A discrete approach for a generalized Beck’s column in parametric resonance

A Galerkin projection based on non-standard bases is conceived to derive an equivalent discrete model of a continuous system under non-conservative forces. The problem of deriving a discrete model capable of describing critical and post-critical scenarios for non-selfadjoint systems is discussed and an heuristic rule for a proper choice of trial functions is given. The procedure is utilized to analyze the effect of non-conservative autonomous and non-autonomous (pulsating) forces acting on a linearly damped Beck’s column involving geometrical nonlinearities. The linear and nonlinear behaviours arising from the analysis of the proposed discrete model are in good agreement with those observed through the unavoidably more involved direct continuous approach. Critical scenarios for the autonomous and non-autonomous cases are investigated and the multiple scales method is applied in order to obtain the bifurcation equations in the neighborhood of a Hopf bifurcation point in primary parametric resonance. A comparison between critical and post-critical continuous and discrete model is performed adopting two control parameters, namely the amplitudes of the static and dynamic components of the forces, playing the role of detuning and bifurcation parameters, respectively.     

Strain Incompatibility in Single Crystals: Kröner’s Formula Revisited

In this paper we prove under precise working assumptions a famous relation originally due to Ekkehart Kröner (Physiques des défauts, Les Houches session XXXV, 1980) relating linear elastic strain incompatibility and dislocation and disclination densities. The 3D case with a set of isolated skew (skew lines are lines which do not intersect but are not parallel either) defect lines in a single crystal is considered according to a mathematical approach taking into account the combined effects of field multivaluedness and concentration properties.     

A two-dimensional polycrystal model in a ‘deep’ space representation

We propose a model which links macroscopic deformations with changes in crystalline texture. The model calls upon microstructural variables to describe the state of the polycrystal; their time-rates, together with the corresponding microstresses, enter the expression of virtual power. Appropriate supplementary terms also appear in the equations of balance of energy and entropy.Schmid's law for single slip systems is invoked to provide one of the additional constitutive equations. A constitutive statement similar to the Taylor-Bishop-Hill criterion for polycrystals is introduced to relate the velocity gradient and the lattice spin via the active spin combination.The model is applied to an example agreeing with a Taylor planar polycrystal for a particular choice of constitutive laws for the microstresses.

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Sommario Proponiamo un modello che collega la deformazione macroscopica all'evoluzione della tessitura cristallina. Questo modello si rifà a variabili microstrutturali per descriverelo stato profondo del policristallo; le velocità di tali variabili entrano nell'espressione della potenza virtuale in uno con le microtensioni ad esse associate. Termini supplementari appropriati appaiono inoltre nelle equazioni di bilancio dell'energia e dell'entropia.Viene utilizzato il criterio di Schmid per i singoli sistemi di scivolamento per scrivere una delle equazioni costitutive necessarie. Viene quindi introdotto un assunto simile al criterio di Taylor-Bishop-Hill per i policristalli per mettere in relazione il gradiente di velocità e la velocità di rotazione del reticolo cristallino via la combinazione di scivolamenti attiva.Il modello viene applicato ad un esempio mostrando buon accordo con un policristallo piano alla Taylor per una scelta particolare dei legami costitutivi delle microtensioni.

     

A Cartesian cut cell based two-way strong fluid–solid coupling algorithm for 2D floating bodies

In a recent paper Kelly et al. (2015) [PICIN: A Particle-In-Cell solver for incompressible free surface flows with two-way fluid–solid coupling. SIAM Journal on Scientific Computing 37 (3), B403–24.] detailed the PICIN full particle Particle-In-Cell (PIC) solver for incompressible free-surface flows. The model described in that paper employed a tailored version of the Distributed Lagrange Multiplier (DLM) method for the strong coupling of fluid–solid interaction. In this paper we propose an alternative strong fluid–solid coupling algorithm based on a modification to the cut cell methodology that is informed by the variational approach. The solid velocity flux/integral on the boundary is expressed purely in terms of pressure leading to a revised pressure Poisson equation that is discretised in a finite volume sense. This approach allows the PICIN model to simulate the motion of floating bodies of arbitrary configuration. 2D test cases involving floating bodies with one or more degrees of freedom (DoF) are used to validate the modified PICIN model. The results presented show that the modified PICIN model is able to both efficiently and robustly predict the motions of surface-piercing floating structures under either regular or extreme wave action.     

Advanced parametric space-frequency separated representations in structural dynamics: A harmonic–modal hybrid approach

This paper is concerned with the solution to structural dynamics equations. The technique here presented is closely related to Harmonic Analysis, and therefore it is only concerned with the long-term forced response. Proper Generalized Decomposition (PGD) is used to compute space-frequency separated representations by considering the frequency as an extra coordinate. This formulation constitutes an alternative to classical methods such as Modal Analysis and it is especially advantageous when parametrized structural dynamics equations are of interest. In such case, there is no need to solve the parametrized eigenvalue problem and the space-time solution can be recovered with a Fourier inverse transform. The PGD solution is valid for any forcing term that can be written as a combination of the considered frequencies. Finally, the solution is available for any value of the parameter. When the problem involves frequency-dependent parameters the proposed technique provides a specially suitable method that becomes computationally more efficient when it is combined with a modal representation.     

Cracks’ closure in 3-D fracture dynamics: the effect of relative location of two coplanar cracks

In this paper, the problem of two equal coplanar cracks with allowance for the crack faces contact interaction was investigated. The problem of the cracks located in homogeneous, isotropic, and linearly elastic solid subjected to normally incident tension–compression wave is solved by the boundary integral equations method. The influence of the distance between two cracks on the stress intensity factors (opening mode and transverse shear mode) is studied for a range of wave numbers. The results are compared with those obtained neglecting cracks’ closure.     

2D DEM simulation of particle mixing in rotating drum： A parametric study

Mixing behaviors of equal-sized glass beads in a rotating drum were investigated by both DEM simula- tions and experiments. The experiments indicated that higher rotation speed can significantly enhance mixing. The particle profiles predicted by 2D DEM simulation were compared with the experimental results from a quasi-2D drum, showing inconsistency due to reduction of contacts in the single-layer 2D simulation which makes the driving friction weaker than that in the quasi-2D test, better results could be reached by specifying a higher frictional coefficient between the particles and the cylinder wall. In order to explore the influences of physical properties （density, size or friction） on mixing behavior, numerical 2D simulations were carried out systematically, in which one examined specific property being examined was exaggerated while the others were kept the same as that in the control group. The DEM simulations reveal that particle density and size are the dominating factors affecting mixing behaviors, while the effect of frictional coefficient is less significant. However, segregation due to any of the factors can be diminished by specifying a proper particle size distribution （multi-size with lower size ratio）. 2009 Chinese Society of Particuology and Institute of Process Engineering, Chinese Academy of Sciences. Published by Elsevier B.V. All rights reserved.     

A new finding of the existence of Feigenbaum’s constants in the fractional-order Chen–Lee system

This paper examines the universal quantitative properties of the fractional- and integer-order Chen?CLee systems. A?series of bifurcation diagrams of the system were generated in order to measure Feigenbaum??s constants. It was found that the measured values of the integer-order system were accurately approaching their universal constants, while the errors between measured values of the fractional-order system and the universal constants were not very large. The results showed that both the fractional- and integer-order Chen?CLee systems belonged to a quadratic map. To the authors?? knowledge, this is the first paper to measure Feigenbaum??s constants in fractional-order systems.     

A 2.5-D dynamic model for a saturated porous medium: Part I. Green’s function

Based on Biot’s theory, the dynamic 2.5-D Green’s function for a saturated porous medium is obtained using the Fourier transform and the potential decomposition methods. The 2.5-D Green’s function corresponds to the solutions for the following two problems: the point force applied to the solid skeleton, and the dilatation source applied within the pore fluid. By performing the Fourier transform on the governing equations for the 3-D Green’s function, the governing differential equations for the two parts of the 2.5-D Green’s function are established and then solved to obtain the dynamic 2.5-D Green’s function. The derived 2.5-D Green’s function for saturated porous media is verified through comparison with the existing solution for 2.5-D Green’s function for the elastodynamic case and the closed-form 3-D Green’s function for saturated porous media. It is further demonstrated that a simple form 2-D Green’s function for saturated porous media can be been obtained using the potential decomposition method.     

Nonlinear dynamics in mechanics and engineering: 40 years of developments and Ali H. Nayfeh’s legacy

Nonlinear Dynamics - Nonlinear dynamics of engineering systems has reached the stage of full maturity in which it makes sense to critically revisit its past and present in order to establish an...     

A construction of new exact periodic wave and solitary wave solutions for the 2D Ginzburg–Landau equation

Using a uniform algebraic method, new exact solitary wave solutions and periodic wave solutions for 2D Ginzburg–Landau equation are obtained. Moreover, three-dimensional and two-dimensional graphics of some solutions have been plotted.     

A new laser vorticity probe — LAVOR: Its development and validation in a turbulent boundary layer

Vorticity measurements, which are scarce at the present time, can provide valuable dynamical information, particularly in unsteady and separated flows. Advances in laser Doppler anemometry and optical techniques have furnished the opportunity for the development of a non-intrusive vorticity probe with very fine spatial and temporal resolutions. The laser vorticity probe (LAVOR), which makes use of minimal laser beams and optical components, is capable of measuring velocity gradients with a separation distances as small as 0.3 mm. Velocity gradients are measured using two points on the same probe volume. However, unlike other techniques, the LAVOR also provides the instantaneous velocity at each point in the probe volume, and so provides additional dynamical information. The LAVOR probe was used in a fully turbulent two-dimensional boundary layer, and the data obtained are compared with the existing hot-wire vorticity data obtained in the same wind tunnel facility and with data obtained in other facilities. The spatial resolution is of the order of three Kolmogorov microscale units. Published online: 19 October 2002     

A Brief Account of G.D.Birkhoff’s Problem in the Problem of Three Bodies

The present article gives a historical survery of G.D.Birkhoff’s seventh problem which is an inquiry about the topological structure of the set of definition of the reduced differential equations of motion.Recent advances in the problem and their meaning have been briefly indicated.The classical 3-body problem concerns how the three particles should move under their mutual Newtonian attraction.By a particle we mean a goometrical point endorsed with a constant positive number m which is called mass.Expressed mathematically,the problem appears as to solving of the following system of differential equations:     

A geometrically exact approach to the overall dynamics of elastic rotating blades—part 2: flapping nonlinear normal modes

The geometrically exact equations of motion about the prestressed state discussed in part 1 (i.e., the nonlinear equilibrium under centrifugal forces) are expanded in the Taylor series of the incremental displacements and rotations to obtain the third-order perturbed form. The expanded form is amenable to a perturbation treatment to unfold the nonlinear features of free undamped flapping dynamics. The method of multiple scales is thus applied directly to the partial-differential equations of motion to construct the backbone curves of the flapping modes and their nonlinear approximations when they are away from internal resonances with other modes. The effective nonlinearity coefficients of the lowest three flapping modes of elastic isotropic blades are investigated when the angular speed is changed from low- to high-speed regimes. The novelty of the current findings is in the fact that the nonlinearity of the flapping modes is shown to depend critically on the angular speed since it can switch from hardening to softening and vice versa at certain speeds. The asymptotic results are compared with previous literature results. Moreover, 2:1 internal resonances between flapping and axial modes are exhibited as singularities of the effective nonlinearity coefficients. These nonlinear interactions can entail fundamental changes in the blade local and global dynamics.     

A correction of the 2D KdV equation of Djordjevic ＆ Redekopp in exponentially stratified fluid＊

To further study the fission laws of initial internal solitons on the continental shelf/slope, we rederive and correct the 2D KdV equation of Djordjevic & Redekopp for exponentially stratified fluid (or ocean) and with two-dimensional topography. Through a combination of theoretical study and numerical experiments, we show that solitons in the odd vertical modes can fission. However, because of the corrections, the fission conditions are different from those of Djordjevic & Redekopp. The even modes cannot fission unless the initial internal solitons propagate from shallow sea to deep sea. This conclusion is entirely opposite to that of Djordjevic & Redekopp.The project supported by the National Natural Science Foundation of China (40276008) and the Grant of Key Laboratory of Marine Science and Numerical Modeling, SOA (0201(2003))The English text was polished by Keren Wang.     

Axial damping in metal-matrix composites. I: A new technique for measuring phase difference to 10−4 radians

A majority of the current experimental techniques for measuring damping employ either flexural or torsional vibrations.In either case the strain field is nonhomogeneous. If the material damping is linear, i.e., strain independent, then the measured quantity equals the intrinsic material damping. if, on the other hand, the material damping is nonlinear, i.e., strain dependent, then the measured quantity also reflects the nonhomogeneity of the strain field and, therefore, is not equal to the intrinsic material damping. In this work we describe a new experimental technique in which the foregoing problem is circumvented by employing a homogeneous strain field, namely, uniform uniaxial tension. Damping is viewed as the phase angle by which the stress leads the strain. The finiteduration, time-harmonic stress and strain signals are transformed to the frequency domain via the use of Fourier transforms. It is shown that if one confines attention to the immediate neighborhood of the excitation frequency then, for all practical purposes, the phase difference between the two sinusoids is equal to the phase difference between their Fourier transforms. We will demonstrate that this phase difference can be measured to an accuracy of 2/2¹⁶ or 9.587×10^–5 radians.Paper was presented at the 1990 SEM Spring Conference on Experimental Mechanics held in Albuquerque, NM on June 3–6.