A comparison between continuous and discrete modelling of cables with bending stiffness

The static configuration in space of an inflexible two-dimensional cable anchored at its endpoints is obtained by the numerical solution of both continuous and discrete models. The results are compared and validated by experiment. It is shown that although the continuous model yields more accurate results than the discrete model, as expected, the effort and cost of numerically integrating the continuous model do not compare favourably with the relative ease and efficiency of solving the discrete model, which yields perhaps surprisingly accurate results.     

On a variational inequality on elasto-hydrodynamic lubrication

We consider the problem of a deformable surface moving over a flat plane. The surfaces are separated by a small gap filled by a lubricant fluid. The mathematical model consists of the Reynolds variational inequality with nonlocal coefficients given by an integral operator which depends on the fluid pressure. The nonlocal operator represents the deformation of the lubricated surfaces. The problem considers the vertical displacement of the elastic surface from its reference configuration. The goal of the paper is to obtain the range of these admissible displacements. We present general results for nonlocal coefficients with applications to particular problems in elasto-hydrodynamic lubrication.     

The diffraction of plane elastic waves by a delaminated rigid inclusion when there is smooth contact in the delamination region

A solution of the problem of the diffraction of harmonic elastic waves by a thin rigid strip-like delaminated inclusion in an unbounded elastic medium, in which the conditions for plane deformation are satisfied, is proposed. We mean by a delaminated inclusion an inclusion, one side of which is completely bonded to the elastic medium, while the second does not interact in any way with it, or this interaction is partial. It is assumed that the conditions for smooth contact are satisfied in the delamination region. The method of solution is based on the use of previously constructed discontinuous solutions of the equations describing the vibrations of an elastic medium under plane deformation conditions. The problem therefore reduces to solving a system of three singular integral equations in the unknown stress and strain jumps at the inclusion. An approximate solution of the latter enabled formulae to be obtained that are convenient for numerical realization when investigating the stressed state in the region of the inclusion and its displacements when acted upon by incident waves.     

A new formalism for the dynamic modelling of cables

This article proposes a new formalism for the dynamic modelling of cables that can even be applied when they are submitted to cross flow of water or air. An important application is the case of umbilical cables used in remotely operated vehicles. The primary basis for the formulation is to assume that the continuous flexibility is represented by a discrete approach, consisting of rigid links connected by elastic joints, allowing movement in three dimensions. Each elastic joint allows three independent movements, called elevation, azimuth and torsion (twist). A significant contribution of the proposed formalism is the development of a compact equation that allows obtaining the Lagrangian of the system directly and automatically, regardless of the number of links chosen to form a chain of rigid bodies connected by flexible joints to represent the continuous flexibility of the cable. This formulation allows the construction of an algorithm for obtaining the equations of the dynamic model of flexible cables.     

Differentiation of the energy functional in the equilibrium problem for a Timoshenko plate with a crack on the boundary of an elastic inclusion

Under consideration is the equilibrium of a composite plate containing a through vertical crack of variable length at the interface between thematrix and the elastic inclusion. The deformation of the matrix is described by the Timoshenko model, and the deformation of the elastic inclusion, by the Kirchhoff–Love model. Some formula is obtained for the derivative of the energy functional with respect to the crack length.     

无限非均质横观各向同性黏弹性介质中具有圆柱孔洞时的振动

在无限介质中,研究了横截面为圆的柱形孔洞表面上瞬时径向力或扭转引起的扰动,讨论了高阶黏弹性和横观各向同性弹性参数的非均匀性对扰动产生的影响．根据高阶黏弹性Voigt模型,将非零应力分量简化为径向位移分量项表示,这对横观各向同性和高阶黏弹性固体介质是合宜的．导出了含有弹性和黏弹性参数以幂律变化时的应力方程．在瞬时力和扭转边界条件下,求解该方程,求得径向位移分量以及和它相关的应力分量,用修正的Bessel函数项来表示．对瞬时径向力作用问题进行了数值分析,并给出了不同阶的黏弹性和非均质性时的位移和应力变化图形．扭转作用时扰动的数值解可以用类似的方法研究,这里不再深入讨论．     

Traveling-Wave Solutions of the Kolmogorov–Petrovskii–Piskunov Equation

We consider quasi-stationary solutions of a problem without initial conditions for the Kolmogorov–Petrovskii–Piskunov (KPP) equation, which is a quasilinear parabolic one arising in the modeling of certain reaction–diffusion processes in the theory of combustion, mathematical biology, and other areas of natural sciences. A new efficiently numerically implementable analytical representation is constructed for self-similar plane traveling-wave solutions of the KPP equation with a special right-hand side. Sufficient conditions for an auxiliary function involved in this representation to be analytical for all values of its argument, including the endpoints, are obtained. Numerical results are obtained for model examples.     

Radiation instability of a circular cylinder in a uniform flow of a two-layer fluid

A solution of the plane linear problem of the oscillations of a horizontal circular cylinder in a uniform flow of a two-layer unbounded fluid is obtained using the method of multipole expansions. The direction of the flow is perpendicular to the cylinder axis. The whole cylinder Ges in the upper or lower layer. The fluid is assumed to be ideal and incompressible, the flow in each layer being a potential one. All the components of the radiation load (the apparent masses and damping coefficients) are determined and the regions of existence of radiation instability are found, depending on the flow velocity for a cylinder suspended by horizontal and vertical elastic links. By solving the integro-differential equation numerically the relative oscillations of the body under specified initial conditions are found.     

Fragments of the theory of nanotransistors: self-switching of a plane transverse hypersonic wave in nonlinearly elastic nanocomposite materials

We present fragments of the theory of nanotransistors concerning general information about transistors, models of nanocomposite materials, effects of nonlinear interaction of waves, and a theoretical analysis of the interaction of cubically nonlinear elastic plane harmonic waves in materials whose nonlinear properties are described by the Murnaghan potential. Using the method of slowly varying amplitudes, we investigate the interaction of two harmonic vertical transverse plane waves. Shortened and evolution equations, and Manley-Rowe relations, are obtained. We analyze analytically and numerically the mechanism of repumping of the energy of a strong pumping wave that propagates at a frequency of ω to a weak signal wave propagating at a frequency of 3ω . The described mechanism of switching of hypersonic waves in a nonlinearly elastic nanomaterial is similar to the mechanism of switching observed in optical and other transistors.     

Minimal curves of constant torsion

The Griffiths formalism is applied to find constant torsion curves which are extremal for arclength with respect to variations preserving torsion, fixing the endpoints and the binormals at the endpoints. The critical curves are elastic rods of constant torsion, which are shown to not realize certain boundary conditions.

     

Mathematical modeling of seismic and acousto-gravitational waves in a heterogeneous earth-atmosphere model

A numerical-analytical solution to problems of seismic and acoustic-gravitational wave propagation is applied to a heterogeneous Earth-Atmosphere model. The seismic wave propagation in an elastic half-space is described by a system of first order dynamic equations of the elasticity theory. The propagation of acoustic-gravitational waves in the atmosphere is described by the linearized Navier-Stokes equations. The algorithm proposed is based on the integral Laguerre transform with respect to time, the finite integral Bessel transform along the radial coordinate with a finite difference solution of the reduced problem along the vertical coordinate. The algorithm is numerically tested for the heterogeneous Earth-Atmosphere model for different source locations.     

Simulation of stage I-crack growth using a 3D-model

A three-dimensional model for stage I-short crack propagation on a single slip plane is presented. It considers elastic plastic material behaviour by allowing sliding on the active slip plane in a defined slip direction. A crack propagation law based on the crack tip slide displacement is implemented to simulate crack propagation. The model is solved numerically using the dislocation loop technique. (© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Nonlinear Oscillations of a Spring Pendulum at the 1 : 1 : 2 Resonance: Theory,Experiment, and Physical Analogies

Nonlinear spatial oscillations of a material point on a weightless elastic suspension are considered. The frequency of vertical oscillations is assumed to be equal to the doubled swinging frequency (the 1 : 1 : 2 resonance). In this case, vertical oscillations are unstable, which leads to the transfer of the energy of vertical oscillations to the swinging energy of the pendulum. Vertical oscillations of the material point cease, and, after a certain period of time, the pendulum starts swinging in a vertical plane. This swinging is also unstable, which leads to the back transfer of energy to the vertical oscillation mode, and again vertical oscillations occur. However, after the second transfer of the energy of vertical oscillations to the pendulum swinging energy, the apparent plane of swinging is rotated through a certain angle. These phenomena are described analytically: the period of energy transfer, the time variations of the amplitudes of both modes, and the change of the angle of the apparent plane of oscillations are determined. The analytic dependence of the semiaxes of the ellipse and the angle of precession on time agrees with high degree of accuracy with numerical calculations and is confirmed experimentally. In addition, the problem of forced oscillations of a spring pendulum in the presence of friction is considered, for which an asymptotic solution is constructed by the averaging method. An analogy is established between the nonlinear problems for free and forced oscillations of a pendulum and for deformation oscillations of a gas bubble. The transfer of the energy of radial oscillations to a resonance deformation mode leads to an anomalous increase in its amplitude and, as a consequence, to the break-up of a bubble.     

A nonlinear elasticity model of macromolecular conformational change induced by electrostatic forces

In this paper we propose a nonlinear elasticity model of macromolecular conformational change (deformation) induced by electrostatic forces generated by an implicit solvation model. The Poisson-Boltzmann equation for the electrostatic potential is analyzed in a domain varying with the elastic deformation of molecules, and a new continuous model of the electrostatic forces is developed to ensure solvability of the nonlinear elasticity equations. We derive the estimates of electrostatic forces corresponding to four types of perturbations to an electrostatic potential field, and establish the existence of an equilibrium configuration using a fixed-point argument, under the assumption that the change in the ionic strength and charges due to the additional molecules causing the deformation are sufficiently small. The results are valid for elastic models with arbitrarily complex dielectric interfaces and cavities, and can be generalized to large elastic deformation caused by high ionic strength, large charges, and strong external fields by using continuation methods.     

Effect of wing deformation on highlift systems of transport aircraft

An elastic multibody model is developed that combines highlift system models consisting of flap and slat mechanisms with an elastic wing model. It describes the motions and deformations of the highlift systems with airloads applied and, in particular, the effect of wing deformations on the highlift systems. In addition the model contains a flexible wing with the engine and the middle section of the fuselage that is fixed at the symmetry plane of the aircraft. While previously wing deformation was taken into account in static FE simulations only the developed multibody model provides the investigation of the motions and inner loads of the highlift mechanisms with consideration of the wing deformations with dynamical effects. (© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Dynamic modelling of a multi-cable driven parallel platform with guiding devices

ABSTRACT

In this paper, a mathematical model is presented to numerically simulate the dynamical responses in a multi-cable suspension platform taking into account the slack cables and guiding devices. The state change of the cable (slack versus tensioned) is considered and is described mathematically by a complementary condition equation, and the interactions between the guiding wheels and the shaft wall are described by the Heaviside step function. The Lagrange’s equation with constraints is used to derive the dynamic equations of the system, and a non-smooth generalized-α algorithm for non-smooth phenomena of multibody dynamics is applied to numerically solve the equations. The simulation results have shown the dynamic responses of the platform and the cable tension characters when different cables are excited by different longitudinal excitations. Moreover, the results have illustrated how the cable tension differences may affect the pressure on the shaft wall applied by the guiding devices.     

On the stability of periodic trajectories of a planar Birkhoff billiard

The inertial motion of a material point is analyzed in a plane domain bounded by two curves that are coaxial segments of an ellipse. The collisions of the point with the boundary curves are assumed to be absolutely elastic. There exists a periodic motion of the point that is described by a two-link trajectory lying on a straight line segment passed twice within the period. This segment is orthogonal to both boundary curves at its endpoints. The nonlinear problem of stability of this trajectory is analyzed. The stability and instability conditions are obtained for almost all values of two dimensionless parameters of the problem.     

Reflection and transmission of couple longitudinal waves at a plane interface between two dissimilar half-spaces of thermo-elastic materials with voids

The problem of reflection and transmission of elastic waves due to incident plane couple longitudinal waves at a plane interface between two dissimilar half-spaces of thermo-elastic materials with voids has been investigated. Using the theory of Iesan (1986), the propagation of couple longitudinal waves in the thermo-elastic materials with voids has been explained. The expressions of the reflection and transmission coefficients and energy ratios corresponding to reflected and transmitted waves are obtained. These coefficients and energy ratios of the various reflected and transmitted waves are computed numerically for a specific model.     

Analysis of elastomeric composites based on fiber-reinforced systems. 2. Unidirectional composites

The elastic properties of unidirectionally reinforced composite materials under large deformations are studied. The applied model for deformation of materials is based on the structural macroscopic theory of stiff and soft composites, including micro- and macromechanical levels of analysis of composite media. The properties of unidirectional elastomeric composites are studied in tension and shear in the plane of reinforcement. The microscopic fields in the structural components of composites having poorly compressible and compressible matrices are also analyzed. Changes in the parameters of macroscopic deformation of the composites are examined as functions of the loading parameters and initial conditions of the structure. The evolution of the structural changes in deformed composite materials is described.State Metallurgical Academy of Ukraine, Dnepropetrovsk, Ukraine. Translated from Mekhanika Kompozitnykh Materialov, Vol. 35, No. 1, pp. 29–50, January–February, 1999.     

Plane wave decomposition in the unit disc: Convergence estimates and computational aspects

This paper deals with the numerical simulation of time-harmonic wave fields using progressive plane waves. It is shown that a plane wave travelling in arbitrary direction can be numerically recovered with an accuracy of the order of the machine precision with a collocation formulation and the square root of the machine precision with a least-square formulation. However, strongly evanescent and nearly singular wave fields cannot be properly recovered with standard double-precision floating-point arithmetic. Some of the ideas are applied to the elastic wave equation and a simple optimization algorithm is proposed to find a good compromise between the accuracy and the number of plane waves.