Zenon J. G. N. del Prado

Nonlinear Oscillations and Stability of Parametrically Excited Cylindrical Shells

Based on Donnell shallow shell equations, the nonlinear vibrations and dynamic instability of axially loaded circular cylindrical shells under both static and harmonic forces is theoretically analyzed. First the problem is reduced to a finite degree-of-freedom one by using the Galerkin method; then the resulting set of coupled nonlinear ordinary differential equations of motion are solved by the Runge–Kutta method. To study the nonlinear behavior of the shell, several numerical strategies were used to obtain Poincaré maps, Lyapunov exponents, stable and unstable fixed points, bifurcation diagrams, and basins of attraction. Particular attention is paid to two dynamic instability phenomena that may arise under these loading conditions: parametric excitation of flexural modes and escape from the pre-buckling potential well. Calculations are carried out for the principal and secondary instability regions associated with the lowest natural frequency of the shell. Special attention is given to the determination of the instability boundaries in control space and the identification of the bifurcational events connected with these boundaries. The results clarify the importance of modal coupling in the post-buckling solution and the strong role of nonlinearities on the dynamics of cylindrical shells.

Transient Stability of Empty and Fluid-Filled Cylindrical Shells

In the present work a qualitatively accurate low dimensional model is used to study the non-linear dynamic behavior of shallow cylindrical shells under axial loading. The dynamic version of the Donnell non-linear shallow shell equations are discretized by the Galerkin method. The shell is considered to be initially at rest, in a position corresponding to a pre-buckling configuration. Then, a harmonic excitation is applied and conditions to escape from this configuration are sought. By defining steady state and transient stability boundaries, frequency regimes of instability may be identified such that they may be avoided in design. Initially a steady state analysis is performed; resonance response curves in the forcing plane are presented and the main instabilities are identified. Finally, the global transient response of the system is investigated in order to quantify the degree of safety of the shell in the presence of small perturbations. Since the initial conditions, or even the shell parameters, may vary widely, and indeed are often unknown, attention is given to all possible transient motions. As parameters are varied, transient basins of attraction can undergo quantitative and qualitative changes; hence a stability analysis which only considers the steady-state and neglects this global transient behavior, may be seriously non-conservative. Keywords : Cylindrical shells, fluid-structure interaction, parametric instability, nonlinear vibrations

Influence of physical and geometrical system parameters uncertainties on the nonlinear oscillations of cylindrical shells

This work investigates the influence of physical and geometrical system parameters uncertainties and excitation noise on the nonlinear vibrations and stability of simply-supported cylindrical shells. These parameters are composed of both deterministic and random terms. Donnells non-linear shallow shell theory is used to study the non-linear vibrations of the shell. To discretize the partial differential equations of motion, first, a general expression for the transversal displacement is obtained by a perturbation procedure which identifies all modes that couple with the linear modes through the quadratic and cubic nonlinearities. Then, a particular solution is selected which ensures the convergence of the response up to very large deflections. Finally, the in-plane displacements are obtained as a function of the transversal displacement by solving the in-plane equations analytically and imposing the necessary boundary, continuity and symmetry conditions. Substituting the obtained modal expansions into the equation of motion and applying the Galerkins method, a discrete system in time domain is obtained. Several numerical strategies are used to study the nonlinear behavior of the shell considering the uncertainties in the physical and geometrical system parameters. Special attention is given to the influence of the uncertainties on the parametric instability and escape boundaries.

Influence of Modal Coupling on the Nonlinear Vibration of Simply Supported Cylindrical Panels

The aim of this paper is to analyse the influence of the nonlinear modal coupling on the nonlinear vibrations of a simply supported cylindrical panel excited by a time dependent transversal load. The cylindrical panel is modeled by the Donnell nonlinear shallow shell theory and the lateral displacement field is based on a perturbation procedure. The axial and circumferential displacement are described in terms of the obtained lateral displacement, generating a precise low-dimensional model that satisfies all transversal boundary conditions. The discretized equations of motion in time domain are determined by applying the standard Galerkin method. Various numerical techniques are employed to obtain the cylindrical panel resonance curves, bifurcation scenario and basins of attraction. The results show the influence of geometry and the nonlinear modal coupling on the nonlinear response of the cylindrical panel.

Nonplanar Dynamics of Fixed-Free Beams With Low Torsional Stiffness

The three-dimensional motions of a clamped-free, inextensible beam subject to lateral harmonic excitation are investigated in this paper. Special attention is given to the nonlinear oscillations of beams with low torsional stiffness and its influence on the bifurcations and instabilities of the structure, a problem not tackled in the previous literature on this subject. For this, the nonlinear integro-differential equations describing the flexural-flexural-torsional couplings of the beam are used, together with the Galerkin method, to obtain a set of discretized equations of motion, which are in turn solved by numerical integration using the Runge-Kutta method. Both inertial and geometric nonlinearities are considered in the present analysis. By varying the beam stiffness parameters, and using several tools of nonlinear dynamics, a complex dynamic behavior of the beam is observed near the region where a 1:1:1 internal resonance occurs. In this region several bifurcations leading to multiple coexisting solutions, including planar and nonplanar motions are obtained. Finally, the paper shows how the tools of nonlinear dynamics can help in the understanding of the global integrity of the model, thus leading to a safe design.Copyright

Theoretical and Experimental Study on Large Amplitude Vibrations of Clamped Viscoelastic Plates

In this paper, the large amplitude vibrations of clamped-clamped thin viscoelastic rectangular plates due to a concentrated transversal harmonic load are investigated both theoretically and experimentally. Clamped boundary condition on all edges and von Karman nonlinear strain-displacement relationships are considered while rotary inertia, geometric imperfections, and shear deformation are neglected. In the theoretical study, the viscoelastic behaviour of the material is modelled using the Kelvin-Voigt model. In-plane loads applied during the assembly of the plate are taken into account and clamped boundary conditions are modelled using artificial rotational springs. The nonlinear ordinary differential equations for the considered Kelvin-Voigt model are obtained using the generalized energy approach. These equations contain quadratic and cubic nonlinear viscoelasticity terms in addition to quadratic and cubic stiffness terms. Non-dimensionalization of variables is carried out and each second order equation is converted into two first order equations. The resulting system of equations is solved using AUTO (software based on the arclength continuation method that allows bifurcation analysis), to get the frequency-response curves at various force levels. Moreover numerical time integration of equations was also performed using the fourth-order Runge-Kutta method to understand the time response of the structure. In the experimental study, two rubber plates with different material and thicknesses were considered; a silicone plate with 0.0015 m thickness and a neoprene plate with 0.003 m thickness. The plates were fixed on a heavy rectangular metal frame thereby ensuring the clamped boundary condition on all edges. Linear experimental modal analysis was carried out as a first step to estimate the mode shapes and natural frequencies. In the second step, the nonlinear vibration response of the plate around its first resonance was measured at various harmonic force levels. At each force level, the amplitude of the harmonic excitation was kept constant by LMS Data Acquisition System and Test.Lab Stepped Sine software module while slowly varying the frequency of excitation to get the frequency-response curves. Laser Doppler Vibrometry was used to measure the response from the plate as it eliminates the possible mass loading effect introduced by any contact type sensors. A maximum amplitude of more than three times the thickness of the plate was achieved. The nonlinear response curves showed a typical hardening type nonlinearity along with sudden jumps as expected for plates. Experimental frequency-response curves were compared with theoretical results and a good agreement was found. The influence of nonlinear viscoelastic damping terms was clearly noticed on the response curves of the plate. The retardation time, measured in seconds decreases with increasing excitation force and larger amplitude vibrations.Copyright

GEOMETRY EFFECTS ON THE NONLINEAR OSCILLATIONS OF VISCOELASTIC CYLINDRICAL SHELLS

In this work the influence of geometry, load and material properties on the nonlinear vibrations of a simply supported viscoelastic circular cylindrical shell subjected to lateral harmonic load is studied. Donnell’s non-linear shallow shell theory is used to model the shell, assumed to be made of a Kelvin-Voigt material type, and a modal solution with six degrees of freedom is used to describe the lateral displacements. The Galerkin method is applied to derive a set of coupled non-linear ordinary differential equations of motion. Obtained results show that the viscoelastic dissipation parameter has significant influence on the instability loads and resonance curves.

Nonlinear Dynamics of Functionally Graded Cylindrical Shells With Internal Fluid

This work analyzes the nonlinear vibrations of a simply supported functionally graded cylindrical shell considering the effects of an internal fluid and static preloading. The cylindrical shell is subjected to a time dependent axial loading. The fluid is considered to be incompressible, non-viscous and irrotational and its effect on the shell wall is obtained using the potential flow theory. The shell is modeled by Donnell nonlinear shallow shell theory. The axial and circumferential displacement fields are described in terms of lateral displacement, thus generating a low-dimensional model, while the lateral displacement field is determined by a perturbation procedure which provides a general expression for the nonlinear vibration modes. These modal expansions satisfy the boundary and symmetry conditions of the problem. The discretized equations of motion are obtained by applying the Galerkin method. Various numerical techniques are employed to obtain the resonance curves and time responses of the cylindrical shell, showing the influence of the geometry, the internal fluid, static preloading and functionally graded material law on the shell dynamics and stability.Copyright

UM ESTUDO DAS OSCILAÇÕES NÃO LINEARES DE VIGAS ESPACIAIS

Resumo: As equacoes nao lineares integro-diferenciais ordinarias que descrevem o movimento tridimensional de uma viga inextensivel engas tada-livre, sujeita a excitacao harmonica lateral, sao aprese ntadas neste trabalho. Aplicando-se o metodo de Galerkin, as equacoes de movimento sao di scretizadas e usadas pa ra investigar as oscilacoes nao lineares e i dentificar os tipos de bifurcacoes associados com seu movimento.

ON THE GALERKIN - ITERATIVE METHOD APPLIED TO THE NON-LINEAR VIBRATIONS OF RECTANGULAR PLATES

In this work, by using the Galerkin-iterative method, the non-linear free and forced vibrations of rectangular plates are investigated. This method gives the eigenfunctions for the lateral displacements of plates with different sets of boundary conditions. Special attention is given to the frequency-amplitude relations and to the parametric instability boundaries of rectangular plates in the force control space.