Finite-amplitude Oscillations Research Articles

Nonlinear anelasticity of magnesium

An approximate solution of the equation of motion of a nonlinear anelastic reed at or near resonance is presented. The steady state solution reproduces the well-known nonlinear resonances. The solution also predicts the existence of automodulations, i.e., self-excited modulations of the amplitude and phase at constant power of excitation of the reed. Numerical examples of such automodulations are presented for an antisymmetric deformation potential. Experimental studies of finite amplitude oscillations of a magnesium reed vibrating at 72 and 431 Hz at room temperature confirm the existence of automodulations. The experimental results can be semiquantitatively described in terms of the solution given. The assumption that finite deformation by twinning represents the essential nonlinearity leads to a self-consistent interpretation. The relaxation time of twinning is obtained from the analysis of the automodulation and is 22 msec in the sample investigated. It is proposed that point defects control the relaxation process.

The complex Lorenz equations

We have undertaken a study of the complex Lorenz equations x ̇ = −σx + σy . y ̇ = (r − z)x − ay . z ̇ = −bz + 1 2 (x ∗y + xy ∗) . where x and y are complex and z is real. The complex parameters r and a are defined by r = r 1 + ir 2; a = 1 − ie and σ and b are real. Behaviour remarkably different from the real Lorenz model occurs. Only the origin is a fixed point except for the special case e + r 2 = 0. We have been able to determine analytically two critical values of r 1, namely r1 c and r1 c . The origin is a stable fixed point for 0 < r 1 < r 1 c , but for r 1 > r 1 c , a Hopf bifurcation to a limit cycle occurs. We have an exact analytic solution for this limit cycle which is always stable if σ < b + 1. If σ > + 1 then this limit is only stable in the region r 1 c < r 1 < r lc . When r 1 > r lc , a transition to a finite amplitude oscillation about the limit cycle occurs. The nature of this bifurcation is studied in detail by using a multiple time scale analysis to derive the Stuart-Landau amplitude equation from the original equations in a frame rotating with the limit cycle frequency. This latter bifurcation is either a sub- or super-critical Hopf-like bifurcation to a doubly periodic motion, the direction of bifurcation depending on the parameter values. The nature of the bifurcation is complicated by the existence of a zero eigenvalue.

A theoretical study of limit cycle oscillations of plenum air cushions

Air cushion vehicles (ACV) are prone to the occurrence of dynamic instabilities which frequently appear as stable finite amplitude oscillations. The aim of this work is to ascertain if the non-linearities characteristics of ACV dynamics generate limit cycle oscillations for cushion systems operating at conditions for which a linear theory predicts instability. The types of non-linearity that can occur are discussed, and an analysis is presented for a single cell flexible skirted plenum chamber constrained to move in pure heave only. Two cushion feed cases are considered: a plenum box supply and a duct. The results obtained by a Galerkin/describing function analysis are compared with those generated by a full numerical simulation. For the plenum box supply system, it is shown that the limit cycles can be suppressed by using a piston to introduce high frequency small amplitude volume oscillations into the plenum chamber.

Free Oscillations in a Climate Model with Ice-Sheet Dynamics

A study of stable periodic solutions to a simple nonlinear model of the ocean-atmosphere-ice system is presented. The model has two dependent variables: ocean-atmosphere temperature and latitudinal extent of the ice cover. No explicit dependence on latitude is considered in the model. Hence all variables depend only on time and the model consists of a coupled set of nonlinear ordinary differential equations. The globally averaged ocean-atmosphere temperature in the model is governed by the radiation balance. The reflectivity to incoming solar radiation, i.e., the planetary albedo, includes separate contributions from sea ice and from continental ice sheets. The major physical mechanisms active in the model are (1) albedo-temperature feedback, (2) continental ice-sheet dynamics and (3) precipitation-rate variations. The model has three-equilibrium solutions, two of which are linearly unstable, while one is linearly stable. For some choices of parameters, the stability picture changes and sustained, finite-amplitude oscillations obtain around the previously stable equilibrium solution. The physical interpretation of these oscillations points to the possibility of internal mechanisms playing a role in glaciation cycles.

Slowly Varying Jump and Transition Phenomena Associated with Algebraic Bifurcation Problems

Parameter-dependent equilibrium solutions are analyzed as the parameter slowly varies through critical values corresponding to a bifurcation or to a jump phenomena. At these critical times, interior nonlinear transition layers are necessary. Depending on the particular situation, local scaling analysis yields the first and a second Painleve transcendent among other generic equations. In specific cases the resulting boundary layer solutions either increase algebraically or explode (via a singularity). The algebraic growth corresponds to a smooth transition to a bifurcated equilibrium. When a jump phenomena is expected, an explosion can occur. In this case, the solution of first-order differential equations approaches the equilibrium, describing the slow evolution through such a jump. However, second-order differential equations have finite amplitude oscillations around the new equilibrium.

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.

MHD Oscillatory Flow past a Semi-Infinite Plate

Theme L IGHTHILL studied effects of small amplitude oscillations and Lin studied effects of finite amplitude oscillations on the boundary-layer flow. Lighthill's results were experimentally confirmed by Hill and Stenning. They assumed the freestream to consist of a constant mean velocity over which is superimposed a purely time-dependent oscillatory flow. Kestin et al. and Patel studied the effects of oscillatory progressive wave type of freestream on the boundary-layer flow. In the energy equation, the viscous dissipation effects were neglected by Kestin et al. So taking into account viscous dissipative heat, Kestin's problem was solved again by Takhar and Soundalgekar. We now consider the effects of a transverse magnetic field on the flow past a semi-infinite plate, taking into account the progressive wave type of disturbance in the freestream. By assuming the disturbance in the form (/oo(0 =U0(1 + te), a similar problem was solved by Evans. Evans solved it by assuming a series in powers of two parameters s(wx/w0), the Strouhal number, and \(oByx/pU0 = Ha//Re), the magnetic field parameter. We have solved the problem by assuming a series in powers of Vs only and x > 1 • The induced magnetic field is assumed to be negligible (Cowling) which is true when magnetic Reynolds numer is small.

Free and forced finite amplitude oscillation of a spherical shell of radially isotropic elastic material

A spherical elastic shell with radial transverse isotropy is considered. The periods of finite amplitude radial oscillation of the shell have been obtained in two cases, namely (i) when both the surfaces of the shell are free from traction, and (ii) when the shell boundaries are uniformly loaded in such a way that the pressure difference between inner and outer surfaces is constant with respect to time. It is observed that for free oscillation to take place it is necessary to impose a new restriction on the strain energy function in addition to those already obtained for finite amplitude oscillation of an isotropic elastic shell. In the forced oscillation case however the required conditions are of the same form as in the corresponding case for isotropic media.

Parametric oscillations of liquid in communicating vessels

It is well known that a periodic change in the equilibrium or flow parameters of an incompressible liquid exerts a material influence on the hydrodynamic stability. As an example we may quote the parametric excitation of surface waves (gravitational-capillary [1], electrohydrodynamic [2], magnetohydrodynamic [3]) and the oscillations of liquid in communicating vessels [4, 5]. The chief object of the foregoing experimental investigations was that of determining the boundaries of the regions of unstable equilibrium with respect to small perturbations. In the present investigation we made an experimental study of the parametric resonance and finite-amplitude parametric oscillations arising in a liquid-filled U-tube subject to alternating vertical overloadings. We shall describe two forms of oscillations in the liquid, and we shall determine the corresponding ranges of unstable equilibrium with respect to small random perturbations (self-excitation) and also to finite-amplitude perturbations. We shall study nonlinear modes of excitation and mutual transitions between the two forms of oscillations. We shall find the ranges of existence of steady-state oscillations.

Non-linear theory of cyclotron oscillations near a stable state of the plasma

A non-linear equation is derived for an individual mode of electrostatic oscillations of small but finite amplitude in a plasma in a homogeneous magnetic field. The Larmor radius is assumed arbitrary. The method is based on a solution of the transport equation by perturbation theory to terms of third order inclusively. As examples, the non-linear evolution of the anisotropic, loss-cone and Dory-Guest-Harris instabilities of a non-equilibrium plasma is considered near the stability boundary of the ion cyclotron oscillations. It is shown that non-linearity (self-influence) stabilizes these instabilities. The typical maximal values of the amplitude of stable non-linear oscillations are found.

Finite-amplitude stability and draw resonance in isothermal melt spinning

The spinning of fibers from polymer melts is sometimes limited at high takeup rates by the onset of a flow instability known as “draw resonance,” which is characterized by large amplitude fluctuations in the takeup area and the force. In this work, the onset and growth of the spinning instability is analyzed for an isothermal Newtonian liquid by examining the nonlinear dynamics of the most unstable spatial mode. At draw ratios below the critical value D R = 20·21 computed from linear theory the system is stable to finite-amplitude perturbations. At higher draw ratios all disturbances approach a stable finite-amplitude oscillation which agrees in period and magnitude with experimental observations of draw resonance.

Instabilities of finite-amplitude oscillations of a plasma in a magnetic field in the presence of trapped particles

A study is made of the instabilities of a plane monochromatic electrostatic wave of finite amplitude in a constant magnetic field due to particles trapped by the electric field of this wave under conditions of Cherenkov resonance or cyclotron resonances. If the particles are trapped near a centre on the phase plane with the transverse or longitudinal velocity and the helical phase as coordinates in the case n not=0 or on the phase plane with the longitudinal velocity and the helical phase as coordinates in the case n=0, and if the particles have a small spread with respect to the velocities and helical phase, there arise coherent instabilities excited by all the trapped particles. If the wave vector k of the unstable oscillations is not near k0, there arises an instability of the two-stream type. If k approximately=k0, there arises a parametric satellite instability. In both cases, the maximal growth rate is gamma max approximately=(n1/n0)1/3 Omega 0, where n1 is the number of trapped particles in unit volume and n0 is the density of the plasma.

Mass transport sustained by oscillations of finite amplitude in a stratified fluid

A general theory for the mass transport sustained by oscillations of finite amplitude in a stratified fluid in the field of gravity is presented. The mass transport is given by the Lagrangian mean velocity, calculated to the second order in the oscillatory perturbation. The vertical component of this mean velocity, the divergence, and the vertical component of the vorticity are computed. An arbitrary equation of state is assumed, and the effects of viscosity and thermal conductivity are taken into account. The theory is applied to standing waves. The results seem to provide a qualitative explanation of experiments by Schaaffs and Haun (1968 Acustica 20 , 348–359) and others.

Nonlinear penetrative convection

Convection in water above ice penetrates into the stably stratified region above the density maximum at 4 °C. Two-dimensional penetrative convection in a Boussinesq fluid confined between free boundaries has been studied in a series of numerical experiments. These included cases with a constant temperature at both boundaries as well as cases with a fixed average flux at the lower boundary. Steady convection occurs at Rayleigh numbers below the critical value predicted by linear theory. At high Rayleigh numbers, resonant coupling between convection and gravitational modes in the stable layer excites finite amplitude oscillations. The problem can be described by a simplified model which allows for distortion of the mean temperature profile and balances the convected and conducted flux. This model explains the finite amplitude instability and predicts the Nusselt number as a function of Rayleigh number. These predictions are in excellent agreement with the computed results.

Two-dimensional Rayleigh-Benard convection

Two-dimensional convection in a Boussinesq fluid confined between free boundaries is studied in a series of numerical experiments. Earlier calculations by Fromm and Veronis were limited to a maximum Rayleigh number R 50 times the critical value R, for linear instability. This range is extended to 1000Rc. Convection in water, with a Prandtl number p = 6·8, is systematically investigated, together with other models for Prandtl numbers between 0·01 and infinity. Two different modes of nonlinear behaviour are distinguished. For Prandtl numbers greater than unity there is a viscous regime in which the Nusselt number . At higher Rayleigh numbers advection of vorticity becomes important and N ∞ R0·365. When p = 6·8 the heat flux is a maximum for square cells; steady convection is impossible for wider cells and finite amplitude oscillations appear instead, with periodic fluctuations of temperature and velocity in the layer. For p < 1 it is also found that N ∞ R0·365, with a constant of proportionality equal to 1·90 when p [Lt ] 1 and decreasing slowly as p is increased. The physical behaviour in these regimes is analysed and related to astrophysical convection.

Dynamically forced neutral waves in a rotating stratified fluid

A mathematical model for finite but nonamplifying waves in a rotating stratified air layer that respond to dynamic forcing through two parallel horizontal boundaries is formulated and solved as a boundary value problem. The model derives from a first-order perturbation analysis of the equations of motion, continuity, and energy with the extraction of only neutrally stable solutions whose existence is owed to the forced mass influxes through the boundaries. The classification of the equation defining the perturbed state is shown to depend on the frequency of forcing relative to inertial and adiabatic (Brunt-Vaisala) oscillation frequencies and the degree of hydrodynamic stability of the mean state. Analytic solutions for a certain range of subinertial frequencies are derived. The amplitudes of the responding waves are found to be larger the closer the frequency is to the inertial resonance and the closer the mean state is to hydrodynamic neutrality. The detailed spatial geometry of the responding waves is, however, a complicated function of the mean state and the time and space scales of the forcing. For near-neutral conditions and near-resonant frequencies, solutions are suggestive of the finite-amplitude diurnal oscillations in the wind at upper levels over the great plains region, which have been noted to occur in association with the low-level jet.

Non-linear surface wave convolution theory and materials : Lim, T.C., Kraut, B.R. Vol SU-19 No 3 (July 1972) p 399

A general theory for the mass transport sustained by oscillations of finite amplitude in a stratified fluid in the field of gravity is presented. The mass transport is given by the Lagrangian mean velocity, calculated to the second order in the oscillatory perturbation. The vertical component of this mean velocity, the divergence, and the vertical component of the vorticity are computed. An arbitrary equation of state is assumed, and the effects of viscosity and thermal conductivity are taken into account. The theory is applied to standing waves. The results seem to provide a qualitative explanation of experiments by Schaaffs and Haun (1968 Acustica 20, 348–359) and others.

A finite-amplitude oscillation theory of the classical problem of a particle in interaction with a scalar field. Application to the polaron model

An exact, particular but nontrivial, solution of the classical polaron problem is obtained. In this solution, the electron rotates on an orbit of finite radius around the polaron centre which is itself in translation, so that the resulting motion is helicoidal. By comparison with this solution, it is shown that the perturbation and small-oscillation methods involve an increase in the order of the ultraviolet divergences of the self-energy and the frequency of rotation. Moreover, the presence of a pole at α = 0 in the energy of the exact solution, function of the coupling constant a, explains why the perturbation method is no longer valid in the classical situation, if the radius of the orbit is not identically zero.

A finite-amplitude oscillation theory of the classical problem of a particle in interaction with a scalar field. Application to the polaron model

A finite-amplitude oscillation theory of the classical problem of a particle in interaction with a scalar field. Application to the polaron model

Some Elementary Applications of the Virial Theorem to Stellar Dynamics

The dynamical evolution of spherical and spheroidal systems of mass points is examined with the aid of the scalar and the tensor forms of the virial theorem. Spherical systems with positive total energy tend to disperse to infinity while those with negative total energy execute periodic oscillations of finite amplitude. Spheroidal systems with positive total energy exhibit expansion like the spherical systems; but they also become less oblate (and sometimes actually become prolate) if they are initially oblate, and less prolate if they are initially prolate. Spheroidal systems with negative total energy collapse to smaller volumes while enhancing their initial oblateness or prolateness.