 4. Conluson Ths work deals the dynam analyss o mpated sandwh plate. For ths purpose, a numeral method s developed n order to estmate the mpat ore and determne the dynam behavour o the struture under mpat. In low veloty mpat problem, Hertzan ontat law s approprate to study the mpat on sandwh plate at small ndentaton. The parametr study o the sandwh plate shows the nluene o the layers thkness on dynam behavour. In at the numeral results o the dynam analyss onrmed the apaty o attenuaton o the vbratons o a sandwh plate. Fnally, t was shown that the present approah ould be used to study the low-veloty mpat response analyss problem. Reerenes: [1] Goldsmth. W., Impat, dward Arnold Publ., 196, London,. [] Johnson. K.L., Contat Mehans, Cambrdge Unversty Press, 1985, Cambrdge. [3] Stronge. W.J., Impat Mehans, Cambrdge Unversty Press,, Cambrdge. [4] Cho Ik Hyeon, Lm Cheol Ho, Low-veloty mpat analyss o omposte lamnates usng lnearzed ontat law, Composte strutures, 66, 4, [5] Setoodeh A.R., Malekzadeh P., Nkbn K., Low veloty mpat analyss o lamnated omposte plates usng a 3D elastty based layerwse FM, Materal an Desgn, 3, 9, [6] Malekzadeh K., Khall M.R., Mttal R.K., Response o omposte sandwh panels wth transversely lexble ore to low-veloty transverse mpat: A new dynam model, Internatonal Journal o Impat ngneerng, 34, 7, [7] Abrate Serge, Modelng o mpats on omposte strutures, Composte Strutures, 51, 1, [8] Hammam L., Fenna S., Abdennadher M., Haddar M., Vbro-Aoust Analyss o a Double Sandwh Panels System, Internatonal Journal o engneerng smulaton, Volume 6, Numéro1, 5, ISSN , Wolverhampton (UK). [9] Nagheh G.R., Jn Z.M., Rahnejat H., Contat haratersts o vsoelast bonded layers, Appled Mathematal Modellng,, 1998, [1] Meuner M., Sheno R.A., Dynam analyss o omposte sandwh plates wth dampng modelled usng hgh-order shear deormaton theory, Composte Strutures, 54, 1, [11] Hammam L., Zghal B., Fakhakh T., Haddar M., Charaterzaton o modal dampng o sandwh plates, J. Vbr.Aoust., 17, 5, [1] Werner Shehlen, Robert Sered, Peter berhard, lastoplast phenomena n multbody mpat dynams, Comput. Methods Appl. Meh. ngrg., 195, 6, ISSN: Issue 1, Volume 6, January 11 8

9 [13] Khall M.R., Malekzadeh K., Mttal R.K., et o physal and geometral parameters on transverse low-veloty mpat response o sandwh panels wth a transversaly lexble ore, Composte Strutures, 77, 7, [14] Lu Z.S., Lee H.P., Lu C., Strutural ntensty study o plates under low-veloty mpat, Int. J. o mpat engneerng, 3, 5, Appendx A.: Nomenlature b sde o the square plate C vsous dampng matrx o sandwh plate Cjkl ( t-τ ) relaxaton modulus D hysteret dampng matrx 1,, I Young s modulus P eetve equvalent transverse Young s modulus o the mpated plate omplex Young s modulus F G 1, G G nodal mpat ore vetor shear modulus omplex shear modulus h thkness o plate h 1, h thkness o two aes layers h thkness o the ore layer k stness ontat K stness matrx o sandwh plate K omplex stness matrx M mass matrx o the plate M I mass o the sphere R I radus o the sphere t tme u m, v m longtudnal dsplaements o a pont loated on the axs o the plate u 1, v 1, u, v longtudnal dsplaements o the two skns plates u, v dsplaement o a pont loated on the th layer axs u, v longtudnal dsplaements o the ore [u], [v] relatve longtudnal dsplaements o the two skns V ntal mpat veloty w transversal dsplaement W ( t) dsplaement vetor W ( t) veloty vetor W ( t) aeleraton vetor W r, W mehanal work Z I dsplaement o the mpator z normal oordnate z dstane between the th skn layer axs and plate axs z m z oordnate o the mddle plan o the plate β x, β y skns rotatons ϕ x, ϕ y ore rotatons δ ndentaton ν 1, ν, ν, ν I Posson s rato ν P eetve transverse Posson s rato o the mpated plate ε yy, ε xx strans γ xy, γ xz, γyz shear strans σ xx, σ yy normal stresses σ xy, σ xz, σ yz shear stresses τ relaxaton tme η φ th modal dampng loss ator egenvetors Φ mass-normalzed egnevotor matrx Appendx B.: The equaton o moton or sandwh plate an be wrtten as: M W t + K W t = F t ( ) ( ) ( ) where, K = K + j D The egenvalue problem an be expressed as: λ M + K + jd φ = = 1, n ( ) Solvng the above egenproblem leads to the soluton ontanng omplex egenvalues λ and egenvetors φ The egensoluton proesses the orthogonalty propertes, whh are dened by the equatons t t Φ M Φ= I and Φ ( K + jd) Φ= λ ISSN: Issue 1, Volume 6, January 11 9

10 Where Φ s the mass-normalzed egnevotor matrx and λ s dagonal matrx o omplex egenvalues wh an be expressed as: λ = ω 1+ jη ( ) Where η s th modal dampng loss ator The realsonshp between vsous and hysteret dampng models s modelled by [11]: ξ = η Then The dynam equaton may be wrtten as : M W t + CW t + KW t = F t ( ) ( ) ( ) ( ) ISSN: Issue 1, Volume 6, January 11 1