Nonlinear flutter instability of thin damped plates: A solution by the analog equation method

Abstract

We investigate the nonlinear flutter instability of thin elastic plates of arbitrary geometry subjected to a combined action of conservative and nonconservative loads in the presence of both internal and external damping and for any type of boundary conditions. The response of the plate is described in terms of the displacement field by three coupled nonlinear partial differential equations (PDEs) derived from Hamilton's principle. Solution of these PDEs is achieved by the analog equation method (AEM), which uncouples the original equations into linear, quasistatic PDEs. Specifically, these are a biharmonic equation for the transverse deflection of the plate, that is, the bending action, plus two linear Poisson's equations for the accompanying in-plane deformation, that is, the membrane action, under time-dependent fictitious loads. The fictitious loads themselves are established using the domain boundary element method (D/BEM). The resulting system for the semidiscretized nonlinear equations of motion is first transformed into a reduced problem using the aeroelastic modes as Ritz vectors and then solved by a new AEM employing a time-integration algorithm. A series of numerical examples is subsequently presented so as to demonstrate the efficiency of the proposed methodology and to validate the accuracy of the results. In sum, the AEM developed herein provides an efficient computational tool for realistic analysis of the admittedly complex phenomenon of flutter instability of thin plates, leading to better understanding of the underlying physical problem.

Nonlinear Vibrations of Viscoelastic Plates of Fractional Derivative Type: An AEM Solution

“…The associated boundary conditions result as [27] V n + D c V n + N n w, n +N t w, t +k T w =V n or w = w on…”

“…Step of the AEM Eqs (24), (27) and (28) give the displacements w(x,t),u(x,t) , v(x,t) and their derivatives provided that the three fictitious sources b(t), b (1) (t), b (2) (t) are first established. This is achieved by working as following.…”

“…Collocating the fractional PDEs (10) at the M internal nodal points and substituting the expressions for the transverse deflection, Eqs (24), and the membrane displacements, Eqs (27) and (28), we obtain the following system of 3M nonlinear fractional ordinary differential equations for…”

“…In this investigation the eigenmodes of the linear problem are selected as Ritz vectors [27]. Using Eq.…”

An initial stability of plates in various conservative load conditions by the boundary element method

“…A similar approach was applied by Chinnaboon, Chucheepsakul and Katsikadelis [5] to solve buckling analysis of plates. Katsikadelis and Babouskos [6] applied AEM in combination with the BEM to describe and solve the nonlinear flutter instability problem of thin dumped plates. In order to simplify the assembly of a set of algebraic equations and calculation procedures, Guminiak, Sygulski [7] and Guminiak [8] proposed a modified, simplified formulation of the boundary integral equations for a thin plate.…”