Solution of free vibration equations of semi-rigid connected Reddy–Bickford beams resting on elastic soil using the differential transform method

The literature regarding the free vibration analysis of Bernoulli–Euler and Timoshenko beams under various supporting conditions is plenty, but the free vibration analysis of Reddy–Bickford beams with variable cross-section on elastic soil with/without axial force effect using the Differential Transform Method (DTM) has not been investigated by any of the studies in open literature so far. In this study, the free vibration analysis of axially loaded and semi-rigid connected Reddy–Bickford beam with variable cross-section on elastic soil is carried out by using DTM. The model has six degrees of freedom at the two ends, one transverse displacement and two rotations, and the end forces are a shear force and two end moments in this study. The governing differential equations of motion of the rectangular beam in free vibration are derived using Hamilton’s principle and considering rotatory inertia. Parameters for the relative stiffness, stiffness ratio and nondimensionalized multiplication factor for the axial compressive force are incorporated into the equations of motion in order to investigate their effects on the natural frequencies. At first, the terms are found directly from the analytical solutions of the differential equations that describe the deformations of the cross-section according to the high-order theory. After the analytical solution, an efficient and easy mathematical technique called DTM is used to solve the governing differential equations of the motion. The calculated natural frequencies of semi-rigid connected Reddy–Bickford beam with variable cross-section on elastic soil using DTM are tabulated in several tables and figures and are compared with the results of the analytical solution where a very good agreement is observed.     

Simulation of free shear flow by the continuous vortex sheet method

A method is developed for simulating the motion and deformation of a tangential discontinuity with stability characteristics similar to the real ones. As distinct from the discrete vortex method, which forms the basis of the method proposed, the motion of a continuous vortex sheet of finite thickness is considered. The equations of motion are derived on the basis of an analysis of the physical reasons for the stability of this sheet with respect to small-scale perturbations. Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 1, pp. 42–50, January–February, 1999.     

On the Chaotic Dynamics of a Spherical Pendulum with a Harmonically Vibrating Suspension

The equations of motion for a lightly damped spherical pendulum are considered. The suspension point is harmonically excited in both vertical and horizontal directions. The equations are approximated in the neighborhood of resonance by including the third order terms in the amplitude. The stability of equilibrium points of the modulation equations in a four-dimensional space is studied. The periodic orbits of the spherical pendulum without base excitations are revisited via the Jacobian elliptic integral to highlight the role played by homoclinic orbits. The homoclinic intersections of the stable and unstable manifolds of the perturbed spherical pendulum are investigated. The physical parameters leading to chaotic solutions in terms of the spherical angles are derived from the vanishing Melnikov–Holmes–Marsden (MHM) integral. The existence of real zeros of the MHM integral implies the possible chaotic motion of the harmonically forced spherical pendulum as a result from the transverse intersection between the stable and unstable manifolds of the weakly disturbed spherical pendulum within the regions of investigated parameters. The chaotic motion of the modulation equations is simulated via the 4th-order Runge–Kutta algorithms for certain cases to verify the analysis.     

Finite element analysis for train–track–bridge interaction system

This article deals with the dynamic analysis of train–track–bridge interaction system using the finite element method. In this interaction system, each four-wheelset vehicle in the train is modeled by a mass–spring–damper system with 10 degrees of freedom; the rails and the bridge decks are modeled as a number of Bernoulli–Euler beam elements, while the elasticity and damping properties of the rail bed are represented by continuous springs and dampers. The equation of motion for the interaction system is presented in matrix form with time-dependent coefficients. The correctness of the proposed procedure is illustrated by a comparison with the numerical result from the existing literature. Several numerical examples are chosen to investigate the effect of two types of vehicle models, two types of bridge models and three damping values of bridge on the maximum dynamic responses of train, track and bridges.     

Thermo-mechanical nonlinear vibration analysis of a spring-mass-beam system

Thermo-mechanical vibrations of a simply supported spring-mass-beam system are investigated analytically in this paper. Taking into account the thermal effects, the nonlinear equations of motion and internal/external boundary conditions are derived through Hamilton’s principle and constitutive relations. Under quasi-static assumptions, the equations governing the longitudinal motion are transformed into functions of transverse displacements, which results in three integro-partial differential equations with coupling terms. These are solved using the direct multiple-scale method, leading to closed-form solutions for the mode functions, nonlinear natural frequencies and frequency–response curves of the system. The influence of system parameters on the linear and nonlinear natural frequencies, mode functions, and frequency–response curves is studied through numerical parametric analysis. It is shown that the vibration characteristics depend on the mid-plane stretching, intra-span spring, point mass, and temperature change.     

Symmetry Reduced Dynamics of Charged Molecular Strands

The equations of motion are derived for the dynamical folding of charged molecular strands (such as DNA) modeled as flexible continuous filamentary distributions of interacting rigid charge conformations. The new feature is that these equations are nonlocal when the screened Coulomb interactions, or Lennard–Jones potentials between pairs of charges, are included. The nonlocal dynamics is derived in the convective representation of continuum motion by using modified Euler–Poincaré and Hamilton–Pontryagin variational formulations that illuminate the various approaches within the framework of symmetry reduction of Hamilton’s principle for exact geometric rods. In the absence of nonlocal interactions, the equations recover the classical Kirchhoff theory of elastic rods. The motion equations in the convective representation are shown to arise by a classical Lagrangian reduction associated to the symmetry group of the system. This approach uses the process of affine Euler–Poincaré reduction initially developed for complex fluids. On the Hamiltonian side, the Poisson bracket of the molecular strand is obtained by reduction of the canonical symplectic structure on phase space. A change of variables allows a direct passage from this classical point of view to the covariant formulation in terms of Lagrange–Poincaré equations of field theory. In another revealing perspective, the convective representation of the nonlocal equations of molecular strand motion is transformed into quaternionic form.     

Estimating the Hydraulic Conductivity of Two-Dimensional Fracture Networks Using Network Geometric Properties

Based on a modified Darcy–Brinkman–Maxwell model, a linear stability analysis of a Maxwell fluid in a horizontal porous layer heated from below by a constant flux is carried out. The non-oscillatory instability and oscillatory instability with different hydrodynamic boundaries such as rigid and free surfaces at the bottom are studied. Compared with the rigid surface cases, onset of fluid motion due to non-oscillatory instability and oscillatory instability is found to occur both more easily for the system with a free bottom surface. The critical Rayleigh number for onset of fluid motion due to non-oscillatory instability is lower with a constant flux heating bottom than with an isothermal heating bottom, but the result due to oscillatory instability is in contrast. The effects of the Darcy number, the relaxation time, and the Prandtl number on the critical Rayleigh number are also discussed.     

Comparison of near- and far-fault ground motion effect on the nonlinear response of dam–reservoir–foundation systems

In this paper, it is aimed to compare the near- and far-fault ground motion effects on the nonlinear dynamic response of dams including dam–reservoir–foundation interaction. Two different types of dams, which are concrete arch and concrete faced rockfill dams, are selected to investigate the near- and far-fault ground motion effects on the dam responses. The behavior of reservoir water is taken into account using Lagrangian approach. The Drucker–Prager material model is employed in nonlinear analyses. Near and far-fault strong ground motion records, which have approximately identical peak ground accelerations, of Loma Prieta (1989) earthquake are selected for the analyses. Displacements, maximum and minimum principal stresses are determined using the finite element method. The displacements and principal stresses obtained from the analyses of dams subjected to each fault effect are compared with each other. It is clearly seen that there is more seismic demand on displacements and stresses when the dam is subjected to near-fault ground motion.     

Gas-dynamic analogy for vortex free-boundary flows

The classical shallow-water equations describing the propagation of long waves in flow without a shear of the horizontal velocity along the vertical coincide with the equations describing the isentropic motion of a polytropic gas for a polytropic exponent γ = 2 (in the theory of fluid wave motion, this fact is called the gas-dynamic analogy). A new mathematical model of long-wave theory is derived that describes shear free-boundary fluid flows. It is shown that in the case of one-dimensional motion, the equations of the new model coincide with the equations describing nonisentropic gas motion with a special choice of the equation of state, and in the multidimensional case, the new system of long-wave equations differs significantly from the gas motion model. In the general case, it is established that the system of equations derived is a hyperbolic system. The velocities of propagation of wave perturbations are found. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 48, No. 3, pp. 8–15, May–June, 2007.     

Thermal convection initiated by the quasistatic component of the microacceleration field on the “Mir” orbital station

The three-dimensional unsteady thermal convection developing in a cubic cavity on board the “Mir” station under the action of the quasistatic component of the microacceleration caused by the earth’s gravity field strength gradient and the motion of the station about the centre of mass is investigated numerically. The calculations are carried out for two real time intervals of motion of the station about the centre of mass using actual values of the quasistatic component. Moscow, Sankt-Peterburg. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 3, pp. 39–45, May–June, 2000.     

Effect of anisotropy of porous media on the onset of buoyancy-driven convection

A theoretical analysis of buoyancy-driven instability under transient basic fields is conducted in an initially quiescent, fluid-saturated, horizontal porous layer. Darcy’s law is used to explain characteristics of fluid motion and the anisotropy of permeability is considered. Under the Boussinesq approximation and the principle of exchange of stabilities, the stability equations are derived by using the linear stability theory and the energy method. The linear stability equations are analyzed numerically by using the frozen-time model and the linear amplification theory and the global stability limits are obtained numerically from the energy method. For the various anisotropic ratios, the critical times are predicted as a function of the Darcy–Rayleigh number and the critical Darcy–Rayleigh number is also obtained. The present predictions are compared each another and with existing theoretical ones.     

Numerical solution of the problem of motion of a load on a cracked ice sheet

A mathematical model is constructed for the motion of a load on a cracked ice sheet. Examples of calculation of ice deflections are given, and the calculation results are analyzed. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 49, No. 3, pp. 151–156, May–June, 2008.     

Vibration of beams with general boundary conditions due to a moving random load

Summary  The transverse vibrations of elastic homogeneous isotropic beams with general boundary conditions due to a moving random force with constant mean value are analyzed. The boundary conditions considered are: pinned–pinned, fixed–fixed, pinned–fixed, and fixed–free. Based on the Bernoulli beam theory, the problem is described by means of a partial differential equation. Closed-form solutions for the variance and the coefficient of variation of the beam deflection are obtained and compared for three types of force motion: accelerated, decelerated and uniform. The effects of beam damping and speed of the moving force on the dynamic response of beams are studied in detail. Received 3 December 2001; accepted for publication 30 April 2002     

Cavitation-induced particle–wall interaction in Newtonian and non-Newtonian fluids

A novel hydrodynamic effect, namely, slow contactless motion of a heavy spherical particle along an inclined wall, accompanied by the formation of a finite particle–wall clearance under the action of a cavitation-induced lift force, is investigated. Similarity parameters controlling the particle motion, determined using the dimensionality theory, are validated experimentally. These parameters are related to the atmospheric pressure, the surface tension on the liquid–air interface, the density of the air dissolved in the fluid, the particle weight in the fluid, and the viscoelastic properties of the fluid.This paper was presented at the AERC 2005.     

Soil Behavior at the Interface with a Rigid Projectile During Penetration

The motion and state of soil at the interface with a penetrating rigid projectile is studied by numerical solution of the problem of a cylindrical projectile which expands and at the same time moves translationally along its axis in soil. The soil behavior is described using the model of a compressible elastoplastic medium with transition to a plastic state depending on the pressure in it. It is shown that a thin layer of soil at the interface with the projectile nose should be set in motion and move together with the projectile without sliding. An analysis is performed of the validity of using the dry friction law to determine the shear stresses on the projectile surface during penetration. The heat release in the soil layer at the interface due to internal friction and its possible effect on the penetration are estimated. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 46, No. 6, pp. 116–127, November–December, 2005.     

The Onset of Double-Diffusive Nanofluid Convection in a Layer of a Saturated Porous Medium

The onset of convection in a horizontal layer of a porous medium saturated by a nanofluid is studied analytically. The model used for the nanofluid incorporates the effects of Brownian motion and thermophoresis. For the porous medium, the Brinkman model is employed. Three cases of free–free, rigid–rigid, and rigid–free boundaries are considered. The analysis reveals that for a typical nanofluid (with large Lewis number), the prime effect of the nanofluids is via a buoyancy effect coupled with the conservation of nanoparticles, whereas the contribution of nanoparticles to the thermal energy equation is a second-order effect. It is found that the critical thermal Rayleigh number can be reduced or increased by a substantial amount, depending on whether the basic nanoparticle distribution is top-heavy or bottom-heavy, by the presence of the nanoparticles. Oscillatory instability is possible in the case of a bottom-heavy nanoparticle distribution.     

Numerical study of initial pulsed motion of a powder layer in a duct under the action of a compressed gas

One-dimensional plane pulsed joint motion of a gas phase and a disperse phase in the presence of friction of the latter against the duct walls are studied numerically using the model of a heterogeneous medium. It is established that two qualitatively different regimes of motion are possible, depending on the initial conditions in a high-pressure chamber and the value of the friction factor. It is found that the powder exit velocity is self-similar with respect to friction. Mozhaiskii Military Space Engineering Academy, St. Petersburg 197082. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 40, No. 6, pp. 122–127, November–December, 1999.     

Conformational evolution under steady shear flow: a comparison between cyclic and linear block copolymers

Cyclic block copolymer is a special type of block copolymer having no free ends. Comparison of the dynamic behavior between cyclic and linear block copolymers enables an understanding of the role of chain ends in dynamics of the latter. In relation to this point, analysis was made on the conformational dynamics for a cyclic bead-spring type diblock copolymer chain, AoB, under the steady shear flow. Further comparison was made on the conformational behavior of the AoB chain and that of two symmetric linear triblock copolymer chains, A–B–A and B–A–B. For these chains, the mobility was set to be higher for the A segments than the B segments. Thus, for the AoB chain under the steady shear flow, the segments of the A block exhibit less orientational anisotropy than those of the B block. This orientational contrast is enhanced for the A–B–A chain partly because the constraint for the motion of the segments is less near the chain ends than near the chain center. Nevertheless, for the B–A–B chain, the segmental orientation over the A block becomes more anisotropic than that over the B block. Detailed analysis shows that this result is attributable to a high orientational correlation for the segments of two end B blocks, in particularly for those near the block junctions. The correlated B segments exert a tensile force on the A block thereby significantly enhancing the orientational anisotropy of the A segments.     

Characteristics, conservation laws, and symmetries of the kinetic equations of motion of bubbes in a fluid

Generalized characteristics and Riemann invariants that are preserved along the characteristics are found for a kinetic model of motion of bubbles in a fluid. Conditions that ensure the hyperbolicity of a set of equations of a bubbly flow are obtained. It is shown that the set of equations of motion has an infinite number of conservation laws. An infinite series of generalized symmetries admitted by the equations is constructed. Solutions that are invariant under the generalized symmetries of solution and describe the propagation of running and simple waves in a bubbly fluid are found. Lavrent’ev Institute of Hydrodynamics, Siberian Division, Russian Academy of Sciences, Novosibirsk 630090. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 40, No., 2. pp. 86–100, March–April, 1999.     

Quasi-Periodic Attractors in Celestial Mechanics

We prove that KAM tori smoothly bifurcate into quasi-periodic attractors in dissipative mechanical models, provided external parameters are tuned with the frequency of the motion. An application to the dissipative spin–orbit model of celestial mechanics (which actually motivated the analysis in this paper) is presented.