NON-LINEAR DYNAMIC BEHAVIOR OF THIN RECTANGULAR PLATES PARAMETRICALLY EXCITED USING THE ASYMPTOTIC METHOD, PART 2: COMPUTATION OF THE PHASE ANGLE

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1 Proceedings of the 9th WSEAS International Conference on Alied Mathematics, Istanbul, Turkey, May 7-9, 6 (9-99 NON-LINEAR DYNAMIC BEHAVIOR OF THIN RECTANGULAR PLATES PARAMETRICALLY EXCITED USING THE ASYMPTOTIC METHOD, PART : COMPUTATION OF THE PHASE ANGLE MIHAI BUGARU EUGEN TRANĂ ADRIAN ROTARIU SORIN GHEORGHIAN Deartment of Mechanics, Politehnica University of Bucharest Slaiul Indeendentei 1, Bucharest 776 ROMANIA htt:// Deartment of Mechanics Military Technical Academy George Coşbuc 81-8, Bucharest ROMANIA htt:// Abstract: - The aer reveals recent develoments of the influence of the geometric imerfections on the hase angle of the non-linear vibrations of thin rectangular lates arametrically excited. In the region of rincial arametric resonance, starting from the temoral non-linear differential equation that describes the oscillatory movement and using the second order aroximation of the asymtotic method was comuted the hase angle as function of system arameters and geometric imerfections. By varying the intensity of the geometric imerfections was obtained their influence uon the hase angle for the stationary non-linear dynamic resonse. Key words: Parametric vibrations, thin late, asymtotic method, hase angle Nomenclature A 1, A, B 1, B unknown functions in asymtotic exansion; C viscous daming coefficient; D flexural rigidity of late; E Young s modulus; M coefficient of the non-linear term; N y (t external in-lane loading er unit width; N y static in-lane loading er unit width; N yt amlitude of harmonic in-lane loading er unit width; N cr critical buckling load of the late, defined as in [1]. 5; W amlitude of the arametric vibration; a length of late in x-direction; b length of late in y-direction; f(x,y,t Airy s stress function; h late thickness; ttime; w(x,y,tlateral mid-surface dislacement in z-direction; w (x,y initial geometric imerfection in z-direction; decrement of daming; Λ(t instantaneous frequency of the external in-lane excitation, Λ dθ/dt; Ω free vibration circular frequency of a rectangular late loaded by a constant comonent of in-lane force; Ω free vibration circular frequency of a rectangular late, with initial geometric imerfections, loaded by a constant comonent of in-lane force; ε small ositive arameter in asymtotic exansion, <ε<<1;

2 Proceedings of the 9th WSEAS International Conference on Alied Mathematics, Istanbul, Turkey, May 7-9, 6 (9-99 θ (t total hase angle of harmonic excitation; µ load arameter of the late; ν Poisson s ratio; ρ mass density er unit volume of late; τ slowing time in asymtotic analysis; ψ (t hase angle of the arametric vibration; double iterated Lalace oerator in R ; ( differentiation with resect to time; (, ξ artial differentiation with resect to ξ. 1. Introduction Extensive efforts and considerable amount of research has been concentrated on the rediction of the non-linear dynamic behavior of rectangular lates with small deviation from flatness called initial geometric imerfection. Excellent reviews on the subject can be found in articles written by Hui [-8]. Studies of the effect of geometric imerfection on the smallamlitude vibration frequencies of simly suorted rectangular lates have been done by Hui and Leissa Figure [], 1 Ilanko and Dickinson [9] and Bugaru [1]. They found out that geometric imerfections of the order of the late thickness may raised the vibration frequencies and may even cause the structures to exhibit soft-sring behavior [7]. The survey of the literature reveals that the work on the subject has been devoted to the investigation of various tyes of shaes, loadings, and boundary conditions [11-1]. N y (t N y o N yt cosθ(t Figure 1 The resent work covers an existing ga in our understanding of the arametric resonance of continuous systems and resents a rational analysis of the influence of geometric imerfections uon the hase angle for the stationary non-linear dynamic resonse.. Concetual Model The model under investigation is an imerfect rectangular late simly suorted along its edges and acted by eriodic in-lane forces uniformly distributed along two oosite edges as shown in figure 1. It is assumed that the late is of uniform thickness, stress free, elastic, homogeneous and isotroic and also the late thickness and the resulting dislacements are small comared with the wavelength of lateral vibration in order to be able to use thin late theory. Consequently, since thin late theory is used in the analysis, the loading frequencies over which lateral vibrations occur are considerably below the natural frequencies of longitudinal vibrations and in-lane inertia forces can be neglected..basic Equations The late theory used in this analysis may be considered as the dynamic analogue of the von Karman largedeflection theory and is derived in terms of Airy s stress function, the lateral dislacement and the initial geometric imerfection. The differential equations governing the non-linear flexural vibrations of the late are: f E [(( w w, xy ( w, xy (1 ( w w ( w w w w ], w h / D[ f, yy, xx ( w w, yy, xx f, xy, xx, yy ( w w, xy f, xx ( w w, yy ρ w, tt ], where D Eh /1(1-ν. The boundary stress conditions (in-lane movable edges are exressed as: f, yy and f, xy along x,a ( f N ( and f along y,b, xx y t, xy

3 Proceedings of the 9th WSEAS International Conference on Alied Mathematics, Istanbul, Turkey, May 7-9, 6 (9-99 The boundary suorting conditions are exressed as: w w, xx ν w, yy along x,a ( w w, yy ν w, xx along y,b. The roblem consists in determining the functions f and w, for a given function w, which satisfy the governing equations (1 together with the boundary conditions ( and (.. Method of Solution Alying the Kantorovich s method to the governing equations ( as in [1], introducing linear daming and taking one term in the exansion for the lateral dislacement, the system is reduced to the following differential equation of motion: w&& C w& Ω [1 µ ( Ω / Ω µ cos( θ ( t Ω ( w d cos( θ ( t] w M w M w ( w d where d is the amlitude of the static deformation of the late [1] and µ N yt / [(N cr N y ]. (5 This is a second-order non-linear differential equation with eriodic coefficients, which may be considered as an extension of the standard Mathieu-Hill s equation. ( 5. Solution of The Temoral Equation of Motion Mathematical techniques for solving such roblems are limited and aroximate methods are generally used. The method of asymtotic exansion in owers of a small arameter ε, elaborated by Krylov and Bogoliubov and develoed by Mitroolskii [1], is a most effective tool for studying non-linear vibrating systems with slowly varying arameters. The solution is develoed in the region of rincial arametric resonance that is defined by Λ Ω, (6 dθ Λ θ. & (7 dt Assuming that the viscous daming and the non-linearity are small and the instantaneous frequency of excitation and the load arameter vary slow with the time i.e. µ εµ, C ε C, M ε M. (8 The equation ( can be written, by denoting Θ θ, in the following asymtotic form: w&& Ω w ε [ C w& µ ( τ cos( Θ( τ Ω ( w w d M w M w ( w d], (9 where τ εt is the slowing time. For the second order of aroximation in ε, we seek a solution for the equation (9 in the following form: w W (τ cos[((1/θψ ] ε u(τ,w, Θ,(1/Θψ, (1 where W, ψ are functions of time defined by the system of differential equations: 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 (11 and dθ(t/dt Λ(τ. Functions u, A 1, A, B 1, B are selected in such a way that the w, given by (1, will reresent a solution of the equation (9, after relacing W and ψ by the functions defined in the system (11. Following the general scheme of constructing asymtotic solutions and erforming numerous transformations and maniulations, we can finally arrive at a system of equations describing the nonstationary resonse of the discretized system. By integrating this system of equations, amlitude W and hase angle ψ can be obtained as functions of time from the following system where

4 Proceedings of the 9th WSEAS International Conference on Alied Mathematics, Istanbul, Turkey, May 7-9, 6 (9-99 dw [ α1w α ( w d W αw dt α W cosψ α W α W, 1 α W 1 α, dψ 1 Ω Λ [ α7w α8( w d α9 ]cosψ dt α sinψ α W α ( w d W where, 1 α1 [( µ Ω M ( Λ ΛΩ 7Ω /( Λ( Λ Ω(Ω Λ Ω ], α 6µ Ω M Λ( Λ Ω Ω µ µ Ω, CM α α τ, α 5, Λ Λ 8 Ω α C, α α, α α, α α, α 7 [( µ Ω M ( Λ 6ΛΩ Ω / /( Λ( Λ Ω(Ω Λ Ω ], 15 M α11, 56 Ω 15 M M α1, α1, Ω 8 Ω µ Ω ( Λ Ω C Λ ( Λ Ω α1. ΩΛ ( Λ Ω, ]sinψ / The solution w of the equation (9 is w( t W cos( (1/ Θ ψ [( µ Ω / ( Λ( Λ Ω] W cos [ M /( Ω ] W cos (1 (1 ((/ Θ ψ (( / Θ ψ [µ Ω / ( Λ Ω ]( w d cos Θ o [M /(Ω ]( w d W o [ M /(Ω ]( w d W o cos ( Θ ψ (1 Analyzing relation (1, the aer reveals, for the first time, new terms not yet mentioned by the researchers in the field. 6. Stationary Resonse The stationary resonse given by the amlitude W and the hase angle ψ, associated with the assumed satial forms of vibration of our system, may be comuted as a secial case of the nonstationary motion in the resonant regime described by the system of equations (1 and equation (1. As mentioned by Ostiguy and Nguyen [1, 1] the solution for simly-suorted lates indicates the resence of rincial arametric resonance, the ossibility of internal resonance and the occurrence of simultaneous resonance but recludes the ossibility of combination resonance. As can be seen in relation (1, the authors founded for the first time, with analytical tools, the influence of the geometric imerfections in the regions of forced, sub-harmonic and sura-harmonic arametric resonance. In this way was found theoretical the resence of internal resonance and the occurrence of simultaneous resonance already mentioned exerimentally by Ostiguy and Nguyen. As mentioned by Ostiguy and Evan-Iwanowski [11] the base width of the stationary arametric resonse is the only region in which vibrations may normally initiate. The hase angle of the stationary arametric resonse can be obtained from the system (1 setting dw /dt, dψ /dt and eliminating the amlitude W. By this way was obtained the stationary hase angle in the region of rincial arametric resonance from the following equation: 1 ψ arcsin{[ C ( CM / Ω W ]/ 8 1 [( [( µ Ω M( Λ ΛΩ 7Ω / /( Λ( Λ Ω(Ω Λ Ω ] W µ Ω [(6µ Ω M /( Λ( Λ Ω ]( Wo d ]}. Λ (15 Equation (15 makes ossible to comute the hase angle of stationary

5 Proceedings of the 9th WSEAS International Conference on Alied Mathematics, Istanbul, Turkey, May 7-9, 6 (9-99 resonse of the late at the rincial arametric resonance by taking into account the geometrical imerfections of the late. 7. Results and Discussions For the comuter rograms develoed to obtain the numerical results the authors used the soft ackages MATLAB. In order to get more insight into various asects of the roblem and to highlight the influence of the initial geometric imerfections on the non-linear dynamic resonse of rectangular lates, numerical evaluation of the solution were erformed for a wide variety of cases. The results shown in figures and are tyical of those obtained. For.1 were founded the hase angle of the vibrations for the late subjected to arametric excitation having moderate imerfections (w o /h.1 and large ones (w o /h.6. By regarding the above-mentioned figures we can conclude that by increasing the imerfections aears the henomena of simultaneous resonance mentioned by Nguyen [1]. This henomena manifests itself by multile salts and the effect of soft sring in the area of [65,85] Hz. This was determined for the first time theoretical while Nguyen discovered it exerimentally. Also from figure we see that in the area of simultaneous resonance the hase angle is constant and in the mean time all over the area is negative therefore the non-linear dynamic resonse of the late is in advance with regard to the excitation. Phase angle [rad] Excitation frequency [Hz] Figure. w o / h.1 Phase angle [rad] Excitation frequency [Hz] Figure. w o / h.6 References [1] Bugaru, M.J., Predoi,M.V., Vibrations of arametrically rectangular excited lates( in Romanian, Bren Publishing House, Bucharest, 1999, ISBN [] Hui, D. and Leissa, A. W., Effects of Geometric Imerfections on Vibrations of Biaxially Comressed Rectangular Flat Plates, ASME Journal of Alied Mechanics, Vol. 5, Dec. 198, [] Hui, D., Large Amlitude Axisymmetric Vibrations of Geometrically Imerfect Circular Plates, J. of Sound and Vibration, Vol. 91, No., 198, [] Hui, D. and Leissa, A.W., Effects of Uni-Directional Geometric Imerfections on Vibrations of Pressurised Shallow Sherical Shells, Int. J. of Non-linear Mechanics, Vol. 18, No., 198, [5] Hui, D., Influence of Geometric Imerfections and In-Plane Constraints on Non-linear Vibrations of Simly Suorted Cylindrical Panels, ASME Journal of Alied Mechanics, Vol. 51, June 198, [6] Hui, D., Effects of Geometric Imerfections on Frequency-Load Interaction of Biaxially Comressed Antisymmetric Angle Ply Rectangular Plates, AIAA Journal, Vol. 1, 198, [7] Hui, D., Effects of Geometric Imerfections on Large-Amlitude Vibrations of Rectangular Plates With

6 Proceedings of the 9th WSEAS International Conference on Alied Mathematics, Istanbul, Turkey, May 7-9, 6 (9-99 Hysteresis Daming, ASME Journal of Alied Mechanics, Vol. 51, March 198, [8] Hui, D., Large Amlitude Vibrations of Geometrically Imerfect Shallow Sherical Shells with Structural Daming, AIAA Journal, Vol. 1, 198, [9] Ilanko, S. and Dickinson, S.M., The Vibration and Post-Buckling of Geometrically Imerfect, Simly Suorted, Rectangular Plates Under Uni- Axial Loading, Part I: Theoretical Aroach, J. of Sound and Vibration, Vol. 118, No., 1987,. 1-6 [1] Mitroolskii, Yu. A., Problems of the Asymtotic Theory of Nonstationary Vibrations. Moscow: Izdatel stovo Nauka, 196; English Translation: (New York D. Davey & Co., [11] Ostiguy, G.L. and Evan-Iwanowski, R.M., Influence of the Asect Ratio on the Dynamic Stability and Non-linear Resonse of Rectangular Plates, ASME Journal of Mechanical Design, Vol. 1, Aril 198, [1] Ostiguy, G.L. and Nguyen, H., Stabilité dynamique et résonance des laques rectangulaires, Mécanique Matériaux Electricité (G.A.M.I, No. 9-95, Oct.-Nov. 198, [1] Ostiguy, G.L. and Nguyen, H., Influence of Boundary Conditions on the Dynamic Stability and Non-linear Resonse of Rectangular Plates, Develoments in Mechanics, Vol. 1, Proc. of the 19th Midwestern Mechanics Conference, The Ohio State University, Set. 1985, [1] Timoshenko, S.P. and Gere, J.M. Theory of Elastic Stability, New York. McGraw-Hill Inc., 1961.