Transversal vibrations of double-plate systems

This paper presents an analytical and numerical analysis of free and forced transversal vibrations of an elastically connected double-plate system. Analytical solutions of a system of coupled partial differential equations, which describe corresponding dynamical free and forced processes, are obtained using Bernoulli’s particular integral and Lagrange’s method of variation constants. It is shown that one-mode vibrations correspond to two-frequency regime for free vibrations induced by initial conditions and to three-frequency regime for forced vibrations induced by one-frequency external excitation and corresponding initial conditions. The analytical solutions show that the elastic connection between plates leads to the appearance of two-frequency regime of time function, which corresponds to one eigenamplitude function of one mode, and also that the time functions of different vibration modes are uncoupled, for each shape of vibrations. It has been proven that for both elastically connected plates, for every pair of m and n, two possibilities for appearance of the resonance dynamical states, as well as for appearance of the dynamical absorption, are present. Using the MathCad program, the corresponding visualizations of the characteristic forms of the plate middle surfaces through time are presented.The English text was polished by Keren Wang.     

Global dynamics of an autoparametric spring–mass–pendulum system

In this paper, a study of the global dynamics of an autoparametric four degree-of-freedom (DOF) spring–mass–pendulum system with a rigid body mode is presented. Following a modal decoupling procedure, typical approximate periodic solutions are obtained for the autoparametrically coupled modes in 1:2 internal resonance. A novel technique based on forward-time solutions for finite-time Lyapunov exponent is used to establish global convergence and domains of attraction of different solutions. The results are compared to numerically constructed domains of attraction in the plane of initial position and initial velocity for the pendulum. Simulations are also provided for a few interesting cases of interest near critical values of parameters. Results also shed some light on the role played by other modes present in a multi-DOF system in shaping the overall system response.     

Forced oscillations of a gas bubble in a spherical volume of a compressible liquid

A spherically symmetric problem of oscillations of a single gas bubble at the center of a spherical flask filled with a compressible liquid under the action of pressure oscillations on the flask wall is considered. A system of differential-difference equations is obtained that extends the Rayleigh-Plesset equation to the case of a compressible liquid and takes into account the pressure-wave reflection from the bubble and the flask wall. A linear analysis of solutions of this system of equations is performed for the case of harmonic oscillations of the bubble. Nonlinear resonance oscillations and nearly resonance nonharmonic oscillations of the bubble caused by harmonic pressure oscillations on the flask wall are analyzed. Ufa Scientific Center, Russian Academy of Sciences, Ufa 450000. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 40, No. 2, pp. 111–118, March–April, 1999.     

On the large amplitude oscillations of a thin elastic beam

This paper is concerned with the finite amplitude, free, planar oscillations of a thin elastic beam. By assuming the motion to be inextensional but at the same time recognizing the existence of a resultant normal force acting on each cross-section of the beam a system of governing equations is derived which is manageable but still meaningful. For the case of the simply-supported beam a finite difference, Galerkin, and (regular) perturbation solutions are explicitly obtained. The results are compared and discussed. In the course of obtaining the various solutions it is found that an additional simplification in the form of the governing equations is possible. This simplification turns out to be quite important from a general point of view of obtaining approximate analytical solutions.     

Identification of Friction Coefficients in the Hinge of a Controlled Physical Pendulum Using the Amplitudes of Steady-State Oscillations

A dynamic model of a controlled physical pendulum is considered. The Pontryagin method of searching for the periodic solutions to near-Hamiltonian systems is used to formulate a programmed law of pendulum oscillations such that the test modes of oscillations become steady and orbitally stable. An approach to identify the friction parameters in the hinge of the pendulum is proposed for the case of the active motor mode. This approach is based on the data available about the integral characteristics of motion. The motion of the system under consideration is numerically simulated.     

Class of differentially invariant solutions of the submodel of axisymmetric flows of an ideal gas

A class of differentially invariant solutions of a problem with the pressure independent of the radial coordinate is considered for a submodel of steady axisymmetric flows of a polytropic gas. The overdetermined system turns out to be compatible and is integrated. All solutions defining transonic and supersonic flows with a limiting surface are found. These solutions are compared with invariant solutions obtained previously.     

Forced oscillations of a viscous incompressible plasma with allowance for the electron inertia in a circular tube

The forced oscillations of a plasma column resulting from harmonic oscillations of the total current at a frequency ω are investigated analytically and numerically. The column plasma is assumed to be quasi-neutral two-component viscous and electroconducting, the electron inertia and the displacement current being completely taken into account. The electrons and ions are considered to be incompressible interpenetrating fluids. It is shown that the oscillations of the total current lead to the appearance of colliding plasma flows in the column, and, as the oscillation frequency ω increases, a skin layer with respect to main plasma parameters (current density, electromagnetic field, and hydrodynamic electron and ion velocities) develops on the boundary of the column. A comparison with the MHD theory is carried out and the role of the electron inertia and the displacement current in the generation of forced oscillations is investigated. The results obtained are used to analyze the plasma compression in apparatuses such as z-pinch and plasma focus.     

Young‐Measure Solutions for a Viscoelastically Damped Wave Equation with Nonmonotone Stress‐Strain Relation

We study the viscoelastically damped wave equation with a nonmonotone stress‐strain relation σ. This system describes the dynamics of phase transitions, which is closely related to the creation of microstructures. In order to analyze the dynamic behavior of microstructures we consider highly oscillatory initial states. Two questions are addressed in this work: How do oscillations propagate in space and time? What can be said about the long‐time behavior? An appropriate tool to deal with oscillations are Young measures. They describe the local distribution or one‐point statistics of a sequence of fast fluctuating functions. We demonstrate that highly oscillatory initial states generate in a unique fashion an evolution in the space of Young measures and we derive the determining equations. Further on we prove a generalized dissipation identity for Young‐measure solutions. As a consequence, it is shown that every low‐energy solution converges to a Young‐measure equilibrium as t→∞. This is a generalization of G. Friesecke's & J. B. McLeod's [FM96] convergence result for classical solutions to the case of Young‐measure solutions. (Accepted November 12, 1997)     

Hopf Bifurcations and Hysteresis in Flow-induced Vibrations of Cylinders

Flow-induced oscillations of rigid cylinders exposed to fully developed flow can be described by a fourth order autonomous system of ordinary differential equations. Its rest solution is the only equilibrium point which is unstable in the entire regime of parameters. It turns out that Hopf bifurcations from the trivial solution occur in regions of comparatively low damping. We found that a wind speed parameter, Ω, controls the bifurcations while the other parameters have been arranged into discrete sets. In the case of two bifurcating solutions with branches of amplitudes tending towards each other, hysteresis occurred. The bifurcating solutions are unstable close to their respective bifurcation points. The branch tending to the left-hand side changes its stability and exhibits high-level amplitudes of synchronized oscillations. This type of solution can also be analysed by means of asymptotic methods. Near the location of the bifurcation, the predictions of bifurcation theory, the multiple scales approach, and numerics are in quite good agreement. As opposed to this, the branch tending to the right-hand side represents synchronized oscillations of somewhat smaller period but much smaller cylinder amplitudes, and these vibrations remain unstable in the entire regime of parameters. This means that keeping the cylinder fixed, starting the wind tunnel, and releasing the cylinder at low wind speeds would lead to a jump of its displacement amplitude from the low, unstable to the comparatively high-stable values. It is shown that the theoretical predictions are in fairly good agreement with the experimental trends of flow-induced synchronized cylinder oscillations.     

Boundary Integral Equations Method for the Analysis of Acoustic Scattering from Line-2 Elastic Targets

The prediction of the acoustic scattering from elastic structures is a recurrent problem of practical importance as, for example, in underwater detection and target identification. We aim at setting out the diffraction problem of a transient acoustic wave by an axisymmetric shell composed of a cylinder bounded by hemispherical endcaps, called Line-2. Its time-dependent response is expanded in terms of the resonance modes of the fluid-loaded structure. The latter are well suited when the structure is submerged in a heavy fluid: it is an alternative to modal methods whose expansions as series of natural modes of the in vacuo shell are much better for describing the interaction between a structure and a light fluid. The resonance frequencies are defined as solutions of the nonlinear eigenvalue problem described by the set of homogeneous equations governing the structure displacement coupled to the acoustic radiated pressure. The resonance modes of the coupled system are the corresponding eigenvectors. Both hemisphere and cylinder equations are modeled by the approximation of Donnel and Mushtari which governs thin shells oscillations. The modeling of the sound pressure by a hybrid potential integral representation leads to a system of integro-differential equations defined on the surface of the structure only (boundary integral equations). The unknowns, the hybrid potential density as well as the shell displacement vector, are developed into Fourier series with respect to the natural cylindrical coordinate. Each angular component of the unknown functions is then expanded as series of Legendre polynomials, the coefficients of which are calculated thanks to a Galerkin method derived from the energetic form of the equations. The whole method can also be applied to predict the response of the coupled structure to a harmonic or a random excitation. This revised version was published online in July 2006 with corrections to the Cover Date.     

Nonlinear normal modes in homogeneous system with time delays

Periodic synchronous regimes of motion are investigated in symmetric homogeneous system of coupled essentially nonlinear oscillators with time delays. Such regimes are similar to nonlinear normal modes (NNMs), known for corresponding conservative system without delays, and can be found analytically. Unlikely the conservative counterpart, the system possesses “oval” modes with constant phase shift between the oscillators, in addition to symmetric/antisymmetric and localized regimes of motion. Numeric simulation demonstrates that the “oval” modes may be attractors of the phase flow. These attractors are particular case of phase-locked solutions, rather ubiquitous in the system under investigation.     

On the Energy Distribution in Fermi–Pasta–Ulam Lattices

This paper presents a rigorous study, for Fermi–Pasta–Ulam (FPU) chains with large particle numbers, of the formation of a packet of modes with geometrically decaying harmonic energies from an initially excited single low-frequency mode and the metastability of this packet over longer time scales. The analysis uses modulated Fourier expansions in time of solutions to the FPU system, and exploits the existence of almost-invariant energies in the modulation system. The results and techniques apply to the FPU α- and β-models as well as to higher-order nonlinearities. They are valid in the regime of scaling between particle number and total energy in which the FPU system can be viewed as a perturbation to a linear system, considered over time scales that go far beyond standard perturbation theory. Weak non-resonance estimates for the almost-resonant frequencies determine the time scales that can be covered by this analysis.     

Bifurcations in a parametrically excited non-linear oscillator

A non-linear parametrically excited oscillator, that includes van der Pol as well as Duffing type non-linearities, is studied for its small non-linear motions using the method of averaging. The averaged equations, which form a dynamical system on the plane and depend on the linear damping and the detuning, are analyzed for their constant and periodic solutions. Bendixon's criterion is used to deduce the existence and the non-existence of limit cycle solutions for various values of the parameters. Then, using local bifurcation theory for “saddle-node”, pitchfork and “Hopf” bifurcations and some results from one and two parameter unfoldings of degenerate singularities, a partial bifurcation set is constructed. Since constant and periodic solutions of the averaged system correspond, respectively, to the periodic solutions and almost periodic or amplitude modulated motions of the original oscillator, the bifurcation set indicates some ways in which periodic solutions can become “entrained” or can break the entrainment for almost periodic oscillations.     

Basic Displacements in the Problem of Core Perturbations of a Thin Isochronous Vortex Ring

Periodic perturbations in the core of a thin isochronous vortex ring in an inviscid incompressible fluid are investigated in the linear approximation. The aim of the study is to construct the system of basic displacements, namely, the complete system of solutions of the Helmholtz equation for vorticity perturbations inside the core of a vortex ring with a given frequency in the form of expansion in the ring thinness parameter μ. The structure of basic displacements depends substantially on the fact to what extent the frequency of the forcing action is close to the resonance frequencies of the system. If the difference between these frequencies is small, then, in addition to the ring thinness μ, the second small parameters arises in the problem. This leads to significant complication of the procedure of obtaining the solution and appearance of considerable corrections in the subsequent approximations of the expansion procedure. The case of isochronous vortex ring in which the periods of revolution of liquid particles are identical is considered. Obtaining the threedimensional oscillations in such flows turns out to be the simplest since there are no perturbations of the continuous spectrum for the isochronous ring. The system of basic displacements is the necessary element in deriving the dispersion relation for the eigen-oscillations of the vortex ring. The solutions obtained can also serve as an instrument to analyze the reaction of flows with curvilinear vortex lines or flows localized in toroidal regions to the external excitation.     

Exact solutions for large elastoplastic deformations of a thick-walled tube under internal pressure

A rate-independent plasticity theory based on the concept of dual variables and dual derivatives is utilized to describe finite elastic-plastic deformations including kinematic and isotropic hardening effects. Application of this theory to the problem of the thick-walled tube under internal pressure leads to a system of partial differential equations of hyperbolic type. The existence and uniqueness of the solution of the boundary value problem is guaranteed, as well as the convergence of its numerical approximation. The exact solution of this problem is calculated by means of an extrapolation technique. This integration method turns out to be applicable for rather general hardening models of rate-independent plasticity. On the basis of the computed solutions the influence of the hardening parameters is investigated. As finite deformations are of special interest, this investigation is carried out not only for the partially yielded tube but also for the completely plastified tube. Furthermore, the onset of secondary plastic flow during unloading as well as residual stress distributions are studied.     

Chaotic flexural oscillations of a spinning nanoresonator

The paper investigates the chaotic flexural oscillations of the spinning nanoresonator. The influence of cubic nonlinearity arising from the van der Waals interactions between two neighboring layers of carbon nanotubes on the structural oscillations of the system is considered. The integral–differential equations describing the flexural displacements of the nanoresonator are discretized into two coupled Duffing-type equations using the Galerkin–Ritz procedures. The linear stiffness can be either positive or negative, depending on the amplitudes of the linear trap rigidity arising from both the van der Waals interactions and the axial tensile loads. The chaotic flexural oscillations of the appropriately excited spinning nanoresonator are predicted theoretically. Using the Nayfeh–Mook multiscale perturbation algorithms, the coupled Duffing-type equations with linear positive stiffness may be transformed into autonomous equations of slowly modulated amplitudes whose equilibrium points and chaotic dynamics are investigated numerically. The potential chaotic oscillations of the elastic nanoresonator can be determined by the Melnikov–Holmes–Marsden (MHM) integral associated with the homoclinic/heteroclinic solutions of the disturbed Hamiltonian systems with linear negative stiffness. The findings are validated through the Poincare sections and Lyapunov exponents.     

Buoyancy-Opposed Darcy’s Flow in a Vertical Circular Duct with Uniform Wall Heat Flux: A Stability Analysis

An analytical and numerical study is presented to show that buoyancy-opposed mixed convection in a vertical porous duct with circular cross-section is unstable. The duct wall is assumed to be impermeable and subject to a uniform heat flux. A stationary and parallel Darcy’s flow with a non-uniform radial velocity profile is taken as a basic state. Stability to small-amplitude perturbations is investigated by adopting the method of normal modes. It is proved that buoyancy-opposed mixed convection is linearly unstable, for every value of the Darcy–Rayleigh number, associated with the wall heat flux, and for every mass flow rate parametrised by the Péclet number. Axially invariant perturbation modes and general three-dimensional modes are investigated. The stability analysis of the former modes is carried out analytically, while general three-dimensional modes are studied numerically. An asymptotic analytical solution is found, suitable for three-dimensional modes with sufficiently small wave number and/or Péclet number. The general conclusion is that the onset of instability selects the axially invariant modes. Among them, the radially invariant and azimuthally invariant mode turns out to be the most unstable for all possible buoyancy-opposed flows.     

Higher-order difference approximations of the Navier-Stokes equations

A discretization scheme is presented which, unlike the standard higher-order finite difference and spline methods, does not give rise to unphysical solution modes and boundary conditions. Practical application of this scheme is achieved via the DCMG algorithm recently developed by the same author, which turns out to be able to find a converged solution of the ψ-ζ Navier-Stokes equations in about the same time for highorder as for low-order discretization schemes. Examples are presented for the driven cavity problem to explore the accuracy of the new method. Finally, a local analysis is performed of the corner singularities which exist in driven cavity flow, and their effect on the overall accuracy of the solutions obtained by polynomial interpolation methods is investigated.     

Numerical solution of stiff systems from nonlinear dynamics using single-term Haar wavelet series

The paper presents single-term Haar wavelet series (STHWS) approach to the solution of nonlinear stiff differential equations arising in nonlinear dynamics. The properties of STHWS are given. The method of implementation is discussed. Numerical solutions of some model equations are investigated for their stiffness and stability and solutions are obtained to demonstrate the suitability and applicability of the method. The results in the form of block-pulse and discrete solutions are given for typical nonlinear stiff systems. As compared with the TR BDF2 method of Shampine and Gill’s method, the STHWS turns out to be more effective in its ability to solve systems ranging from mildly to highly stiff equations and is free from stability constraints.