On the non-linear vibration of a thin ring with steady pressurization

Large amplitude, flexural oscillations of an inextensible, linearly elastic, pressurized ring are analyzed. Non-linear governing equations describing the planar motion of a thin rod curved in its undeformed state and subject to a distributed load applied normal to the neutral axis are developed using Hamilton's extended principle. The equations are specialized to study the behavior of a circular ring, and approximate solutions are obtained for a single mode response by a perturbation technique. Free, undamped oscillations and forced response of the ring near resonance are discussed. The influence of the magnitude of pressurization on the non-linear character of the motion is investigated.     

Oscillations of a viscous fluid in a thin layer between two coaxial disks rotating together

A study is made of the problem of the motion of an incompressible viscous fluid in the space between two coaxial disks rotating together with constant angular velocity under the assumption that the pressure changes in time in accordance with a harmonic law. The problem is solved using the equations of unsteady motion of an incompressible viscous fluid in a thin layer. It is shown that the velocity field in this case is a superposition on a steady field of damped oscillations with cyclic frequency equal to twice the angular velocity of the disks and forced oscillations with cyclic frequency equal to the cyclic frequency of the oscillations of the pressure field. It is shown that the amplitude of the forced oscillations of the velocity field depends strongly on the ratio of the cyclic frequency of the oscillations of the pressure field to the angular velocity of the disks. It is shown that there is a certain value of the ratio at which the amplitude of the forced oscillations has a maximal value (resonance). It is shown that even for very small amplitudes of the pressure oscillations the amplitude of the oscillations of the relative velocity at resonance may reach values comparable with the mean velocity of the main flow.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 166–169, January–February, 1984.     

Oscillations of a vessel containing a viscous fluid

The oscillations of a rigid body having a cavity partially filled with an ideal fluid have been studied in numerous reports, for example, [1–6]. Certain analogous problems in the case of a viscous fluid for particular shapes of the cavity were considered in [6, 7]. The general equations of motion of a rigid body having a cavity partially filled with a viscous liquid were derived in [8]. These equations were obtained for a cavity of arbitrary form under the following assumptions: 1) the body and the liquid perform small oscillations (linear approximation applicable); 2) the Reynolds number is large (viscosity is small). In the case of an ideal liquid the equations of [8] become the previously known equations of [2–6]. In the present paper, on the basis of the equations of [8], we study the free and the forced oscillations of a body with a cavity (vessel) which is partially filled with a viscous liquid. For simplicity we consider translational oscillations of a body with a liquid, since even in this case the characteristic mechanical properties of the system resulting from the viscosity of the liquid and the presence of a free surface manifest themselves.The solutions are obtained for a cavity of arbitrary shape. We then consider some specific cavity shapes.     

FORCED OSCILLATIONS OF BOUNDARY VALUE PROBLEMS OF HIGHER ORDER FUNCTIONAL PARTIAL DIFFERENTIAL EQUATIONS

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Nonlinear Flexural-Flexural-Torsional Dynamics of Inextensional Beams. II. Forced Motions

Abstract

The nonplanar, nonlinear, resonant forced oscillations of a fixed-free beam are analyzed by a perturbation technique with the objective of determining quantitative and qualitative information about the response. The analysis is based on the differential equations of motion developed in Part I of this paper which retain not only the nonlinear inertia but also nonlinear curvature effects. It is shown that the latter play a significant role in the nonlinear flexural response of the beam.     

Resistance and inertia of the flow of liquids in a tube or open canal

Summary When investigating the influence of an acceleration upon the velocity distribution and upon the resistance in a tube or canal, a distinction can be made between slowly varying motions where the resistance dominates, and quickly varying motions where the inertia dominates. When the motion varies quickly, practically all the liquid moves bodily, and the resistance only affects a small region near the walls. When the motion varies slowly, the velocity distribution differs from that of steady flow in that there is a phase lag of the central layers with respect to the peripheral layers.The following special types of motion were studied for laminar flow: forced oscillations in a round tube; forced oscillations in a crevice; the starting motion by a constant drop of potential in a round tube; free oscillations in a round tube. For free oscillations in a U-tube, the theory is checked by comparison with experiments (table I). General slow motion was studied in the following cases: laminar flow in tubes of elliptical (circular) and rectangular (square) cross-section, or in open canals of rectangular cross-section; fully developed turbulent flow in round tubes and wide open canals.The relation between drop of potential and total flow can be represented by the impedance of the tube or canal. For quick laminar motions the impedance is given by (52) and (53) and characterized by the high frequency inertanceH. For slow laminar motions the impedance is given by (49) and (50) and characterized by the resistanceR and the low frequency inertanceL. For slow turbulent motions the relation between drop of potential and total flow is given by (73). The low frequency inertance is always greater than the high frequency inertance. The difference represents the change in the resistance caused by the influence of the acceleration upon the distribution of the velocity. This difference betweenL andH amounts to 33% for laminar flow in a round tube, 38% in a square tube, and 20% in a crevice or a wide open canal. For turbulent flow it ranges from 1 to 8% in a rough round tube, from 1/2 to 3% in a smooth round tube, from 1/2 to 3 1/2% in a rough open canal or crevice, and from 1/4 to 1 1/2% in a smooth open canal or crevice (table III). For laminar flow the ratioL: H is equal to the ratio of the mean square and the square mean velocity in a steady flow.     

Modeling of turbulent flow in the neighborhood of forced and freely oscillating rectangular prisms

The problem of incompressible two-dimensional viscous flow past an infinite prism of rectangular cross section subjected to forced or free oscillation in the transverse direction is considered. The complete Reynolds equations, closed by means of a two-parameter--model of turbulence, are solved numerically in the noninertial coordinate system. The coefficients of the aerodynamic forces acting on the body are calculated as time functions; in the case of free oscillations the paths of the body and the instantaneous velocities of the oscillatory motion are calculated.Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 1, pp. 45–50, January–February, 1995.     

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.     

Evolution of a heavy rigid body rotation under the action of unsteady restoring and perturbation torques

The paper develops an approximate solution to the system of Euler’s equations with additional perturbation term for dynamically symmetric rotating rigid body. The perturbed motions of a rigid body, close to Lagrange’s case, under the action of restoring and perturbation torques that are slowly varying in time are investigated. We describe an averaging procedure for slow variables of a rigid body perturbed motion, similar to Lagrange top. Conditions for the possibility of averaging the equations of motion with respect to the nutation phase angle are presented. The averaging technique reduces the system order from 6 to 3 and does not contain fast oscillations. An example of motion of the body using linearly dissipative torques is worked out to demonstrate the use of general equations. The numerical integration of the averaged system of equations is conducted of the body motion. The graphical presentations of the solutions are represented and discussed. A new class of rotations of a dynamically symmetric rigid body about a fixed point with account for a nonstationary perturbation torque, as well as for a restoring torque that slowly varies with time, is studied. The main objective of this paper is to extend the previous results for problem of the dynamic motion of a symmetric rigid body subjected to perturbation and restoring torques. The proposed averaging method is implemented to receive the averaging system of equations of motion. The graphical representations of the solutions are presented and discussed. The attained results are a generalization of our former works where µ and M_i are independent of the slow time τ and M_i depend on the slow time only.

     

Phase space structure of spinning disks

Using a Hamiltonian formalism and a sequence of canonical transformations, we show that the ordinary differential equations associated with the forced oscillations of rotating circular disks admit the first integral of motion. This reduces the phase space dimension of the governing equations from five to three. The phase space flows of the reduced system are then visualized using Poincaré maps. Our results show that single mode oscillations of rotating disks are subject to chaotic behavior through the emergence of higher-order resonant islands that surround fundamental periodic cycles. We extend our new formalism to imperfect disks and construct adiabatic invariants near to and far from resonances. For low-speed imperfect disks, we find a new kind of bifurcations of the phase space flows as the system parameters vary. We study the effect of structural damping using Hamilton's principle for non-conservative systems and reveal the existence of asymptotically stable limit cycles for the damped system near the 1:1 resonance. We show that a low-speed disk is eventually flattened due to damping effect.     

CFD modelling of the cross-flow through normal triangular tube arrays with one tube undergoing forced vibrations or fluidelastic instability

A CFD methodology involving structure motion and dynamic re-meshing has been optimized and applied to simulate the unsteady flow through normal triangular cylinder arrays with one single tube undergoing either forced oscillations or self-excited oscillations due to damping-controlled fluidelastic instability. The procedure is based on 2D URANS computations with a commercial CFD code, complemented with user defined functions to incorporate the motion of the vibrating tube. The simulation procedure was applied to several configurations with experimental data available in the literature in order to contrast predictions at different calculation levels. This included static conditions (pressure distribution), forced vibrations (lift delay relative to tube motion) and self-excited vibrations (critical velocity for fluidelastic instability). Besides, the simulation methodology was used to analyze the propagation of perturbations along the cross-flow and, finally, to explore the effect on the critical velocity of the Reynolds number, the pitch-to-diameter ratio and the degrees of freedom of the vibrating cylinder.     

Attractors of a rotating viscoelastic beam

We investigate the non-linear oscillations of a rotating viscoelastic beam with variable pitch angle. The governing equations of motion are two coupled partial differential equations for the longitudinal and transversal displacements. Using a perturbation technique and Galerkin's projection, we reduce the equations of motion to a non-autonomous ordinary differential equation. Our regular perturbation technique is based on the expansion of longitudinal displacement and the amplitude of first transversal mode in terms of a small parameter. We numerically generate the Poincaré maps of the reduced equations and reveal that the system exhibits regular and chaotic attractors. The regular attractors are stable limit-cycles that are relevant to stable, short-period oscillations of the beam. A bifurcation analysis has also been performed when the pitch angle is constant.     

Nonsmall liquid oscillations in moving vessels

The system of approximate nonlinear equations describing liquid oscillations in axisymmetric vessels is constructed. The equations are obtained for the case in which two coordinates belonging to the family of generalized coordinates characterizing the liquid motion are not small. This family is selected so that from the resulting nonlinear equations we can obtain as a particular case the nonlinear equations of [1–3], which are valid for the class of cylindrical vessels, and the requirements are satisfied that the resulting nonlinear equations correspond to the widely adopted linearized equations of liquid oscillations [4–6], Nonlinear equations are obtained which describe liquid oscillations in arbitrary vessels of rotation with radial baffles.     

Discrete optimal control for Birkhoffian systems

In this paper, we investigate a discrete variational optimal control for mechanical systems that admit a Birkhoffian representation. Instead of discretizing the original equations of motion, our research is based on a direct discretization of the Pfaff–Birkhoff–d’Alembert principle. The resulting discrete forced Birkhoffian equations then serve as constraints for the minimization of the objective functional. In this way, the optimal control problem is transformed into a finite-dimensional optimization problem, which can be solved by standard methods. This approach yields discrete dynamics, which is more faithful to the continuous equations of motion and consequently yields more accurate solutions to the optimal control problem which is to be approximated. We illustrate the method numerically by optimizing the control for the damped oscillator.     

Finite amplitude oscillations of a hyperelastic spherical cavity

We present numerical results for the finite oscillations of a hyperelastic spherical cavity by employing the governing equations for finite amplitude oscillations of hyperelastic spherical shells and simplifying it for a spherical cavity in an infinite medium and then applying a fourth-order Runge-Kutta numerical technique to the resulting non-linear first-order differential equation.The results are plotted for Mooney-Rivlin type materials for free and forced oscillations under Heaviside type step loading. The results for Neo-Hookean materials are also discussed. Dependence of the amplitudes and frequencies of oscillations on different parameters of the problem is also discussed in length.     

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.     

Impact of Atmospheric Front Parameters on Free and Forced Oscillations of Level and Current in the Sea of Azov

Level and current oscillations in the basin of the Sea of Azov have been studied by hydrodynamic modeling using the Princeton ocean model (POM). The hypothesis on the role of the resonance mechanism in the occurrence of extremely large amplitudes of storm surge and seiche oscillations depending on the velocity and time of motion of atmospheric fronts of the Sea of Azov has been tested. It is found that at the same wind, pressure perturbations moving over the Sea of Azov induce forced oscillations, and after the perturbations cease, free oscillations with amplitudes that are 14% higher than those obtained at constant atmospheric pressure. It is shown that the motion of the atmospheric front (whose velocity and time are selected under the assumption that waves with maximum amplitudes are generated) plays an important but not decisive role in the formation of the structure of currents and level oscillations in the Sea of Azov.     

Periodic and Chaotic Motions of a Rotor-Active Magnetic Bearing with Quadratic and Cubic Terms and Time-Varying Stiffness

In this paper, we use the asymptotic perturbation method to investigate nonlinear oscillations and chaotic dynamics in a rotor-active magnetic bearings (AMB) system with 8-pole legs and the time-varying stiffness. The stiffness in the AMB is considered as the time varying in a periodic form. Because of considering the weight of the rotor, the formulation on the electromagnetic force resultants includes the quadratic and cubic nonlinearities. The resulting dimensionless equations of motion for the rotor-AMB system with the time-varying stiffness in the horizontal and vertical directions are a two-degree-of-freedom nonlinear system with quadratic and cubic nonlinearities and parametric excitation. The asymptotic perturbation method is used to obtain the averaged equations in the case of primary parametric resonance and 1/2 subharmonic resonance. It is found that there exist period-3, period-4, period-6, period-7, period-8, quasiperiodic and chaotic modulated amplitude oscillations in the rotor-AMB system with the time-varying stiffness. It is seen from the numerical results that there are the phenomena of the multiple solutions and the soft-spring type and the hardening-spring type in nonlinear frequency-response curves for the rotor-AMB system. The parametric excitation, or the time-varying stiffness produced by the PD controller is considered to be a controlling force which can control the chaotic response in the rotor-AMB system to a period n motion.     

Influence of heat transfer on the dynamic response of a spherical gas/vapour bubble

The standard approach to analyse the bubble motion is the well known Rayleigh–Plesset equation. When applying the toolbox of nonlinear dynamical systems to this problem several aspects of physical modelling are usually sacrificed. Particularly in vapour bubbles the heat transfer in the liquid domain has a significant effect on the bubble motion; therefore the nonlinear energy equation coupled with the Rayleigh–Plesset equation must be solved. The main aim of this paper is to find an efficient numerical method to transform the energy equation into an ODE system, which, after coupling with the Rayleigh–Plesset equation can be analysed with the help of bifurcation theory. Due to the strong nonlinearity and violent bubble motions the computational effort can be high, thus it is essential to reduce the size of the problem as much as possible. In the first part of the paper finite difference, Galerkin and spectral collocation methods are examined and compared in terms of efficiency. In the second part free and forced oscillations are analysed with an emphasis on the influence of heat transfer. In the case of forced oscillations the unstable branches of the amplification diagrams are also computed.