Inelastic impact dynamics of ships with one-sided barriers. Part II: experimental validation

This paper is the second part of a two-part study of impact interaction of a ship roll motion with one-sided ice barrier. The first part was devoted to analytical and numerical simulations for the case of inelastic impact. The analytical approach was based on Zhuravlev and Ivanov non-smooth coordinate transformations. Extensive numerical simulations were carried out for all initial conditions covered by the ship grazing orbit for different values of excitation amplitude and frequency of external wave-induced roll moment. The basins of attraction of safe operation revealed the coexistence of different response regimes such as non-impact periodic oscillations, modulated impact motion, period added impact oscillations, chaotic impact motion and roll-over dynamics. This part presents an experimental investigation conducted on a small ship model in a tow tank. In particular, the experimental tests reveal complex dynamic response characteristics such as multi-frequency wave motion caused by the wave reflection from the tank end wall. Measured results show a good agreement with the predicted results by for small angles of the barrier relative to the ship unbiased position. However, deviation becomes significant as the angle increases. This deviation is mainly attributed to the uncertainty of the coefficient of restitution, which is found to depend on the velocity of impact in addition to the geometry and material properties of the model and barrier.     

The Okhotsimskii-Pontryagin method in control theory and analytical mechanics. Part I

The Okhotsimskii method for the differentiation of functionals in control theory is discussed on the basis of the Pontryagin formalism. The association of this method with other approaches to solving control problems and with methods of analytical mechanics is studied. Some typical cases of solving optimal control problems are considered.     

Targeted energy transfer with parallel nonlinear energy sinks. Part I: Design theory and numerical results

In this paper, we study a Targeted Energy Transfer (TET) problem between a p degrees-of-freedom (dof) linear master structure and several coupled parallel slave Nonlinear Energy Sink (NES) systems. In detail, each lth dof l=1,2,…,p contains n _l parallel NES; so the linear structure has (n ₁+n ₂+⋅⋅⋅+n _l +⋅⋅⋅+n _p ) NES. We are interested to study analytically the TET phenomenon during the first mode of the compound system. To this end, complexification, averaging, and multiple scales methods are used.     

Non-linear dynamics of curved beams. Part 2, numerical analysis and experiments

The aim of this work is to formulate a model for the study of the dynamics of curved beams undergoing large oscillations. In Part 1, the interest was oriented to the formulation of a consistent analytical model and to obtain the equations of motion in weak form. In Part 2, a case-study is considered and the response for various initial curved configurations, obtained by varying the initial curvature, is analyzed. Both the free and the forced problems are considered: the linear free dynamics are studied to detect how the initial configuration affects the modal properties and to enlighten the typical phenomena of frequency coalescence and avoidance; the forced dynamics are then studied for different internal resonance conditions to enlighten the phenomenon of the dynamic instability under a shear periodic tip follower force and to describe the various classes of post-critical motion. The results of experimental tests conducted on a slightly imperfect straight beam prototype are eventually discussed.     

Analysis of pipe flow transition. Part I. Direct numerical simulation

We have developed an accurate hybrid finite-difference code for the simulation of unsteady incompressible pipe flow. The numerical scheme uses compact finite differences of at least eighth-order accuracy for the axial coordinate, and Chebyshev and Fourier polynomials for the radial and azimuthal coordinates, respectively. Boundary conditions for the incompressible flow are enforced using an influence-matrix technique, and the Poisson equation for pressure is solved using a fast direct method. The code has been used to simulate and analyze the spatial transition process in developed laminar pipe flow at a Reynolds number of Re=2350. Results of the simulation are compared to experimental data given by Han, Tumin and Wygnanski [18]. PACS 47.11.+j, 47.20.Ft, 47.27.Cn     

Some recent developments on integrity assessment of pipes and elbows. Part I: Theoretical investigations

Integrity assessment of piping components is very essential for safe and reliable operation of power plants. Over the last several decades, considerable work has been done throughout the world to develop a system oriented methodology for integrity assessment of pipes and elbows, mainly for application to nuclear power plants. However, there is a scope of further development/improvement of issues, particularly for pipe bends, that are important for accurate integrity assessment of pipings. Considering this aspect, a comprehensive Component Integrity Test Program was initiated in 1998 at Reactor Safety Division (RSD) of Bhabha Atomic Research Centre (BARC), India in collaboration with MPA, Stuttgart, Germany through Indo-German bilateral project. In this program, both theoretical and experimental investigations were undertaken to address various issues related to the integrity assessment of pipes and elbows. The important results of the program are presented in this two-part paper. In the part I of the paper, the theoretical investigations are discussed. Part II will cover the experimental investigations. The theoretical investigations considered the following issues: new plastic (collapse) moment equations of defect-free elbow under combined internal pressure and in-plane closing/opening moments; new plastic (collapse) moment equations of throughwall circumferentially cracked elbow, which are more accurate and closer to the test results; new ‘η_pl’ and ‘γ’ functions of pipes and elbows with various crack configurations under different loading conditions to evaluate J–R curve from test data; and the effect of deformation on the unloading compliance of TPB specimen and throughwall circumferentially cracked pipe to measure crack growth during fracture experiment. These developments would also help to study the effect of stress triaxiality in the transfer of material J–R curve from specimen to component.     

Resonant non-linear normal modes. Part I: analytical treatment for structural one-dimensional systems

Approximations of the resonant non-linear normal modes of a general class of weakly non-linear one-dimensional continuous systems with quadratic and cubic geometric non-linearities are constructed for the cases of two-to-one, one-to-one, and three-to-one internal resonances. Two analytical approaches are employed: the full-basis Galerkin discretization approach and the direct treatment, both based on use of the method of multiple scales as reduction technique. The procedures yield the uniform expansions of the displacement field and the normal forms governing the slow modulations of the amplitudes and phases of the modes. The non-linear interaction coefficients appearing in the normal forms are obtained in the form of infinite series with the discretization approach or as modal projections of second-order spatial functions with the direct approach. A systematic discussion on the existence and stability of coupled/uncoupled non-linear normal modes is presented. Closed-form conditions for non-linear orthogonality of the modes, in a global and local sense, are discussed. A mechanical interpretation of these conditions in terms of virtual works is also provided.     

Inelastic behavior of an AS4/PEEK composite under combined transverse compression and shear. Part I: experiments

The nonlinear behavior in shear and transverse compression of unidirectional AS4/PEEK and their interaction are investigated experimentally. The composite is rate dependent even at room temperature and its rate exponent is similar to that of neat PEEK. The material is tested under pure shear, pure compression and under biaxial loading histories. The biaxial tests are performed in a custom facility on thin strips of the material. The facility allows freedom to choose the loading path in the biaxial stress and strain spaces of interest. Tests are performed for three biaxial loading paths. In the first, the specimen is sheared then compressed while the shear stress is held constant; in the second, the specimen is compressed then sheared while the compressive stress is held constant; and in the third, the specimen is loaded simultaneously by proportional amounts of compression and shear. It was found that the induced deformation is influenced significantly by the loading history followed. Also, initial loading in shear or compression has only a modest effect on subsequent loading of the other type. An unorthodox yielding behavior for the composite results from this lack of interaction. Finally, the stresses at failure are found to trace an elliptical path in the shear–transverse compression plane, but the failure stress state is not significantly affected by the loading path followed.     

Wave propagation in periodic buckled beams. Part I: Analytical models and numerical simulations

Periodic buckled beams possess a geometrically nonlinear, load–deformation relationship and intrinsic length scales such that stable, nonlinear waves are possible. Modeling buckled beams as a chain of masses and nonlinear springs which account for transverse and coupling effects, homogenization of the discretized system leads to the Boussinesq equation. Since the sign of the dispersive and nonlinear terms depends on the level of buckling and support type (guided or pinned), compressive supersonic, rarefaction supersonic, compressive subsonic and rarefaction subsonic solitary waves are predicted, and their existence is validated using finite element simulations of the structure. Large dynamic deformations, which cannot be approximated with a polynomial of degree two, lead to strongly nonlinear equations for which closed-form solutions are proposed.     

Flow boiling heat transfer with HFC mixtures in a smooth horizontal tube. Part I: Experimental investigations

The substitution of working fluids in vapour-compression plants causes major problems, because of a reduced plant performance. Therefore, extremely accurate design procedures are needed, because the relative sizing of each plant-component is essential for the cycle performance. For this reason, the knowledge of the heat-transfer characteristics of new fluids in condensers and evaporators is mandatory. The heat-transfer characteristics of R410A and R404A were experimentally investigated and analyzed as a function of evaporating pressure, heat- and mass flux. The test section was a smooth, horizontal, stainless steel tube (6 mm ID, 6 m length) uniformly heated by the Joule effect. The working parameter was varied within the following ranges: evaporating pressure 3–12 bar, refrigerant mass-flux 290–1100 kg/m² s, heat flux 11–39 kW/m², respectively.     

Nonlinear dynamic analysis of Timoshenko beams by BEM. Part I: Theory and numerical implementation

In this two-part contribution, a boundary element method is developed for the nonlinear dynamic analysis of beams of arbitrary doubly symmetric simply or multiply connected constant cross section, undergoing moderate large displacements and small deformations under general boundary conditions, taking into account the effects of shear deformation and rotary inertia. Part I is devoted to the theoretical developments and their numerical implementation and Part II discusses analytical and numerical results obtained from both analytical or numerical research efforts from the literature and the proposed method. The beam is subjected to the combined action of arbitrarily distributed or concentrated transverse loading and bending moments in both directions as well as to axial loading. To account for shear deformations, the concept of shear deformation coefficients is used. Five boundary value problems are formulated with respect to the transverse displacements, to the axial displacement and to two stress functions and solved using the Analog Equation Method, a BEM based method. Application of the boundary element technique yields a nonlinear coupled system of equations of motion. The solution of this system is accomplished iteratively by employing the average acceleration method in combination with the modified Newton–Raphson method. The evaluation of the shear deformation coefficients is accomplished from the aforementioned stress functions using only boundary integration. The proposed model takes into account the coupling effects of bending and shear deformations along the member, as well as the shear forces along the span induced by the applied axial loading.

     

Analysis and numerical simulation of magnetic forces between rigid polygonal bodies. Part I: Analysis

The mathematical and physical analysis of magnetoelastic phenomena is a topic of ongoing research. Different formulae have been proposed to describe the magnetic forces in macroscopic systems. We discuss several of these formulae in the context of rigid magnetized bodies. In case the bodies are in contact, we consider formulae both in the framework of macroscopic electrodynamics and via a multiscale approach, i.e., in a discrete setting of magnetic dipole moments. We give mathematically rigorous proofs for domains of polygonal shape (as well as for more general geometries) in two and three space dimensions. In an accompanying second article, we investigate the formulae in a number of numerical experiments, where we focus on the dependence of the magnetic force on the distance between the bodies and on the case when the two bodies are in contact. The aim of the analysis as well as of the numerical simulation is to contribute to the ongoing debate about which formula describes the magnetic force between macroscopic bodies best and to stimulate corresponding real-life experiments.      

Elastic behaviour of shrink-fitted forming die assemblies. Part I: Development of an analytical solution

Summary The paper deals with the elastic behaviour of the die assemblies obtained by shrinking a thick walled ring onto a finite hollow right circular cylinder in general of different length. Displacements and stresses in the two components of the assembly are evaluated by using the theoretical general solution on the finite thick-walled hollow elastic right circular cylinder with an axially symmetrical load system in terms of Airy's stress function. Radial and axial displacement influence coefficient technique is used to solve the integral equations governing the two extreme frictionless and slipless contact modes at the mating surface.

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Sommario L'articolo riferisce sul comportamento elastico di accoppiamenti realizzati per forzamento di un anello di grosso spessore su di un cilindro cavo circolare retto di lunghezza finita ed in generale diversa da quella dell'elemento forzante.La valutazione delle componenti di spostamento e di tensione presenti in ciascuno dei due elementi dell'accoppiamento è operata mediante una soluzione teorica, generale, relativa ad un cilindro cavo, circolare, retto, di lunghezza finita e sollecitato da un sistema di carichi distribuiti in modo assialsimmetrico e che è stata sviluppata in termini della funzione delle tensioni di Airy. Il metodo dei coefficienti di influenza applicato alle componenti radiale ed assiale degli spostamenti è utilizzato per la risoluzione delle equazioni, scritte in forma integrale, che governano le modalità di contatto relative alle due situazioni estreme di assenza di attrito e di adesione completa in corrispondenza delle superfici di accoppiamento.

     

The evolution of an initially circular vortex near an escarpment. Part I: analytical results

The motion of a quasigeostrophic, equivalent-barotropic, initially circular vortex patch near an infinitely long topographic escarpment is studied using f-plane dynamics. There are two time scales in the problem: the advective time scale associated with the vortex, and the time scale for topographic vortex stretching. Analytical progress is possible when these two time scales are well-separated and results are presented here.If topographic vortex stretching dominates advection by the vortex the vortex is said to be ‘weak’. The vortex patch remains circular on the topographic time scale, and dispersive topographic waves rapidly propagate the initial disturbance away from the vicinity of the vortex. Subsequently cross-isobath motion is inhibited, and the vortex moves as though the escarpment were a plane wall. The same behaviour was observed for the motion of a weak singular vortex near an escarpment by Dunn, McDonald and Johnson [7], who named the phenomenon the ‘pseudoimage’ of the vortex.If advection dominates over topographic effects, the vortex is said to be ‘intense’. The vortex also remains circular to leading order, but the relative vorticity produced by the swirl of the vortex is less able to escape the vicinity of the vortex. The vortex follows a similar curved trajectory to those observed for intense vortices on the β-plane. The dipolar mechanism for this behaviour is described. Large time solutions are inhibited by the form of the escarpment topography, but examination of the equations leads to the conclusion that the leading order solution may be predict the motion for times beyond its formal range of validity.     

On short-crested waves: experimental and analytical investigations

Analytical and experimental investigations were conducted on short-crested wave fields generated by a sea-wall reflection of an incident plane wave. A perturbation method was used to compute analytically the solution of the basic equations up to the sixth order for capillary-gravity waves in finite depth, and up to the ninth order for gravity waves in deep water. For the experiments, we developed a new video-optical tool to measure the full three dimensional wave field η(x,y,t). A good agreement was found between theory and experiments. The spatio-temporal bi-orthogonal decomposition technique was used to exhibit the periodic and progressive properties of the short-crested wave field.     

An analytical and numerical study of chaotic dynamics in a simple bouncing ball model

Dynamics of a ball moving in gravitational field and colliding with a moving table is studied in this paper. The motion of the limiter is assumed as periodic with piecewise constant velocity-it is assumed that the table moves up with a constant velocity and then moves down with another constant velocity.The Poincaré map,describing evolution from an impact to the next impact,is derived and scenarios of transition to chaotic dynamics are investigated analytically and numerically.     

Time-Stepping Approximation of Rigid-Body Dynamics with Perfect Unilateral Constraints. I: The Inelastic Impact Case

We consider a discrete mechanical system with a non-trivial mass matrix, subjected to perfect unilateral constraints described by the geometrical inequalities ${f_{\alpha} (q) \geqq 0, \alpha \in \{1, \dots, \nu\} (\nu \geqq 1)}$ . We assume that the transmission of the velocities at impact is governed by Newton’s Law with a coefficient of restitution e = 0 (so that the impact is inelastic). We propose a time-discretization of the second order differential inclusion describing the dynamics, which generalizes the scheme proposed in Paoli (J Differ Equ 211:247–281, 2005) and, for any admissible data, we prove the convergence of approximate motions to a solution of the initial-value problem.     

Free vibrations of beam-mass-spring systems: analytical analysis with numerical confirmation

Free vibrations of a beam-mass-spring system with different boundary conditions are analyzed both analytically and numerically.In the analytical analysis,the system is divided into three subsystems and the effects of the spring and the point mass are considered as internal boundary conditions between any two neighboring subsystems.The partial differential equations governing the motion of the subsystems and internal boundary conditions are then solved using the method of separation of variables.In the numerical analysis,the whole system is considered as a single system and the effects of the spring and point mass are introduced using the Dirac delta function.The Galerkin method is then employed to discretize the equation of motion and the resulting set of ordinary differential equations are solved via eigenvalue analysis.Analytical and numerical results are shown to be in very good agreement.     

Experimental and numerical investigations into collapse behaviour of thin spherical shells under drop hammer impact

A study of the collapse behaviour of hemi spherical and shallow spherical shells and their modes of deformation under impact loading are presented in this paper. Aluminium spherical shells of various radii and thicknesses were made by spinning. These were subjected to impact loading under a drop hammer and the load histories were obtained in all the cases. Three-dimensional numerical simulations were carried out for all the tested specimen geometries using LS-DYNA^®. Material, geometric and contact nonlinearities were incorporated in the analysis. The uni-axial stress–strain curve for the material was obtained experimentally and was assumed to be piecewise linear in the plastic region. The results from impact experiments are used for the validation of the numerical simulations. Three distinct modes of deformation, namely local flattening, inward dimpling and formation of multiple numbers of lobes were analysed and influence of various parameters on these modes is discussed.