Non-Linear Dynamic Behavior of Thin Rectangular Plates Parametrically Excited Using the Asymptotic Method, Part 2: Computation of the Phase Angle MIHAI BUGARU * , OVIDIU VASILE * * Department of Mechanics University POLITEHNICA of Bucharest Splaiul Independentei 313, post code 060042, Bucharest ROMANIA Abstract: The paper reveals recent developments of the influence of the geometric imperfections on the phase angle of the non-linear vibrations of thin rectangular plates parametrically excited. In the region of principal parametric resonance, starting from the temporal non-linear differential equation that describes the oscillatory movement and using the second order approximation of the asymptotic method was computed the phase angle as function of system parameters and geometric imperfections. By varying the intensity of the geometric imperfections was obtained their influence upon the phase angle for the stationary non-linear dynamic response. Key-Words: Non-linear dynamics of plates Nomenclature A 1 , A 2 , B 1 , B 2 = unknown functions in asymptotic expansion; C = viscous damping coefficient; D = flexural rigidity of plate; E = Young’s modulus; M = coefficient of the non-linear term; N y (t) = external in-plane loading per unit width; N y0 = static in-plane loading per unit width; N yt = amplitude of harmonic in-plane loading per unit width; N cr = critical buckling load of the plate, defined as in [14] pp. 353; W p = amplitude of the parametric vibration; a = length of plate in x-direction; b = length of plate in y-direction; f(x,y,t) = Airy’s stress function; h = plate thickness; t=time; w(x,y,t)=lateral mid-surface displacement in z-direction; w 0 (x,y) = initial geometric imperfection in z-direction; ∆ = decrement of damping; Λ(t) = instantaneous frequency of the external in-plane excitation, Λ = dθ/dt; Ω = free vibration circular frequency of a rectangular plate loaded by a constant component of in-plane force; Ω =free vibration circular frequency of a rectangular plate , with initial geometric imperfections , loaded by a constant component of in-plane force; ε = small positive parameter in asymptotic expansion, 0<ε<<1; θ (t) = total phase angle of harmonic excitation; µ = load parameter of the plate; ν = Poisson’s ratio; ρ = mass density per unit volume of plate; τ = slowing time in asymptotic analysis; ψ p (t) = phase angle of the parametric vibration; ∆∆ = double iterated Laplace operator in R 2 ; ) ( • = differentiation with respect to time; ( ), ξ = partial differentiation with respect to ξ. 1 Introduction Extensive efforts and considerable amount of research has been concentrated on the prediction of the non-linear dynamic behavior of rectangular plates with small deviation from flatness called initial geometric imperfection. Excellent reviews on the subject can be found in articles written by Hui [2-8]. Studies of the effect of geometric imperfection on the small-amplitude vibration frequencies of simply supported rectangular plates have been done by Hui and Leissa [2], Ilanko and Dickinson [9] and Bugaru [1]. They found out that geometric imperfections of the order of the plate thickness may raised the vibration frequencies and may even cause the structures to exhibit soft-spring behavior [7]. The survey of the literature reveals that the work on the subject has been devoted to the investigation of various types of shapes, loadings, and boundary conditions [11-13]. Proceedings of the 10th WSEAS Interbational Conference on APPLIED MATHEMATICS, Dallas, Texas, USA, November 1-3, 2006 493