Small Harmonic Oscillations Research Articles

Energy and frequency ripple in devices with inertial excitation of oscillations.

We consider vibration devices that consist of softly vibration-isolated rigid bodies subjected to vibrations transmitted by means of inertial vibration exciters (unbalanced rotors) driven into rotation by electric motors. Typically, when designing such devices, it is assumed that the rotors rotate uniformly with a certain circular frequency and the body performs small harmonic oscillations with the same frequency. The present work, using a second-order approximation of their nonlinear coupled differential equations, shows that the rotor and the oscillating body keep exchanging energy. At the same time, the angular velocity of the rotor oscillates with the working frequency as well as with its multiple frequencies during each revolution. As a result, the acceleration of the oscillating body also acquires harmonics with multiple frequencies. This may cause both unwanted and beneficial resonance phenomena. We obtain formulae describing the magnitudes of these ripples. We show that the magnitude of oscillations of the angular frequency can also be estimated using energy considerations. Such estimates are provided for the three most common schemes of dynamic devices. Available experimental data confirm the main conclusions of the theory. We discuss both the harmful effects of these phenomena as well as their possible applications. The latter include design of bi-harmonic vibration exciters and exciters based on vibrational resonance. This article is part of the theme issue 'Vibrational and stochastic resonance in driven nonlinear systems (part 2)'.

Eigenfrequency constrained topology optimization of finite strain hyperelastic structures

This paper incorporates hyperelastic materials, nonlinear kinematics, and preloads in eigenfrequency constrained density–based topology optimization. The formulation allows for initial finite deformations and subsequent small harmonic oscillations. The optimization problem is solved by the method of moving asymptotes, and the gradients are calculated using the adjoint method. Both simple and degenerate eigenfrequencies are considered in the sensitivity analysis. A well-posed topology optimization problem is formulated by filtering the volume fraction field. Numerical issues associated with excessive distortion and spurious eigenmodes in void regions are reduced by removing low volume fraction elements. The optimization objective is to maximize stiffness subject to a lower bound on the fundamental eigenfrequency. Numerical examples show that the eigenfrequencies drastically change with the load magnitude, and that the optimization is able to produce designs with the desired fundamental eigenfrequency.

Automation calculation of the stability of drawing process of quartz optical fibers using different modifications of furnace shape

A one-dimensional model of quartz optical fibers extraction was considered and the problem of drawing parameters stability was solved. More over two numerical experiments demonstrating the states of stability or instability depending on the values of small harmonic oscillations introduced into the system, as well as on the furnace parameters were performed.

Harmonic oscillations of neutral particles in the γ metric

We consider a well-known static, axially symmetric, vacuum solution of Einstein equations belonging to Weyl's class and determine the fundamental frequencies of small harmonic oscillations of test particles around stable circular orbits in the equatorial plane. We discuss the radial profiles of frequencies of the radial, latitudinal (vertical), and azimuthal (Keplerian) harmonic oscillations relative to the comoving and distant observers and compare with the corresponding ones in the Schwarzschild and Kerr geometries. We show that there exist latitudinal and radial frequencies of harmonic oscillations of particles moving along the circular orbits for which it is impossible to determine whether the central gravitating object is described by the slowly rotating Kerr solution or by a slightly deformed static space-time.

In Russian

We suggest a new approach to description of driven diffusive systems in which fluctuations of nano-particle potential energy, being a source of directed motion, are considered as small. The base of the approach is the method of the Green’s functions, which allow writing analytical expressions for the sought-for average nano-particle velocity. These expressions are essentially simplified under the assumption of high temperatures. Within this high-temperature approximation, we consider a system (Brownian motor) with small harmonic oscillations of the particle potential energy of a saw-tooth shape. The obtained representation for the average motor velocity is the product of two functions. One of them depends on the fluctuation frequency and the other includes geometrical parameters of the system (that is of the steady and perturbing potential profiles). It is shown that the frequency dependence of the average velocity is a nonmonotonic function with the maximum position determining by the inverse characteristic diffusion time. In its turn, the average motor velocity as a function of the asymmetry parameter of the nonfluctuating component of the potential profile and the phase shift of the fluctuating one is sign-changing. This result means that there exists a possibility to govern the direction of the motor motion by the phase shift of harmonic oscillations relative to a fixed saw-tooth potential component. The intervals of both the phase shift and the asymmetry parameter values, in which the particle can move to the right or to the left, are diagrammed. The regularities obtained are of great importance to choose the values of parameters which provide effective regimes of Brownian motors operation.

Quasi-harmonic oscillatory motion of charged particles around a Schwarzschild black hole immersed in a uniform magnetic field

In order to test the role of large-scale magnetic fields in quasi-periodic oscillation phenomena observed in microquasars, we study the oscillatory motion of charged particles in the vicinity of a Schwarzschild black hole immersed into an external asymptotically uniform magnetic field. We determine the fundamental frequencies of small harmonic oscillations of charged test particles around stable circular orbits in the equatorial plane of a magnetized black hole, and discuss the radial profiles of frequencies of the radial and latitudinal harmonic oscillations in dependence on the mass of the black hole and the strength of the magnetic field. We demonstrate that assuming relevance of resonant phenomena of the radial and latitudinal oscillations of charged particles at their frequency ratio , the oscillatory frequencies of charged particles can be well related to the frequencies of the twin high-frequency quasi-periodic oscillations observed in the microquasars GRS 1915+105, XTE 1550–564 and GRO 1655–40.

Investigating the magnetic interaction with Geomag and Tracker Video Analysis: static equilibrium and anharmonic dynamics

In this paper, we describe how simple experiments realizable by using easily found and low-cost materials allow students to explore quantitatively the magnetic interaction thanks to the help of an Open Source Physics tool, the Tracker Video Analysis software. The static equilibrium of a ‘column’ of permanents magnets is carefully investigated by working on digital photos, while the anharmonic oscillations of a magnetic bar under the action of gravity and of magnetic repulsion are analysed by using a digital video. A detailed comparison between theoretical expectations and experimental results is discussed. We discuss how the magnetic force falls off with the distance between the permanent magnets following an inverse pth power law. Static and dynamic measurements of the force and of the periods for small amplitude harmonic oscillations yield an experimental value for p ≈ 2. The dynamical system is a good example of an anharmonic oscillator. The experiments need simple and inexpensive material to be realized and address a relevant topic in the physics curriculum; thus, they appear appropriate to be used in high school and undergraduate physics courses.

Natural oscillations of a rotating spherical fluid layer of variable depth

Small harmonic oscillations of the free surface of a thin fluid layer covering a rotating sphere are considered. The fluid is in the central field of sphere gravity and is exposed to the centrifugal and Coriolis forces. It is assumed that the fluid layer depth is independent of the longitude. In this formulation the problem is governed by a differential equation with singular coefficients that generalizes the Laplace tidal equation. The method of local separation of singularities is applied to integrate this equation. The solutions obtained are compared with the corresponding modes of the Laplace tidal equation, that is, the solutions of the problem for a fluid layer of constant depth.

Computation of inertial and damping characteristics of ship sections in shallow water

Hydrodynamics of 2D contours representing ship sections is considered for the case of small harmonic oscillations with a modification of a boundary-integral-equation method implemented earlier for the deep-fluid case. Alterations of the algorithm required by the finite-depth case are described in the present study and a number of numerical results are given. These include comparison with another code for the case of flat horizontal bottom and comparative calculations made for the case of the abrupt change of depth near the ship (stepped bottom). The results can be used for estimation of the bottom's influence on the manoeuvring and seakeeping qualities of ships.

Change in the type of bifurcation resulting in periodic oscillations in delayed feedback diode lasers

The appearance of oscillating regimes in a delayed feedback diode laser is studied analytically and numerically. Based on the Lang—Kobayashi model, the transition of the usual oscillation mechanism, related to the transition through the Hopf bifurcation, to hard excitation of the spike regime is studied. The change in the regime of the instability development has a nature of a phase transition. An explicit expression is derived for the frequency of small harmonic oscillations appearing during the transition through the Andronov—Hopf bifurcation. The boundary between two different regimes of the development of laser power oscillations is determined in the parameter space.

Effect of small harmonic oscillations during the steady rolling of a cylinder on a plane

If a wheel rolling over a rail transmits a tangential traction, frictional microslip occurs in part of the contact area, resulting in energy dissipation and localized wear. If the applied forces oscillate in time, the resulting wear will be non-uniform, resulting in ‘corrugations’ that can grow with progressive passes, depending on the dynamics of the overall system. In this paper, a linear perturbation method is used to obtain closed-form expressions for the receptance of a two-dimensional rolling contact subjected to small oscillations in normal force and rotational speed superposed on a mean value in the limit of large coefficient of friction. Corresponding expressions are also obtained for the amplitude and phase of the energy dissipation in the contact, which is expected to correlate with the local wear rate. The results are compared with a simpler Winkler model of the contact and with other models that have been used for the analysis of rail corrugation. Surprisingly good agreement is obtained with numerical results due to Gross-Thebing for the receptances due to oscillations in rotational speed.

Electromagnetic and Viscous Damping of Core Oscillations

Summary A detailed investigation of the interaction of core oscillations with the main magnetic field is made. Selection rules and interaction coefficients are obtained. Although the strengths of the interactions vary over nearly three orders of magnitude, all are extremely weak. Ohmic and viscous dissipation are considered in parallel calculations. It is found that core oscillations are almost unaffected by either damping mechanism. Most of the Ohmic dissipation takes place within a few skin depths of the boundary. Viscous dissipation is estimated only for the body of the liquid core. Even though it is expected to be much larger near the boundaries, it would seem unlikely to be sufficiently large to be of any significance. Our conclusion is that core oscillations are virtually free of damping. In a subadiabatic core, gravitational oscillations might be persistent enough to sustain an oscillatory geodynamo. 1. Introduction In any discussion of the dynamics of the Earth’s liquid outer core, it is important to know the rate at which energy dissipation takes place. There are two possible dissipative mechanisms in the liquid core, Ohmic and viscous. In this paper we consider their relative importance in the damping of small harmonic oscillations. The full geometries of both the main magnetic field and the oscillatory displacements are taken into account. Electromagnetic damping of elastic waves in the core appears to have been first suggested by Cagniard (1952) and an estimate of the strength of such damping was made by Knopoff (1955). Both Knopoff (1955) and Baiios (1955, 1956) developed theories of the modes of propagation of disturbances in the presence of a uniform magnetic field. Using accepted values of the physical parameters involved, they showed that at seismic frequencies the attenuation would be very weak. A similar result was obtained by Kraut (1965) for the damping of radial oscillations of a fluid sphere immersed in a uniform field. Lilley (1967) was the first to point out that as well as the damping that arises in a uniform magnetic field due to the distortion of the medium, there is an additional electromagnetic dissipation which occurs in non-uniform magnetic fields due to translation of the medium into regions of differing flux density. He confirmed the effect both theoretically and experimentally for elastic waves travelling in magnetic field gradients (Lilley & Smylie 1968; Lilley & Carmichael 1968, 1970). The relevance of the non-uniform field effect to the Earth’s core was only briefly considered by Lilley (1967). He concluded that a Q of about lo9 might characterize

Decoherence and transfer of quantum states of field modes in a one-dimensional cavity with an oscillating boundary

We study the evolution of Wigner functions of arbitrary initial quantum states of field modes in a one-dimensional ideal cavity, whose boundary performs small harmonic oscillations at the frequency ωW = pω1 (where ω1 is the fundamental field eigenfrequency). Special attention is paid to the case of initial even and odd coherent states, which serve as models of the 'Schrodinger cat states'. We show that the strong intermode interaction (due to the Doppler upshift of the fields reflected from the oscillating mirror) results in the decoherence of initial quantum superpositions in selected modes, even in the absence of any external 'environment'. Different quantitative measures of decoherence are discussed. The analytical solutions obtained show that any initial state of the field goes asymptotically to a highly mixed and moderately squeezed state in the 'principal resonance case' p = 2 and to the vacuum state in the 'semiresonance case' p = 1. It is shown that the decoherence process has several stages. In the first one, the interference between the components of the initial superposition is rapidly destroyed during the time of the primary decoherence, which is inversely proportional to the first power of the initial distance between the components, as opposed to the second power in the case of usual dissipative reservoirs. However, some weak traces of coherence (quantumness of states), such as the regions of negativity of the Wigner function, survive for much longer times, which do not depend on the size of the initial superposition.

Dynamics of entanglement between field modes in a one-dimensional cavity with a vibrating boundary

We study the dynamics of the coefficient of quantum separability between resonantly coupled modes of the massless scalar field in a one-dimensional ideal cavity, whose boundary performs small harmonic oscillations at the frequency ω w = pω 1 (where ω 1 is the fundamental field eigenfrequency). A qualitative difference between the cases p = 1 and p = 2 is discovered. In the first case, initially factorized thermal states remain separable in the process of evolution, whereas the dynamics of entanglement of initial entangled states (modelled by two-mode squeezed states) turns out to be more intricate, showing sudden changes from 'classical' to 'nonclassical' regimes. In contrast, in the 'principal resonance case' p = 2 the degree of entanglement changes with time monotonically, and any initially factorized thermal states go asymptotically to mixed quantum states with maximal degree of entanglement. Moreover, for highly thermalized states, the time dependence of the separability coefficient exhibits sharp jumps, which are, nonetheless, not correlated with time dependences of the energies of the field modes.

Response to temporal parameter fluctuations in biochemical networks

Metabolic response coefficients describe how variables in metabolic systems, like steady state concentrations, respond to small changes of kinetic parameters. To extend this concept to temporal parameter fluctuations, we define spectral response coefficients that relate Fourier components of concentrations and fluxes to Fourier components of the underlying parameters. It is also straightforward to generalize other concepts from metabolic control theory, such as control coefficients with their summation and connectivity theorems. The first-order response coefficients describe forced oscillations caused by small harmonic oscillations of single parameters: they depend on the driving frequency and comprise the phases and amplitudes of the concentrations and fluxes. Close to a Hopf bifurcation, resonance can occur: as an example, we study the spectral densities of concentration fluctuations arising from the stochastic nature of chemical reactions. Second-order response coefficients describe how perturbations of different frequencies interact by mode coupling, yielding higher harmonics in the metabolic response. The temporal response to small parameter fluctuations can be computed by Fourier synthesis. For a model of glycolysis, this approximation remains fairly accurate even for large relative fluctuations of the parameters.

Mechanical forcing of the wake of a flat plate

We report experimental results of the forced wake of a thin symmetric flat plate, placed parallel to an uniform air stream, in the range of thickness-based Reynolds number 50< Re e<200. External wake forcing was introduced by small harmonic oscillations of a moving flap, placed at the trailing-edge of the flat plate. When the flap remains in a fixed horizontal position, the mean velocity profiles obtained by hot wire measurements, for different Reynolds numbers, are self similar. In the presence of harmonic forcing, within a certain range of the forcing frequency, the mean velocity profiles change and coherent structures are formed in the wake. Two independent flow-type resonances were observed: (i) when the inverse of the forcing frequency matches the flight time of the fluid particles along the flap. (ii) when the forcing frequency of the flap equals one half of the vortex shedding frequency of the flat plate and flap system. Implications of the two observed resonances on the wake structure are important. The first resonance (i) is associated to a wide but less intense (energy fluctuations) wake flow and the second resonance (ii) generates a thin but intense resultant wake flow.

Thermal expansion of polymers subjected to low-amplitude temperature cycling

The temperature dependence of the thermal expansion of polymers subjected to small harmonic temperature oscillations about certain base temperatures near the glass transition range is considered. The conformational component of the expansion as a function of temperature is calculated in terms of the conformational transition kinetics. The temperature dependences of the expansion and the expansion vs. oscillation phase shift are analyzed with allowance for the vibrationally anharmonic component. The thermal expansion of polyvinylacetate at base temperatures of 295–320 K, oscillation frequencies of 0.3 and 0.1 Hz, and an oscillation amplitude of 0.8 K are studied experimentally. The measured and calculated data for the expansion and expansion vs. temperature phase shift are shown to be in good agreement.

The Use Of Trigonometric Interpolation Functions For Vibration AnalysisOf Beam Structures - Bridging Gap Between FEM And Exact Formulations

The small linear harmonic oscillations of linearly elastic coupled beam structures are addressed. The Finite Element Method (FEM), conventional Hermite beam formulation, and the resulting element matrices are then briefly discussed. The so-called exact Dynamic Stiffness Matrix (DSM) formulation for the longitudinal vibration of axially loaded beams is presented. Finally, the Dynamic Finite Element (DFE) approach is introduced and its application to the axial vibration of beams is displayed. The comparison is made between the standard static and (frequency dependent) Dynamic beam shape functions. The DFE formulation, combines the generality of the FEM and the high precision provided by DSM methods. The weighting functions and shape functions are evaluated referring to the appropriate exact DSM formulation. The DFE approach can be advantageously extended to more complex cases which distinguishes this method from the DSM method.

Oscillations of a Cylinder Induced by Free Surface Flow

The results of experiments in which a cylinder freely located on a fixed shaft performs small harmonic oscillations driven by a subcritical free stream in an open channel are given. Examples of the strong feedback effect of the cylinder oscillations on the hydrodynamic wake and surface gravity waves are obtained.

Co-Orbital Motion with Slowly Varying Parameters

We consider the dynamics of a test particle co-orbital with a satellite of mass ms which revolves around a planet of mass M0 ≫ ms with a mean motion ns and semi-major axis as. We study the long term evolution of the particle motion under slow variations of (1) the mass of the primary, M0, (2) the mass of the satellite, ms and (3) the specific angular momentum of the satellite Js. The particle is not restricted to small harmonic oscillations near L4 or L5, and may have any libration amplitude on tadpole or horseshoe orbits. In a first step, no torque is applied to the particle, so that its motion is described by a Hamiltonian with slowly varying parameters. We show that the torque applied to the satellite, as measured by ∈s = js/(nsJs) induces an distortion of the phase space which is entirely described by an asymmetry coefficient α = ∈s/μ, where μ = ms/M. The adiabatic invariance of action implies furthermore that the long term evolution of the particle co-orbital motion depends only on the variation of msas with time. Applying a constant torque to the particle, as measured by ∈s = js/(nsJp) is then merely equivalent to replacing α = ∈s/μ by α = (∈s − ∈p)/μ. However, if the torque acting on the particle exhibits a radial gradient, then the action is no more conserved and the evolution of the particle orbit is no more controlled by msas only. We show that even mild torque gradients can dominate the orbital evolution of the particle, and eventually decide whether the latter will be pulled towards the stable equilibrium points L4 or L5, or driven away from them. Finally, we show that when the co-orbital bodies are two satellites with comparable masses m1 and m2, we can reduce the problem to that of a test particle co-orbital with a satellite of mass m1 + m2. This new problem has then parameters varying at rates which are combinations, with appropriate coefficients, of the changes suffered by each satellite.