Proceedings of the 2006 IASME/WSEAS International Conference on Continuum Mechanics, Chalkida, Greece, May 11-13, 2006 (pp )

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1 The Effect f Gemetric Imerfectins n the Amlitude and the Phase Angle f the Nn-Linear Dynamic Behavir f Thin Rectangular Plates Parametrically Excited MIHAI BUGARU TUDOR CHERECHEŞ ADRIAN ROTARIU SORIN GHEORGHIAN VICTOR COJOCARI Deartment f Mechanics, Plitehnica University f Bucharest Slaiul Indeendentei 313, Bucharest 776 ROMANIA Deartment f Mechanics Military Technical Academy Gerge Cşbuc 81-83, Bucharest ROMANIA Abstract: The aer reveals recent develments f the influence f the gemetric imerfectins n the amlitude and the hase angle f the nn-linear vibratins f thin rectangular lates arametrically excited. In the regin f rincial arametric resnance, starting frm the temral nn-linear differential equatin that describes the scillatry mvement and using the secnd rder arximatin f the asymttic methd were cmuted the amlitude and the hase angle as functins f system arameters and gemetric imerfectins. By varying the intensity f the gemetric imerfectins was btained their influence un the amlitude and the hase angle fr the statinary nn-linear dynamic resnse. Key wrds: lates, nn-linear arametric vibratins 1 Nmenclature A 1, A, B 1, B = unknwn functins in asymttic exansin; C = viscus daming cefficient; D = flexural rigidity f late; E = Yung s mdulus; M = cefficient f the nn-linear term; N y (t) = external inlane lading er unit width; N y = static in-lane lading er unit width; N yt = amlitude f harmnic in-lane lading er unit width; N cr = critical buckling lad f the late, defined as in [14]. 353; a = length f late in x-directin; b = length f late in y-directin; f(x,y,t) = Airy s stress functin; h = late thickness; s=λ/ω=frequency arameter; t=time; w(x,y,t)=lateral midsurface dislacement in z-directin; w (x,y) = initial gemetric imerfectin in z-directin; = decrement f daming; Λ(t) = instantaneus frequency f the external in-lane excitatin, Λ = dθ/dt; Ω q = free vibratin circular frequency f a rectangular late laded by a cnstant cmnent f in-lane frce; Ω q =free vibratin circular frequency f a rectangular late, with initial gemetric imerfectins, laded by a cnstant cmnent f in-lane frce; ε = small sitive arameter in asymttic exansin, <ε<<1; θ(t) = ttal hase angle f harmnic excitatin; µ = lad arameter f the late; ν = Pissn s rati; ρ = mass density er unit vlume f late; τ = slwing time in asymttic analysis; ψ (t) = hase angle f the arametric vibratin; ω q = free vibratin frequency f unladed rectangular late; = duble iterated Lalace eratr in R ; ( ) = differentiatin with resect t time; ( ), ξ = artial differentiatin with resect t ξ. Intrductin Extensive effrts and cnsiderable amunt f research has been cncentrated n the redictin f

2 the nn-linear dynamic behaviur f rectangular lates with small deviatin frm flatness called initial gemetric imerfectin. Excellent reviews n the subject can be fund in articles written by Hui [-8]. Studies f the effect f gemetric imerfectin n the small-amlitude vibratin frequencies f simly surted rectangular lates have been dne by Hui and Leissa [], Ilank and Dickinsn [9] and Bugaru [1]. They fund ut that gemetric imerfectins f the rder f the late thickness may raised the vibratin frequencies and may even cause the structures t exhibit sft-sring behaviur [7]. The survey f the literature reveals that the wrk n the subject has been devted t the investigatin f varius tyes f shaes, ladings, and bundary cnditins [11-13]. The resent wrk cvers an existing ga in ur understanding f the arametric resnance f cntinuus systems and resents a ratinal analysis f the influence f gemetric imerfectins un the amlitude and the hase angle fr the statinary nnlinear dynamic resnse. 3 Cncetual Mdel N y (t) = N y +N y t use thin late thery. Cnsequently, since thin late thery is used in the analysis, the lading frequencies ver which lateral vibratins ccur are cnsiderably belw the natural frequencies f lngitudinal vibratins and in-lane inertia frces can be neglected. 4 Basic Equatins The late thery used in this analysis may be cnsidered as the dynamic analgue f the Vn Karman large-deflectin thery and is derived in terms f Airy s stress functin, the lateral dislacement and the initial gemetric imerfectin. The differential equatins gverning the nn-linear flexural vibratins f the late are: f=e[((w+w ),xy ) -(w,xy ) -(w+w ),xx (w+w ),yy + +w,xx. w,yy ] (1) w = h/d[f,yy. (w+w ),xx -. f,xy (w+w ),xy + +f,xx (w+w ),yy - ρ. w,tt ] where D = Eh 3 /1(1-ν ). The bundary stress cnditins (in-lane mvable edges) are exressed as: f,yy = and f,xy = alng x =,a () f,xx = -N Y (t) and f,xy = alng y =,b The bundary surting cnditins are exressed as: w = w,xx + ν. w,yy = alng x =,a (3) w = w,yy + ν. w,xx = alng y =,b The rblem cnsists in determining the functins f and w, fr a given functin w, which satisfy the gverning equatins (1) tgether with the bundary cnditins () and (3). Fig. 1 The mdel under investigatin is an imerfect rectangular late simly surted alng its edges and acted by eridic in-lane frces unifrmly distributed alng tw site edges as shwn in figure 1. It is assumed that the late is f unifrm thickness, stress free, elastic, hmgeneus and istric and als the late thickness and the resulting dislacements are small cmared with the wavelength f lateral vibratin in rder t be able t 5 Methd Of Slutin Alying the Kantrvich s methd t the gverning equatins (3) as in [1], intrducing linear daming and taking ne term in the exansin fr the lateral dislacement, the system is reduced t the fllwing differential equatin f mtin: w +. C. w + Ω [1 -. µ. (Ω/Ω) cs[θ(t)]]. w -. µ. cs[θ(t)]ω (w +d) + Mw 3 + 3M w. (w +d) =, (4) where d is the amlitude f the static defrmatin f the late and µ q = N yt / [(N cr N y )]. (5)

3 This is a secnd-rder nn-linear differential equatin with eridic cefficients, which may be cnsidered as an extensin f the standard Mathieu-Hill s equatin. 6 Slutin f the Temral Equatin f Mtin Mathematical techniques fr slving such rblems are limited and arximate methds are generally used. The methd f asymttic exansin in wers f a small arameter ε, develed by Mitrlskii [1], is a mst effective tl fr studying nn-linear vibrating systems with slwly varying arameters. Assuming that the viscus daming and the nnlinearity are small and the instantaneus frequency f excitatin and the lad arameter vary slw with the time the equatin (5) can be written, by denting W=w and Θ=θ, in the fllwing asymttic frm: W+Ω. W = ε[µ. cs[θ(t)]. Ω (W +W +d) -. C. W - MW 3-3MW (W +d) (6) W ( t) = W cs + [ M /(3 ((1/ ) Θ + ψ ) _ [( µ Ω ) / ( Λ( Λ + Ω))] Ω 3 )] W cs W cs( (3/ ) Θ + ψ ) ((3/ ) Θ + 3ψ ) + (9) [µ Ω / ( Λ Ω )]( W + d ) cs Θ [3M /(Ω )]( W + d ) W + + [ M /(Ω )]( W + d ) W cs( Θ + ψ ) The slutin (9) was cmuted fr the regin f rincial arametric resnance. The arametric resnance ccurs when the excitatin frequency is arximately equal t twice the natural frequency and can be written as: Λ Ω (1) Analysing relatin (9), the aer reveals, fr the first time, new terms nt yet mentined by the researchers in the field. where τ = εt is the slwing time. Fr the secnd rder f arximatin in ε, we seek a slutin fr the equatin (6) in the fllwing frm: W=W (τ) cs[((1/)θ+ψ )]+ε u(τ,w, Θ,(1/)Θ+ψ ), (7) where W, ψ are functins f time defined by the system f differential equatins: dw /dt = ε A 1 (τ, W, ψ ) + ε A (τ, W, ψ ) dψ /dt = Ω - (1/)Λ + ε B 1 (τ, W, ψ ) + +ε B (τ, W, ψ ) (8) and dθ(t)/dt = Λ(t). Functins u, A 1, A, B 1, B are selected in such a way that the W, given by (7), will reresent a slutin f the equatin (6), after relacing W and ψ by the functins defined in the system (8). Fllwing the general scheme f cnstructing asymttic slutins and erfrming numerus transfrmatins and maniulatins, we can finally arrive at a system f equatins describing the nnstatinary resnse f the discretized system. By integrating this system f equatins, amlitude W and hase angle ψ can be btained as functins f time. The slutin W f the equatin (6) is 7 Statinary Resnse The statinary resnse given by the amlitude W and the hase angle ψ, assciated with the assumed satial frms f vibratin f ur system, may be cmuted as a secial case f the nnstatinary mtin in the resnant regime described by the systems f equatins (8) and (9). As mentined by Ostiguy and Nguyen [1, 13] the slutin fr simly-surted lates indicates the resence f rincial arametric resnances, the ssibility f internal resnances and the ccurrence f simultaneus resnances but recludes the ssibility f cmbinatin resnances. As can be seen in relatin (9), the authrs funded fr the first time, with analytical tls, the influence f the gemetric imerfectins in the regins f frced, sub-harmnic and sura-harmnic arametric resnances. In this way was fund theretical the resence f internal resnances and the ccurrence f simultaneus resnances already mentined exerimentally by Ostiguy and Nguyen. Statinary rincial arametric resnse, assciated with varius satial frms f vibratin, are given by the system (8) setting ε = 1, dw /dt =, dψ /dt = and eliminating ψ frm this system f equatins. Thus the statinary amlitude W can be btained as functin f external excitatin frequency and reresents the slutin f the fllwing equatin

4 7 1 (i 1) β W = (11) i = 1 i As mentined by Ostiguy and Evan-Iwanwski [11] the base width f the statinary arametric resnse is the nly regin in which vibratins may nrmally initiate. The hase angle f the statinary arametric resnse can be btained frm the same system (8) setting dw /dt =, dψ /dt = and eliminating the amlitude W. By this way was btained the statinary hase angle in the regin f rincial arametric resnance frm the fllwing equatin: 1 3 ψ = arcsin{[ C ( CM / Ω ) W ]/ 8 1 /[( )[( µ Ω M( Λ + ΛΩ + 7Ω )) 3 /( Λ( Λ + Ω)(4Ω Λ) Ω ] W [(6µ Ω M)/( Λ( Λ µ Ω ]} Λ Ω ))]( W + d) (1) Equatins (11) and (1) make it ssible t cmute the statinary resnse f the late at the rincial arametric resnance by taking int accunt the gemetrical imerfectins f the late. 8 Results and Discussins Fr the cmuter rgrams develed t btain the numerical results the authrs used the sft ackages MATLAB. In rder t get mre insight int varius asects f the rblem and t highlight the influence f the initial gemetric imerfectins n the nnlinear dynamic resnse f rectangular lates, numerical evaluatin f the slutin were erfrmed fr a wide variety f cases. The results shwn in figures and 3 are tyical f thse btained. Fr =.1 were funded the amlitude and the hase angle f the vibratins fr the late subjected t arametric excitatin having mderate Amlitude / h Amlitude / h Excitatin frequency [Hz] a. w / h = Excitatin frequency [Hz] b. w / h =.6 Fig. imerfectins (w /h=.1) and large nes (w /h=.6).by regarding the abve-mentined figures we can cnclude that by increasing the imerfectins aears the henmena f simultaneus resnances mentined by Nguyen [13]. This henmena manifests itself by multile salts and the effect f sft sring in the area f [65,85] Hz. This was determined fr the first time theretical while Nguyen discvered it exerimentally. Als frm figure 3 we see that in the area f simultaneus resnances the hase angle is cnstant and in the mean time all ver the area is negative therefre the nn-linear dynamic resnse f the late is in advance with regard t the excitatin.

5 Phase angle [rad] Phase angle [rad] Excitatin frequency [Hz] a. w / h = Excitatin frequency [Hz] b. w / h =.6 Fig. 3 References [1] Bugaru, M.J., The Influence f Gemetric Imerfectins n The Nn-linear Dynamical Behaviur f Parametrically Excited Plates, Internatinal Jurnal f Acustics and Vibratin, Vl.3, n.1, 1998, [] Hui, D. and Leissa, A. W., Effects f Gemetric Imerfectins n Vibratins f Biaxially Cmressed Rectangular Flat Plates, ASME Jurnal f Alied Mechanics, Vl. 5, Dec. 1983, [3] Hui, D., Large Amlitude Axisymmetric Vibratins f Gemetrically Imerfect Circular Plates, J. f Sund and Vibratin, Vl. 91, N., 1983, [4] Hui, D. and Leissa, A.W., Effects f Uni- Directinal Gemetric Imerfectins n Vibratins f Pressurised Shallw Sherical Shells, Int. J. f Nn-linear Mechanics, Vl. 18, N. 4, 1983, [5]Hui, D., Influence f Gemetric Imerfectins and In-Plane Cnstraints n Nn-linear Vibratins f Simly Surted Cylindrical Panels, ASME Jurnal f Alied Mechanics, Vl. 51, June 1984, [6] Hui, D., Effects f Gemetric Imerfectins n Frequency-Lad Interactin f Biaxially Cmressed Antisymmetric Angle Ply Rectangular Plates, AIAA Jurnal, Vl. 1, 1983, [7] Hui, D., Effects f Gemetric Imerfectins n Large-Amlitude Vibratins f Rectangular Plates With Hysteresis Daming, ASME Jurnal f Alied Mechanics, Vl. 51, March 1984, [8] Hui, D., Large Amlitude Vibratins f Gemetrically Imerfect Shallw Sherical Shells with Structural Daming, AIAA Jurnal, Vl. 1, 1983, [9] Ilank, S. and Dickinsn, S.M., The Vibratin and Pst-Buckling f Gemetrically Imerfect, Simly Surted, Rectangular Plates Under Uni- Axial Lading, Part I: Theretical Arach, J. f Sund and Vibratin, Vl. 118, N., 1987, [1] Mitrlskii, Yu. A., Prblems f the Asymttic Thery f Nnstatinary Vibratins. Mscw: Izdatel stv Nauka, 1964; English Translatin: (New Yrk) D. Davey & C., [11] Ostiguy, G.L. and Evan-Iwanwski, R.M., Influence f the Asect Rati n the Dynamic Stability and Nn-linear Resnse f Rectangular Plates, ASME Jurnal f Mechanical Design, Vl. 14, Aril 198, [1] Ostiguy, G.L. and Nguyen, H., Stabilité dynamique et résnance des laques rectangulaires, Mécanique Matériaux Electricité (G.A.M.I), N , Oct.-Nv. 198, [13] Ostiguy, G.L. and Nguyen, H., Influence f Bundary Cnditins n the Dynamic Stability and Nn-linear Resnse f Rectangular Plates, Develments in Mechanics, Vl. 13, Prc. f the 19th Midwestern Mechanics Cnference, The Ohi State University, Set. 1985, [14] Timshenk, S.P. and Gere, J.M. Thery f Elastic Stability, New Yrk. McGraw-Hill Inc., 1961.