Three-dimensional dynamics of a cantilevered pipe conveying fluid, additionally supported by an intermediate spring array

In this paper, the three-dimensional (3-D) non-linear dynamics of a cantilevered pipe conveying fluid, constrained by arrays of four springs attached at a point along its length is investigated. In the theoretical analysis, the 3-D equations are discretized via Galerkin's technique. The resulting coupled non-linear differential equations are solved numerically using a finite difference 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 Poincaré maps for five different spring configurations. Interesting dynamical phenomena, such as 2-D or 3-D flutter, divergence, quasiperiodic and chaotic motions, have been observed with increasing flow velocity. Experiments were performed for the cases studied theoretically, and good qualitative and quantitative agreement was observed. The experimental behaviour is illustrated by video clips (electronic annexes). The effect of the number of beam modes in the Galerkin discretization on accuracy of the results and on convergence of the numerical solutions is discussed.     

Nonlinear dynamics of a pipe conveying pulsating fluid with parametric and internal resonances

In this paper, the nonlinear planar vibration of a pipe conveying pulsatile fluid subjected to principal parametric resonance in the presence of internal resonance is investigated. The pipe is hinged to two immovable supports at both ends and conveys fluid at a velocity with a harmonically varying component over a constant mean velocity. The geometric cubic nonlinearity in the equation of motion is due to stretching effect of the pipe. The natural frequency of the second mode is approximately three times the natural frequency of the first mode for a range of mean flow velocity, resulting in a three-to-one internal resonance. The analysis is done using the method of multiple scales (MMS) by directly attacking the governing nonlinear integral-partial-differential equations and the associated boundary conditions. The resulting set of first-order ordinary differential equations governing the modulation of amplitude and phase is analyzed numerically for principal parametric resonance of first mode. Stability, bifurcation, and response behavior of the pipe are investigated. The results show new zones of instability due to the presence of internal resonance. A wide array of dynamical behavior is observed, illustrating the influence of internal resonance.     

A cantilever conveying fluid: coherent modes versus beam modes

A semi-analytical approach to obtain the proper orthogonal modes is described for the non-linear oscillation of a cantilevered pipe conveying fluid. Theoretically, while the spatial coherent structures are the eigenfunctions of the time-averaged spatial autocorrelation functions, it emerges that once the Galerkin projection of the proper orthogonal modes is carried out using the uniform cantilever-beam modes, the spatial dependency of the integral eigenvalue problem can be eliminated by analytical manipulation which avoids any spatial discretization error. As the solution of the integral equation is obtained semi-analytically by linearly projecting the proper orthogonal modes on the cantilever-beam modes, any linear or non-linear operation can be carried out analytically on the proper orthogonal modes. Furthermore, the reduced-order eigenvalue problem minimizes the numerical pollution which leads to spurious eigenvectors, as may arise in the case of a large-scale eigenvalue problem (without the Galerkin projection of the eigenvectors on the cantilever-beam modes). This methodology can conveniently be used to study the convergence of the numerically calculated proper orthogonal modes obtained from the full-scale eigenvalue problem.     

Vortex-induced vibration of pipes conveying fluid using the method of multiple scales

The nonlinear dynamics of supported pipes conveying fluid subjected to vortex-induced vibration is evaluated using the method of multiple scales. Frequency response portraits for different internal fluid velocities under lock-in conditions are obtained and the stability of steady-state responses is discussed. Results show that the internal fluid velocity has a prominent effect on the oscillation amplitude and that the steady-state responses incorporating unstable solutions in the lock-in region are also obtained. In addition, the effects of two kinds of fluctuating lift coefficients on the steady-state responses are compared with each other.     

Three-dimensional oscillations of a cantilever pipe conveying fluid

The aim of the study described in this paper is to investigate the two-dimensional (2-D) and three-dimensional (3-D) flutter of cantilevered pipes conveying fluid. Specifically, by means of a complete set of non-linear equations of motion, two questions are addressed: (i) whether for a system losing stability by either 2-D or 3-D flutter the motion remains of the same type as the flow velocity is increased substantially beyond the Hopf bifurcation precipitating the flutter; (ii) whether the bifurcational behaviour of a horizontal system and a vertical one (sufficiently long for gravity to have an important effect on the dynamics) are substantially similar. Stability maps and tables are used to delineate areas in a flow velocity versus mass parameter plane where 2-D or 3-D motions occur, and limit-cycle motions are illustrated by phase-plane plots, PSDs and cross-sectional diagrams showing whether the motion is circular (3-D) or planar (2-D).     

Differential quadrature method to stability analysis of pipes conveying fluid with spring support

It is a new attempt to extend the differential quadrature method (DQM) to stability analysis of the straight and curved centerline pipes conveying fluid. Emphasis is placed on the study of the influences of several parameters on the critical flow velocity. Compared to other methods, this method can more easily deal with the pipe with spring support at its boundaries and asks for much less computing effort while giving aceptable precision in the numerical results. Supported by National Key Project of China (No. PD9521907) and the National Science Foundation of China (No. 19872025).     

Geometrically non-linear free and forced vibrations of simply supported circular cylindrical shells: A semi-analytical approach

The non-linear free and forced vibrations of simply supported thin circular cylindrical shells are investigated using Lagrange's equations and an improved transverse displacement expansion. The purpose of this approach was to provide engineers and designers with an easy method for determining the shell non-linear mode shapes, with their corresponding amplitude dependent non-linear frequencies. The Donnell non-linear shell theory has been used and the flexural deformations at large vibration amplitudes have been taken into account. The transverse displacement expansion has been made using two terms including both the driven and the axisymmetric modes, and satisfying the simply supported boundary conditions. The non-linear dynamic variational problem obtained by applying Lagrange's equations was then transformed into a static case by adopting the harmonic balance method. Minimisation of the energy functional with respect to the basic function contribution coefficients has led to a simple non-linear multi-modal equation, the solution of which gives in the case of a single mode assumption an expression for the non-linear frequencies which is much simpler than that derived from the non-linear partial differential equation obtained previously by several authors. Quantitative results based on the present approach have been computed and compared with experimental data. The good agreement found was very satisfactory, in comparison with previous old and recent theoretical approaches, based on sophisticated numerical methods, such as the finite element method (FEM), the method of normal forms (MNF), and analytical methods, such as the perturbation method.     

Effect of geometric imperfections on non-linear stability of circular cylindrical shells conveying fluid

Circular cylindrical shells conveying incompressible flow are addressed in this study; they lose stability by divergence when the flow velocity reaches a critical value. The divergence is strongly subcritical, becoming supercritical for larger amplitudes. Therefore the shell, if perturbed from the initial configuration, has severe deformations causing failure much before the critical velocity predicted by the linear threshold. Both Donnell's non-linear theory retaining in-plane displacements and the Sanders-Koiter non-linear theory are used for the shell. The fluid is modelled by potential flow theory but the effect of steady viscous forces is taken into account. Geometric imperfections are introduced and fully studied. Non-classical boundary conditions are used to simulate the conditions of experimental tests in a water tunnel. Comparison of numerical and experimental results is performed.     

Airy stress function for describing the non-linear effect of simply supported plates with movable edges

The general form of the solution of the Airy function for the stress distributions that describe the non-linear effect developed from the large deflection of simply supported plates with movable edges are found by superposition of the Airy functions, which satisfy the large deflection condition and the boundary conditions of the edges. Each term of the Airy function consists of a particular solution and a homogeneous one. The particular solution satisfying the large deflection condition is classified into six cases, depending on the combinations of the modal numbers of the comparison functions. The corresponding homogeneous solution is found to make each Airy function satisfy the boundary condition by using the Fourier series method. The solution is applied to the non-linear analysis of the deflection of the simply supported plates with movable edges under transverse loading, and is verified by comparison with other investigation.     

A non-linear model for the dynamics of open cross-section thin-walled beams—Part I: formulation

A non-linear one-dimensional model of inextensional, shear undeformable, thin-walled beam with an open cross-section is developed. Non-linear in-plane and out-of-plane warping and torsional elongation effects are included in the model. By using the Vlasov kinematical hypotheses, together with the assumption that the cross-section is undeformable in its own plane, the non-linear warping is described in terms of the flexural and torsional curvatures. Due to the internal constraints, the displacement field depends on three components only, two transversal translations of the shear center and the torsional rotation. Three non-linear differential equations of motion up to the third order are derived using the Hamilton principle. Taking into account the order of magnitude of the various terms, the equations are simplified and the importance of each contribution is discussed. The effect of symmetry properties is also outlined. Finally, a discrete form of the equations is given, which is used in Part II to study dynamic coupling phenomena in conditions of internal resonance.     

A non-linear model for the dynamics of open cross-section thin-walled beams—Part II: forced motion

The discrete equations developed in Part I are here used to analyze the non-linear dynamics of an inextensional shear indeformable beam with given end constraints. The model takes into account the non-linear effects of warping and of torsional elongation. Non-linear 3D oscillations of a beam with a cross-section having one symmetry axis is examined. Only terms of higher magnitude are retained in the equations, which exhibit quadratic, cubic and combination resonances. A harmonic load acting in the direction of the symmetry axis and in resonance with the corresponding natural frequency, is considered. Steady-state solutions and their stability are studied; in particular the effects of non-linear warping and of torsional elongation on the response are highlighted.     

Solutions of magnetohydrodynamic problems based on a conventional computational fluid dynamics code

It is pointed out that there exists a hidden analogy between magnetohydrodynamic (MHD) and conventional computational fluid dynamic (CFD) equations. This allows the generalization of any conventional CFD code so that the effects of MHD can be accounted for. This generalization is actually made for the FLUENT CFD code. Although this generalized FLUENT code can easily be adjusted to any MHD environment, it has been specifically designed for metallurgical applications. Predictions of the code are validated against the analytical solutions for the Poiseuille-Hartmann flow and for the shielding of magnetic field oscillations by a conducting medium (skin effect).     

A new approach to studies of non-linear dynamics of kinks activated in inhomogeneous polynucleotide chains

A new approach has been proposed to study the non-linear dynamics of local conformational distortions (kinks) activated in DNA polynucleotide chains that are inhomogeneous. The dependence of the dynamic characteristics of kinks on the chain composition has been obtained. The result has been applied to estimate the size, energy, density of energy, and velocity of kinks activated in chains having the binary sequence or the sequence similar to the sequences of promoters A₁, A₂, and A₃ of the bacteriophage T7 genome.     

A new method for the non-linear deflection analysis of an infinite beam resting on a non-linear elastic foundation

The aim of this paper is to develop a new method of analyzing the non-linear deflection behavior of an infinite beam on a non-linear elastic foundation. Non-linear beam problems have traditionally been dealt with by semi-analytical approaches that involve small perturbations or by numerical methods, such as the non-linear finite element method. In this paper, in contrast, a transformed non-linear integral equation that governs non-linear beam deflection behavior is formulated to develop a new method for non-linear solutions. The proposed method requires an iteration to solve non-linear problems, but is fairly simple and straightforward to apply. It also converges quickly, whereas traditional non-linear solution procedures are generally quite complex in application. Mathematical analysis of the proposed method is performed. In addition, illustrative examples are presented to demonstrate the validity of the method developed in the present study.     

A reappraisal of the computational modelling of carbon nanotubes conveying viscous fluid

By using the Euler–Bernoulli classical beam theory to model the nanotube as a continuum structure, a reevaluation of the computational modelling of carbon nanotubes conveying viscous fluid is undertaken in this paper, with some fresh insights as to if the viscosity of flowing fluid does influence the free vibration of the nanotube. It is found that during the flow of a fluid through a nanotube, modelled as a continuum beam, the effect of viscosity of flowing fluid on the vibration and instability of CNTs can be ignored.     

Experimental study on dynamics of an oblique-impact vibrating system of two degrees of freedom

The paper presents a detailed experimental study of an oblique-impact vibration system of two degrees of freedom. The primary objective of the study is to verify the hypothesis of instantaneous impact in the oblique-impact process of two elastic bodies such that the incremental impulse method works for computing the nonlinear dynamics of the oblique-impact vibrating systems. The experimental setup designed for the objective consists of a harmonically excited oscillator and a pendulum, which obliquely impacts the oscillator. In the study, the dynamic equation of the experimental setup was established first, and then the system dynamics was numerically simulated by virtue of the incremental impulse method. Afterwards, rich dynamic phenomena, such as the periodic vibro-impacts, chaotic vibro-impacts and typical bifurcations, were observed in a series of experiments. The comparison between the experimental results and the numerical simulations indicates that the incremental impulse method is reasonable and successful to describe the dynamics during an oblique-impact process of two elastic bodies. The study also shows the limitation of the hypothesis of instantaneous impact in an oblique-impact process. That is, the hypothesis only holds true in the case when the impact angle is not too large and the relative approaching velocity in the normal direction is not too low. Furthermore, the paper gives the analysis of the tangential rigid-body slip on the contact surface in the case of a large impact angle, and explains why there exist some discrepancies between the numerical simulations and the experimental results.     

Development of a parallel computational fluid dynamics algorithm on a hypercube computer

One of the main factors limiting the widespread use of computational fluid dynamics codes for engineering design is their very large requirements both in terms of computer memory and CPU time. Distributed memory parallel computers offer both the potential for a dramatic improvement in cost/performance over conventional supercomputers and the scalability to large numbers of processors that is required if performance beyond that of current supercomputers is to be achieved. As part of an evaluation to explore the potential of such machines for computational fluid mechanics applications, a concurrent algorithm for the solution of the Navier-Stokes equations has been developed and demonstrated on a hypercube parallel computer. The algorithm is based on a domain decomposition of a well-established serial pressure correction algorithm. The algorithm is demonstrated on both a 32-node scalar and eight-node vector Intel iPSC/2 for complicated two-dimensional laminar and turbulent flow problems with different grid sizes and numbers of processors. Speed-ups relative to a single processor of 12.9 with 16 processors and 20.2 with 32 processors are achieved on a scalar iPSC/2, demonstrating the parallel efficiency of the algorithm. Measured performance on a 32-node scalar iPSC/2 exceeds one-sixth that of a Cray X-MP running the original serial algorithm. The performance of the algorithm on an eight-node vector iPSC/2 exceeds that of the larger scalar hypercube and is about one-fifth that of the Cray X-MP. With cost/performance more than 10 times better than the Cray, these results dramatically show the cost effectiveness of vector hypercubes for this class of fluid mechanics algorithm.     

A decoupled modal analysis for nonlinear dynamics of an orthotropic thin laminate in a simply supported boundary condition subject to thermal mechanical loading

In this paper, we analyze the nonlinear dynamic response of an orthotropic laminate in a simply supported boundary condition subject to thermal and mechanical loading. The equation of motion for the laminate’s deflection is obtained in a decoupled Duffing equation by means of a Galerkin-type method without Berger’s approximations. The Duffing equation incorporates an arbitrary thermal field, with both the in-plane and transverse temperature variations in a steady-state and a transient state. The formulation indicates that the transverse temperature variation produces an additional pressure load, while the in-plane temperature variation affects the system frequency. The equation allows for characterization of the laminate behaviors in nonlinear thermal buckling, thermal vibration and thermal mechanical response.