M.P. Paı̈doussis

Effect of the geometry on the non-linear vibration of circular cylindrical shells

The non-linear vibration of simply supported, circular cylindrical shells is analysed. Geometric non-linearities due to finite-amplitude shell motion are considered by using Donnells non-linear shallow-shell theory; the effect of viscous structural damping is taken into account. A discretization method based on a series expansion of an unlimited number of linear modes, including axisymmetric and asymmetric modes, following the Galerkin procedure, is developed. Both driven and companion modes are included, allowing for travelling-wave response of the shell. Axisymmetric modes are included because they are essential in simulating the inward mean deflection of the oscillation with respect to the equilibrium position. The fundamental role of the axisymmetric modes is confirmed and the role of higher order asymmetric modes is clarified in order to obtain the correct character of the circular cylindrical shell non-linearity. The effect of the geometric shell characteristics, i.e., radius, length and thickness, on the non-linear behaviour is analysed: very short or thick shells display a hardening non-linearity; conversely, a softening type non-linearity is found in a wide range of shell geometries.

Coherent structures and their influence on the dynamics of aeroelastic panels

Abstract A panel forced by a supersonic unsteady flow is numerically investigated using a finite difference method, a Galerkin approach, and proper orthogonal decomposition (POD). The aeroelastic model investigated is based on piston theory for modeling the flow-induced forces, and von Karman plate theory for modeling the panel. Structural non-linearity is considered, and it is due to the non-linear coupling between bending and stretching. Several novel facets of behavior are explored, and key aspects of using a Galerkin method for modeling the dynamics of the panel exhibiting limit cycle oscillations and chaos are investigated. It is shown that multiple limit cycles may co-exist, and they are both symmetric and asymmetric. Furthermore, the level of spatial coherence in the dynamics is estimated by means of POD. Reduced order models for the dynamics are constructed. The sensitivity to initial conditions of the non-linear aeroelastic system in the chaotic regime limits the capability of the reduced order models to identically model the time histories of the system. However, various global characteristics of the dynamics, such as the main attractor governing the dynamics, are accurately predicted by the reduced order models. For the case of limit cycle oscillations and stable buckling, the reduced order models are shown to be accurate and robust to parameter variations.

A compact limit-cycle oscillation model of a cantilever conveying fluid

A low-dimensional model for the planar nonlinear dynamics of a fluid-conveying cantilever is constructed using the proper orthogonal decomposition method (PODM) in the post-flutter region. Firstly, the nonlinear partial differential equation (PDE) of motion is converted into a finite set of coupled ordinary differential equations (ODEs) by a Galerkin projection scheme using the cantilever beam modes as a basis. A finite difference method based on Houbolts scheme is used to obtain the stable solution of the nonlinear ODEs. A complex eigenvalue analysis is also carried out to determine the region of flutter instability with increasing flow velocity. Secondly, an efficient projection basis for the Galerkin scheme is constructed by using PODM for the low-dimensional representation of the original PDE describing the dynamics of the system. The important question regarding the capability of the reduced-order model to capture the principal features of the original system is addressed. Interestingly, the reduced-order basis constructed using PODM at a specific flow velocity can efficiently reproduce the system response at a range of flow rates involving limit-cycle oscillations (LCO) in the proximity of the flutter point. Furthermore, a weighted POD basis is derived subsequently in order to enhance the efficacy of the reduced-order model over a wider range of flow velocity, in the case when the LCO amplitude exhibits considerable variation beyond the flutter velocity.

Dynamics of a Fluid-Conveying Cantilevered Pipe With Intermediate Spring Support

The aim of this study is to investigate the three-dimensional (3-D) nonlinear dynamics of a fluid-conveying cantilevered pipe, additionally supported by an array of four springs attached at a point along its length. In the theoretical analysis, the 3-D equations are discretized via Galerkin’s technique, yielding a set of coupled nonlinear differential equations. These equations are solved numerically using a finite difference technique along with the Newton-Raphson method. The dynamic behaviour of the system is presented in the form of bifurcation diagrams, along with phase-plane plots, time-histories, PSD plots, and Poincare maps for two different spring locations and inter-spring configurations. Interesting dynamical phenomena, such as planar or circular flutter, divergence, quasiperiodic and chaotic motions, have been observed with increasing flow velocity. Experiments were conducted for the cases studied theoretically, and good qualitative and quantitative agreement was observed.Copyright

Analyses for random flow-induced vibration of cylindrical structures subjected to turbulent axial flow

In this paper, stochastic analytical equations for obtaining the vibratory response of bundles of cylinders in turbulent axial flow, with various degrees of computational efficiency, are presented. Lateral components of the turbulent fluid force-per-unit-length cross-spectral densities in a bundle of cylinders are obtained by the integration of differential wall-pressure fluctuations around the circumferences of the cylinders. These quantities are used as excitation in the calculation of random vibration response spectral densities of the cylindrical structures. Properties of symmetry applicable to lateral forces in bundles of symmetrically arranged cylinders are also discussed.

A Numerical Investigation on the Dynamics of Two-Dimensional Cantilevered Flexible Plates in Axial Flow

A cantilevered plate immersed in an otherwise uniform flow may lose stability at a high enough flow velocity; flutter takes place when the flow velocity exceeds a critical value. In the current research, a nonlinear equation of motion of the plate is developed using the inextensibility condition. Also, an unsteady lumped vortex model is used to calculate the pressure difference across the plate. The pressure difference is then decomposed into a lift force and an inviscid drag force. The fluid loads are coupled with the plate equation of motion and a numerical model of the fluid-structure system is developed. Analysis of the system dynamics is carried out in the time-domain. Both the stability and the post-critical behaviour of the system are studied. The flutter boundary and the vibration modes predicted by the current theory are found to be in good agreement with published experimental data.Copyright

Reduced-Order P.O.D. Models for Nonlinear Vibrations of Cylindrical Shells

The nonlinear (large-amplitude) response of perfect and imperfect, simply supported circular cylindrical shells to harmonic excitation in the spectral neighbourhood of some of their lowest natural frequencies is investigated. The shell is assumed to be completely filled with an incompressible and inviscid fluid at rest. Donnell’s nonlinear shallow-shell theory is used, and the solution is obtained by the Galerkin method. The proper orthogonal decomposition (POD) method is used to extract proper orthogonal modes that describe the system behaviour from time-series response data. These time-series have been obtained via the conventional Galerkin approach (using normal modes as a projection basis) with an accurate model involving 16 degrees of freedom, validated in previous studies. The POD method, in conjunction with the Galerkin approach, permits a lower-dimensional model with respect to the conventional Galerkin approach. Different proper orthogonal modes computed from time-series at different excitation frequencies are used and solutions are compared. Some of these modes are capable of describing the system behaviour over the whole frequency range around the fundamental resonance with good accuracy and with only three degrees of freedom. They allow a drastic reduction in the computational effort, as compared to using the 16 degrees-of-freedom model necessary when the conventional Galerkin approach is used.© 2003 ASME

Phase-Shift Determination in Coriolis Flowmeters With Added Masses

In this study, an approximate analytic solution for phase-shift (and thus mass flow) prediction along the length of the measuring tube of a Coriolis flowmeter is investigated. A single, straight measuring tube is considered; added masses at the sensor locations, are included in the model, and thus in the equation of motion. The method of multiple timescales, an approximate analytical technique, has been applied directly to the equation of motion, and the equations of order one and epsilon have been obtained analytically for the system at resonance. The solution of the equation of motion is obtained by satisfying the solvability condition (making the solution of order epsilon free of secular terms). The measuring tube is excited by the driver, and the phase-shift is measured at two symmetrically located points on either side of the mid-length of the tube. The effects of system parameters on the measured phase-shift are discussed.Copyright

Dynamics of a Free-Clamped Cylinder in Confined Axial Flow

The dynamics of a flexible cantilevered cylinder in confined axial flow is studied theoretically and experimentally, in the case where the flow is directed from the free end towards the clamped end. First, the dynamics is described, as observed in specially conducted experiments with air flow; the system developed small-amplitude first-mode vibrations at low flow velocities, which could be flutter, and at higher flows the oscillatory behaviour was succeeded by a static divergence. A simple, linear theoretical model is also developed, and the theoretically predicted behaviour is compared to the experimental one. The model captures the essentials of the observed behaviour, but requires improvement before quantitative prediction can be considered to be adequate.Copyright

Hyperchaotic Behaviour of Shells Subjected to Flow and External Force

This study treats the nonlinear behaviour of cylindrical shells subjected to internal fluid flow and to an external periodic transverse point force. The shell is supported at both ends by axial and rotational springs capable of simulating boundary conditions ranging from clamped to simple supports. This complex boundary condition configuration is preferred in our analysis in order to be able to compare theoretical findings with water-tunnel experiments available in the literature. The external concentrated point force is applied at mid-length of the immersed shell structure acting in the radial direction and the excitation frequency values lie within the spectral neighbourhood of one of the shell’s lowest frequencies for different flow velocities. The structural model is based on the full nonlinear Donnell shell equations of motion including the effect of the in-plane inertia and accounting for geometric imperfections. The fluid is assumed to be incompressible and inviscid and the flow isentropic and irrotational; it is modelled using potential flow theory with the addition of unsteady viscous terms obtained from the time-averaged Navier-Stokes equations. The coupled system is discretized using a solution expansion based on trigonometric functions satisfying the shell boundary conditions exactly. Numerical results show the nonlinear response at different flow velocities for (i) a fixed excitation amplitude and variable excitation frequency, and (ii) fixed excitation frequency varying the excitation amplitude. Bifurcation diagrams of Poincare maps obtained from direct time integration are presented, as well as the maximum Lyapunov exponent, in order to classify the system dynamics. In particular, periodic, quasi-periodic, sub-harmonic and chaotic responses have been detected. The full spectrum of the Lyapunov exponents and the Lyapunov dimension have been calculated for the chaotic response; they reveal the occurrence of large-dimension hyperchaos.Copyright