Nonlinear Forced Vibration Analysis for Thin Rectangular Plate on Nonlinear Elastic Foundation

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1 Researc Journal of Applied Sciences, ngineering and Tecnolog 5(6: 6-67, 0 ISSN: ; e-issn: Maxwell Scientific Organization, 0 Submitted: Jul, 0 Accepted: September 0, 0 Publised: Februar, 0 Nonlinear Forced Vibration Analsis for Tin Rectangular Plate on Nonlinear lastic Foundation Zong Zengqiang, Xiao Yonggang and Yang Cuiping Scool of Civil ngineering and Arcitect, Cangsa Universit of Science and Tecnolog, Cangsa, 404, Cina Abstract: Nonlinear forced vibration is analzed for tin rectangular plate wit four free edges on nonlinear elastic foundation. Based on Hamilton variation principle, equations of nonlinear vibration motion for tin rectangular plate under armonic loads on nonlinear elastic foundation are establised. In te case of four free edges, viable expressions of trial functions for tis specification are proposed, satisfing all boundar conditions. Ten, equations are transformed to a sstem of nonlinear algebraic equations b using Galerkin metod and are solved b using armonic balance metod. In te analsis of numerical computations, te effect on te amplitude-frequenc caracteristic curve due to cange of te structural parameters of plate, parameters of foundation and parameters of excitation force are discussed. Kewords: Amplitude-frequenc caracteristic, forced vibration, nonlinear elastic foundation, tin rectangular plate INTRODUCTION Rectangular plates are readil seen in man civil engineering applications suc as igwa concrete out laers, airport runwas, building foundations and so fort. Researces about its mecanical beaviors ave been undertaken b numerous experts, optimized Kantorovic metod and analzed te self-vibration caracteristics of tickness-varing rectangular plates wit one edge free b potesizing on its vibration mode function (Sonzogni et al., 990. Researced te vibration patterns of moderate tickness rectangular plates wit multifarious foundation models: considered coupled effect of elastic foundation, a nonlinear constant load analsis for moderate rectangular plate was conducted (Xiao et al., 004, a nonlinear free vibration equation for moderate tickness cracked plates ad been set and solved (Xiao et al., 005, its self-vibrating amplitude-frequenc caracteristic curve was ten analzed; researced nonlinear vibration of disconnected tin plate on elastic foundation (Xiao and Fu, 006 and conducted a analsis of disconnected tin rectangular plates self-vibration on nonlinear elastic foundation (Xiao and Yang, 0, meanwile, te constant load caracteristics of four free edges rectangular plate was discussed wit consideration of nonlinear elastic foundation (Xiao and Zong, 009. stablised an approac to nonlinear vibration of ortotropic tin rectangular plate on elastic foundation b using ortogonal collocation and solved its nonlinear eigenvalue wit iterative metod (Baskar and Dumir, 988. Also, tere were researces of rectangular plates vibration on linear elastic foundation (Qu and Liang, 996 and circle plates bifurcation and caotic beavior on nonlinear elastic foundation (Qiu and Wang, 00. So far, muc attention of examinations of plate vibration on elastic foundation as been drawn to te realm of free vibration, wereas forced vibration was discussed less frequentl. Hence, tis stud presents a researc of te nonlinear forced vibration caracteristics for tin rectangular plates on nonlinear elastic foundation to extend a discussion of forced vibration caracteristics of plates. Hamilton energ differentiation principles as been utilized for building a nonlinear forced vibrating equation of tin rectangular plates under armonic load, later it is solved b exploiting Galerkin metod and Harmonic balance metod. ffect of variables like geometric and mecanical parameters of plates, response modus of foundation, varing stimulus force on amplitudefrequenc caracteristic curve of plates forced vibration, teir resonance caracteristics were also analzed. Te outcomes serve as teoretical merit to te direction of construction programs. CONTROL QUATIONS Assume a tin rectangular wit lengt α, breadt b, tickness and four free edges. Lateral distribution of te armonic load is described as q(x,, t = q 0 (x, cos θt, in wic q 0 (x, is te amplitude of excitement, Corresponding Autor: Zong Zengqiang, Scool of Civil ngineering and Arcitect, Cangsa Universit of Science and Tecnolog, Cangsa, 404, Cina 6

2 Res. J. Appl. Sci. ng. Tecnol., 5(6: 6-67, 0 N xx, N, u, tt N, x N, v, tt Qxx, Q, w, ttq p ( N xw, xx N w, N w, M x, x M, Qx Jv, xtt M, x M, Q Ju, tt ( Fig. : Loaded tin rectangular plate θ is te frequenc of exciting force. A nonlinear Winker foundation is considered, p = - k w - k w in wic w, k and K are vertical dnamic deflection of plate, linear and nonlinear rigidit coefficients of elastic foundation, respectivel. Figure sows a loaded tin rectangular plate. Te total potential energ of tis sstem is gained b Π = Π + Π - Π, as Hamiltion energ differentiation principle interprets, wen te sstem reaces to a stead balance, its total potential energ as te minim value, alas δπ = 0, or δπ + δπ - δπ = 0Π is te strain energ of elastic plate, its value can be obtained b: ( xz xz z z dxddz x x z z Π is te stud due to motor inertia, te particular value is calculated b: [ ( u, ttu v, ttv w, ttw J ( u u v v ] dxd, tt,, xtt, x were, P, J are te densit and moment of inertia of elastic plate, respectivel. Π is te stud done b external force: ( q p wdxd ( N u A C N v N w M v M u ds z x, x, were, Nx, N, Nz, Mx, M are te known force and torment on te boundar of elastic plate, wile v u uvw,,,,, are te known displacement on x te boundar. Terefore, te motor control equations of tin rectangular plate on nonlinear elastic foundation are: x 64 were, N x, N, N membrane internal force of plate and teir relationsip are can be described as: [ x,, ] [ x,, ] N N N dz M x, M, M are internal torques of plate wilst fitting in te following relationsip: [M x,m,m ] [ x,, ]z dz ; Q x, Q Are transversal sear wic can be calculated b: [ x, ] [ xz, z ] Q Q dz For tin rectangular plate wit free edges, te boundar conditions of its surface forces are: x 0, a Mx Mx, M M, Qx Qx, Nx Nx, N N; 0, b M M, M M, Q Q, N N, N N ( According to te interrelationsip between tin rectangular plate s tin film stress, its bending stress and its displacement, se control equation of forced vibration of four-edge-free tin rectangular plate on nonlinear elastic foundation are:, xxxx, x, ( w, w, xxw, 4 D w kw k w w, tt (, w, xx, xxw,, w, q ( were, te flexural rigidit of plate is D = / ( - μ ; elastic modus and Poisson s ration are, μ respectivel; ø is te stress function. To clarif equations, a few dimensionless parameters are introduced:

3 Res. J. Appl. Sci. ng. Tecnol., 5(6: 6-67, 0 x w,, W a b F,, K k K k 4 ( K, t ab ( a ( Q0 q0, D a b ( (, were, Q = Q 0 (ξ, η cos Θτ (4 Hence, te dimensionless control equations of forced vibration of four-edge-free tin rectangular plate are sown as: F F F F, W, F, W, Q 4 4,,, ( W, W, W, W, W, W, KW K W W, K ( F, W, (5 Te sape functions distinctivel satisf all boundar conditions To introduce q. (7 to (5 te residual value is acquired, ten te ortogonalit relation is preset b combining Galerkin metod and Vibration igen functions of beam, nonlinear ordinar differential equations F mn (τ, W (τ are sown as: mn st aij Fmn ( aij W ( Wst ( 0 (8 aijw ( a4ijw ( mn a5ij Fmn ( W ( a6ij cos were, α mn ij to α 6ij are constant coefficients wose values are mentioned elsewere. In general q. (8 are solved wit armonic balance metod, tus unknown functions F mn (τ, W (τ are expended as Cosine Fourier series for time parameter τ: ( k Fmn ( Fmn cosk k 0 ( k W ( W cosk k 0 (9 Its dimensionless boundar conditions are: 0, M 0, M 0, Q 0, F, 0, F, 0; 0, M 0, M 0, Q 0, F, 0, F, 0 SOLUTION TO TH QUATION (6 Based on plate s boundar conditions solution to q. (5 is assumed in terms of separated functions: F(,, Fmn( Xm ( Yn ( m n W(,, W ( p ( q ( p q (7 were, coefficients F (k mn, W (k are te k-t amplitude of armonic waves for F mn (τ and W (τ, respectivel; ω, te dimensionless frequenc of plate s forced vibration, is associated wit circular frequenc of forced vibration : ( (, 0 (, 0 were, ω 0 and 0 are dimensionless and dimensional base frequenc for linear forced vibration of plate, respectivel. To combine q. (9 and (8 te control equation for plate s frequenc of forced vibration is present as: were, X (cos a cos a b (sin a m m m m m sin a m Yn (cos an cos an bn(sin an sin a n (m,n=,,..., (, ( (cos a cos a b (sin a sin a p p p p p (cos a cos a b (sin a sin a q q q q q (p,q=,4,5... q p 65 a a a M M st mn ij 5ij ij ( mn k0 aij k0 4ij 6ij cos k 0 a k M a (0 were, M = W (k cos kωτ Steps for calculation are as followed: First off enter te geometric and mecanical parameters of plate, elastic modulus of foundation and exciting force, ten using q. (0 to ield ω, q. (9 for one set values of F mn, W, F(ξ, η, τ and W(ξ, η, τ is calculated troug q. (7. Finall, te amplitude-frequenc curve is

4 Res. J. Appl. Sci. ng. Tecnol., 5(6: 6-67, 0 Fig. : Amplitude-frequenc curve of various tick plates Fig. : Amplitude-frequenc curve of different K illustrated for tin rectangular plate on nonlinear elastic foundation wen plate s parameters and foundation s parameters var. XAMPL ANALYSIS Witout consideration of exciting force effect, tis issue witdraws to issue of free vibration, wose teoretical analses and solution were given b Xiao and Yang (0. Tose analses verif te accurate and suitable coices of algoritm, te trial function and solution. Assuming base soil is common coesive soil, parameters for elastic plate are a =.8 m, b =.5 m, = 0. m, = 0 4 MN/m, μ = 0.5, ρ = 450 kg/m ; te rigid coefficients of foundation are k = 0 MN/m, k = 40 MN/m ; dimensionless amplitude of exciting force is Q 0 (ξ, η = 0.0, dimensionless frequenc ratio is s = Θ/ω = Tree plates wit tickness of 0.05, 0. and 0.5 m were selected wile oter parameters are identical. Figure indicates te effect of tickness on Amplitudefrequenc curve. Wen amplitude of forced vibration sta te same, te frequenc of plate s forced vibration increases drasticall as tickness became greater. Figure and 4 illustrate foundation parameters K and K s impact on Amplitude-frequenc curve of plate s forced vibration. Wit a stable amplitude increment of K and K lead to enance in frequenc wic indicates moderate elevation of foundation s response modulus elps to increase te forced vibration frequenc of plate. Figure 5 illustrates te effect of exciting force s amplitude on Amplitude-frequenc curve of plate s forced vibration. As exciting force amplitude abates from.5 to 0.5 wile oter parameter still, forced vibration frequenc increases. 66 Fig. 4: Amplitude-frequenc curve of different K Fig. 5: Amplitude-frequenc curve of different exciting force amplitude Seen in Fig. 6 wen s = Θ/ω approaces to te value, namel wen frequenc nearl equal to

5 Res. J. Appl. Sci. ng. Tecnol., 5(6: 6-67, 0 caracteristic can be exploited in structure and pavement demolition, as adjusting load frequenc to sstem s fixed frequenc. RFRNCS Fig. 6: ffect of exciting force frequenc on amplitudefrequenc curve of plate sstem s fixed frequenc, te amplitude of forced vibration w drasticall rises, te resonance region approximatel lies between s = 0.7 to.. Also, as foundation s linear coefficient strengtens, nonlinear caracteristic of sstem fades. CONCLUSION Tis stud presents an analsis of armonicexcited forced vibration of tin rectangular plate on nonlinear elastic foundation, including te consequential effects of mecanical parameters of plate, response modulus of foundation and cange in exciting force on Amplitude-frequenc curve. Te results sow, in general, frequenc of forced vibration augments along wit increment of plate s amplitude. Wen amplitude remains stable, increments in plate s tickness and response modulus of foundation result in rise of elastic plate s forced vibration frequenc; wile iger amplitude of exciting force leads to lower forced vibration frequenc. speciall wen frequenc of exciting force is 0.7 to. times of sstem s fixed frequenc forced vibration amplitude intensivel expends a clearl caracteristic of resonance effect. Terefore in civil engineering application suc Baskar, A. and P.C. Dumir, 988. Non-linear vibration of ortotropic tin rectangular plates on elastic foundations. J. Sound. Vib., 5(: -. Qiu, P. and X. Wang, 00. Bifurcation and caos problem of tin circular plate on nonlinear elastic foundation. Appl. Mat. Mec., 4(8: Qu, Q., X. Liang, 996. Nonlinear analsis of free rectangular plate on elastic foundation. ng. Mec., (: Sonzogni, V.. and S.R. Idelson 990. Free vibration of rectangular plates of exponentiall varing tickness and wit a free edge. J. Sound. Vib., 40(: 5-5. Xiao, Y. and Y. Fu, 006. Nonlinear vibration of disconnected XIAO moderatel tick rectangular plate on elastic foundation. J. Vib. ng. 9(: 6-6. Xiao, Y. and J. Zong, 009. Nonlinear static analsis of moderate tickness rectangular plate wit four free edges on nonlinear elastic foundation. ng. Mec. 6(4: Xiao, Y. and C. Yang, 0. Free vibration analsis for disconnected tin rectangular plate wit four free edges on nonlinear elastic foundation. Appl. Mec. Mat., 5-54: Xiao, Y., Y. Fu and C.H.A. Xudong, 004. Nonlinear static analsis of moderatel tick rectangular plate considering te coupled effect of elastic foundation. ngineering Mecanics, (4: Xiao, Y., Y. Fu and X. Za, 005. Nonlinear vibration for moderate tickness rectangular cracked plates including coupled effect of elastic foundation. Appl. Mat. Mec., 6(8: