Nonlinear vibrations and damping of fractional viscoelastic rectangular plates

Nonlinear Dynamics - A new model-free robust control scheme for payload swing angle attenuation of two-dimensional crane systems with varying rope length is introduced in this work. The proposed...     

Non-linear vibrations of doubly curved shallow shells

Large amplitude (geometrically non-linear) vibrations of doubly curved shallow shells with rectangular base, simply supported at the four edges and subjected to harmonic excitation normal to the surface in the spectral neighbourhood of the fundamental mode are investigated. Two different non-linear 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 non-linear 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 discussed. Shell stability under static and dynamic load is also investigated by using continuation method, bifurcation diagram from direct time integration and calculation of the Lyapunov exponents and Lyapunov dimension. Interesting phenomena such as (i) snap-through instability, (ii) subharmonic response, (iii) period doubling bifurcations and (iv) chaotic behaviour have been observed.     

Experiments and simulations for large-amplitude vibrations of rectangular plates carrying concentrated masses

Large-amplitude (geometrically nonlinear) forced vibrations of a stainless-steel thin rectangular plate carrying different concentrated masses are experimentally studied. The experimental boundary conditions are close to those of a clamped plate. The plate is vertically and horizontally tested in order to investigate the gravity effect. Harmonic excitation is applied by using electrodynamic exciter and the plate vibration is measured by using a laser Doppler vibrometer with displacement decoder. The harmonic excitation is controlled in closed-loop in order to keep constant the desired force and is increased (or decreased) by very small discrete steps. Numerical simulations on reduced-order models, obtained by using Von Kármán nonlinear plate theory and global discretization, are also carried out and compared to experiments in order to better understand the system. Results show that concentrated masses have no effect on the trend of nonlinearity of the vertical plate, while they play a role in case of horizontal plate due to the static flexural deflection caused by gravity, which reduces the hardening-type nonlinearity. Initial geometric imperfection (deviation from flat surface in vertical position) of the plate is measured and taken into account; it plays a significant role.     

Nonlinear resonant behavior of microbeams over the buckled state

The present study investigates the nonlinear resonant behavior of a microbeam over its buckled (non-trivial) configuration. The system is assumed to be subjected to an axial load along with a distributed transverse harmonic load. The axial load is increased leading the system to lose the stability via a pitchfork bifurcation; the postbuckling configuration is obtained and the nonlinear resonant response of the system over the buckled state is examined. More specifically, the nonlinear equation of motion is obtained employing Hamilton’s principle along with the modified couple stress theory. The continuous system is truncated into a system with finite degrees of freedom; the Galerkin scheme is employed to discretize the nonlinear partial differential equation of motion into a set of ordinary differential equations. This set of equations is solved numerically employing the pseudo-arclength continuation technique; first a nonlinear static analysis is performed upon this set of equations so as to obtain the onset of buckling (supercritical pitchfork bifurcation) and the buckled configuration of the microbeam. The frequency-response and force-response curves of the system are then constructed over the buckled configurations. A comparison is made between the frequency-response curves obtained by means of the modified couple stress and the classical theories. The effect of different system parameters on the frequency-response and force-response curves is also examined.     

Coupled nonlinear size-dependent behaviour of microbeams

The nonlinear resonant behaviour of a microbeam, subject to a distributed harmonic excitation force, is investigated numerically taking into account the longitudinal as well as the transverse displacement. Hamilton’s principle is employed to derive the coupled longitudinal-transverse nonlinear partial differential equations of motion based on the modified couple stress theory. The discretized form of the equations of motion is obtained by applying the Galerkin technique. The pseudo-arclength continuation technique is then employed to solve the discretized equations of motion numerically. Different types of bifurcations as well as the stability of solution branches are determined. The numerical results are presented in the form of frequency-response and force-response curves for different sets of parameters. The effect of taking into account the longitudinal displacement is highlighted.     

Theory and experiments for large-amplitude vibrations of empty and fluid-filled circular cylindrical shells with imperfections

The large-amplitude response of perfect and imperfect, simply supported circular cylindrical shells to harmonic excitation in the spectral neighbourhood of some of the lowest natural frequencies is investigated. Donnell's non-linear shallow-shell theory is used and the solution is obtained by the Galerkin method. Several expansions involving 16 or more natural modes of the shell are used. The boundary conditions on the radial displacement and the continuity of circumferential displacement are exactly satisfied. The effect of internal quiescent, incompressible and inviscid fluid is investigated. The non-linear equations of motion are studied by using a code based on the arclength continuation method. A series of accurate experiments on forced vibrations of an empty and water-filled stainless-steel shell have been performed. Several modes have been intensively investigated for different vibration amplitudes. A closed loop control of the force excitation has been used. The actual geometry of the test shell has been measured and the geometric imperfections have been introduced in the theoretical model. Several interesting non-linear phenomena have been experimentally observed and numerically reproduced, such as softening-type non-linearity, different types of travelling wave response in the proximity of resonances, interaction among modes with different numbers of circumferential waves and amplitude-modulated response. For all the modes investigated, the theoretical and experimental results are in strong agreement.     

Nonlinear vibrations of a fluid-filled,soft circular shell: experiments and system identification

Nonlinear Dynamics - Vibration experiments are carried out on a slightly corrugated circular cylindrical shell made of polyethylene terephthalate fabric. The shell is liquid-filled, it is...     

Rayleigh Quotient, Ritz Method and Substructuring to Study Vibrations of Structures Coupled to Heavy Fluids: Potential of the Artificial Spring Method

The linear study of free vibrations of structures coupled to incompressible and inviscid fluids are studied by using the Rayleigh-Ritz method. The system is modelled by using different components. The artificial spring method is used to synthesise these components. The advantage is that admissible functions are defined in each component and the continuity condition of translational and rotational displacements between the rigid joints of the structure is no longer required. The fluid-structure interaction can be accurately described by using this method, including the effect of the free surface waves and the dynamic interaction among structural components via the fluid medium. An application of the method to a vertical circular tank partially filled with water is also presented in order to show the potential of the method. This revised version was published online in July 2006 with corrections to the Cover Date.     

Coupled vibrations of a partially fluid-filled cylindrical container with an internal body including the effect of free surface waves

Internal bodies (baffles) are used as damping devices to suppress the fluid sloshing motion in fluid-structure interaction systems. An analytical method is developed in the present article to investigate the effects of a rigid internal body on bulging and sloshing frequencies and modes of a cylindrical container partially filled with a fluid. The internal body is a thin-walled and open-ended cylindrical shell that is coaxially and partially submerged inside the container. The interaction between the fluid and the structure is taken into account to calculate the sloshing and bulging frequencies and modes of the coupled system using the Rayleigh quotient, Ritz expansion and Galerkin method. It is shown that the present formulation is an appropriate and new approach to tackle the problem with good accuracy. The effects of fluid level, number of nodal diameters, internal body radius and submergence ratio on the dynamic characteristics of the coupled system are also investigated.