Vibrations Of Circular Cylindrical Panels Research Articles

Theory and experiments for large-amplitude vibrations of circular cylindrical panels with geometric imperfections

Large-amplitude vibrations of circular cylindrical panels (open shells) subjected to harmonic excitation are numerically and experimentally investigated. The Donnell nonlinear strain–displacement relationships are used to describe the geometric nonlinearity; in-plane inertia is taken into account. Specific boundary conditions, with zero transverse displacement at the panel edges and free or elastically restrained in-plane displacements, not previously considered, have been introduced in order to model the experimental boundary conditions. The nonlinear equations of motion are obtained by the Lagrange equations with multi-mode approach, and are studied by using a code based on the pseudo-arclength continuation method. Two thin circular cylindrical panels of different dimensions and made of stainless steel have been experimentally tested in the laboratory for several excitation amplitudes in order to characterize the nonlinearity. The dimensions of the two panels have been chosen in order to have the fundamental mode with one and two circumferential half-waves, respectively. Numerical results are able to reproduce the experimental results with high accuracy for both panels. The effect of geometric imperfections on the trend of nonlinearity and on natural frequencies is shown; convergence of the solution with the number of generalized coordinates is numerically verified.

Effect of Boundary Conditions on Nonlinear Vibrations of Circular Cylindrical Panels

Geometrically nonlinear vibrations of circular cylindrical panels with different boundary conditions and subjected to harmonic excitation are numerically investigated. The Donnell’s nonlinear strain–displacement relationships are used to describe geometric nonlinearity; in-plane inertia is taken into account. Different boundary conditions are studied and the results are compared; for all of them zero normal displacements at the edges are assumed. In particular, three models are considered in order to investigate the effect of different boundary conditions: Model A for free in-plane displacement orthogonal to the edges, elastic distributed springs tangential to the edges and free rotation; Model B for classical simply supported edges; and Model C for fixed edges and distributed rotational springs at the edges. Clamped edges are obtained with Model C for the very high value of the stiffness of rotational springs. The nonlinear equations of motion are obtained by the Lagrange multimode approach, and are studied by using the code AUTO based on the pseudo-arclength continuation method. Convergence of the solution with the number of generalized coordinates is numerically verified. Complex nonlinear dynamics is also investigated by using bifurcation diagrams from direct time integration and calculation of the Lyapunov exponents and the Lyapunov dimension. Interesting phenomena such as (i) subharmonic response; (ii) period doubling bifurcations; (iii) chaotic behavior; and (iv) hyper-chaos with four positive Lyapunov exponents have been observed.

Nonlinear vibrations of circular cylindrical panels

Large-amplitude (geometrically nonlinear) vibrations of circular cylindrical panels with rectangular base, simply supported at the four edges and subjected to radial harmonic excitation in the spectral neighbourhood of the lowest resonances are investigated. Two different nonlinear strain–displacement relationships, from the Donnell's and Novozhilov's shell theories, are used to calculate the elastic strain energy. In-plane inertia and geometric imperfections are taken into account. The solution is obtained by Lagrangian approach. The nonlinear equations of motion are studied by using (i) a code based on arclength continuation method that allows bifurcation analysis and (ii) direct time integration. Numerical results are compared to those available in the literature and convergence of the solution is shown. Interaction of modes having integer ratio among their natural frequencies, giving rise to internal resonances, is also discussed.