Theory Of Small Oscillations Research Articles

Molecular vibrations in the presence of velocity-dependent forces.

A semiclassical theory of small oscillations is developed for nuclei that are subject to velocity-dependent forces in addition to the usual interatomic forces. When the velocity-dependent forces are due to a strong magnetic field, novel effects arise-for example, the coupling of vibrational, rotational, and translational modes. The theory is first developed using Newtonian mechanics and we provide a simple quantification of the coupling between these types of modes. We also discuss the mathematical structure of the problem, which turns out to be a quadratic eigenvalue problem rather than a standard eigenvalue problem. The theory is then re-derived using the Hamiltonian formalism, which brings additional insight, including a close analogy to the quantum-mechanical treatment of the problem. Finally, we provide numerical examples for the H2, HT, and HCN molecules in a strong magnetic field.

Asymptotics of the Spectrum of Variational Problems Arising in the Theory of Fluid Oscillations

This work is a survey of results on the asymptotics of the spectrum of variational problems arising in the theory of small oscillations of a fluid in a vessel near the equilibrium position. The problems were posed by N. D. Kopachevsky in the late 1970s and cover various fluid models. The formulations of problems are given both in the form of boundary-value problems for eigenvalues in the domain R3, which is occupied by the fluid in the equilibrium state, and in the form of variational problems on the spectrum of the ratio of quadratic forms. The common features of all the problems under consideration are the presence of an elliptic constraint (the Laplace equation for an ideal fluid or a homogeneous Stokes system for a viscous fluid), as well as the occurrence of the spectral parameter in the boundary condition on the free (equilibrium) surface . The spectrum in the considered problems is discrete; the spectrum distribution functions have power-law asymptotics.

О дифференциальных уравнениях в банаховых алгебрах

We consider higher-order linear differential equations with constant coefficients in Banach algebras (this is a direct generalization of higher-order matrix differential equations). The presentation is based on higher algebra, differential equations and functional analysis. The results obtained can be used in the study of matrix equations, in the theory of small oscillations in physics, and in the theory of perturbations in quantum mechanics. The presentation is based on the original research of the authors.

Estudio de los modos de oscilación en un Péndulo Físico Simétrico usando el potencial efectivo

En este artículo presentamos un estudio unificado de la dinámica del péndulo físico simétrico en el espacio de fase de los distintos modos de oscilación que se presentan en este sistema: modo plano, modo elíptico y modo cónico, usando un potencial efectivo que va a depender de la coordenada de nutación y depende también paramétricamente de la energía y del momento angular del sistema. Presentamos también una deducción formal de la aproximación de Allais para la precesión ápsidal del péndulo físico simétrico usando la teoría de pequeñas oscilaciones alrededor del movimiento estacionario o modo de oscilación cónico.

Some Connections Between the Classical Calogero–Moser Model and the Log-Gas

In this work we discuss connections between a one-dimensional system of $N$ particles interacting with a repulsive inverse square potential and confined in a harmonic potential (Calogero-Moser model) and the log-gas model which appears in random matrix theory. Both models have the same minimum energy configuration, with the particle positions given by the zeros of the Hermite polynomial. Moreover, the Hessian describing small oscillations around equilibrium are also related for the two models. The Hessian matrix of the Calogero-Moser model is the square of that of the log-gas. We explore this connection further by studying finite temperature equilibrium properties of the two models through Monte-Carlo simulations. In particular, we study the single particle distribution and the marginal distribution of the boundary particle which, for the log-gas, are respectively given by the Wigner semi-circle and the Tracy-Widom distribution. For particles in the bulk, where typical fluctuations are Gaussian, we find that numerical results obtained from small oscillation theory are in very good agreement with the Monte-Carlo simulation results for both the models. For the log-gas, our findings agree with rigorous results from random matrix theory.

Damping behavior of a harmonic oscillator in self-induced magnetic field of a solenoid

Most oscillators experience damping in their normal operation. In this paper, the damping behaviour of a simple harmonic oscillator made using a low-cost apparatus is examined. The damping coefficient for the (i) self induced magnetic field in a cylindrical solenoid and (ii) air are studied. The magnitude of damping force is evaluated. The results obtained in the experiment are in accordance with the theory of small oscillations. The results indicate that the under damping process is caused in the above two cases. This technique is relevant in magnetic vehicles.

El estudio y enseñanza de los modos vibracionales de la molécula triatómica

En este trabajo se muestra que a través del dominio de la teoría general de pequeñas oscilaciones es posible resolver analíticamente los modos vibracionales de la molécula triangular, y que a la luz de resultados comparativos entre la molécula lineal y triangular se puede evaluar la dinámica de la molécula triangular y simplificar el análisis. Para ilustrar lo anterior, se modela mecánicamente una molécula triatómica, utilizando la formulación de Lagrange y asumiendo un tipo de molécula en la cual las masas (átomos) están ubicadas en los vértices de un triángulo rectángulo. Para un desarrollo analítico del problema se utiliza la teoría de pequeñas oscilaciones, considerando las masas y las constantes de acoplamiento iguales. Esto último permite escribir el Lagrangiano del sistema en forma adecuada y posteriormente determinar las ecuaciones diferenciales de movimiento, que describen el comportamiento de la molécula y permiten determinar las frecuencias propias de vibración. También se analiza y discute el caso de una molécula triatómica lineal, determinando los correspondientes modos normales de vibración. Finalmente, se comparan las soluciones de ambos sistemas.

Theoretical Global Seismology

Tony Dahlen and Jeroen Tromp have written a treatise, Theoretical Global Seismology, that is authoritative and comprehensive. The subject is developed from the theory of small oscillations in classical mechanics, in this case the free oscillations of the Earth. In other words, the conceptual pathway chosen by the authors is to represent the seismic transient response of the Earth in terms of free oscillations, or standing waves. This is followed by representation in terms of traveling waves, rays, and pulses. These representations taken together give the reader broad insight into global seismology and the methods used to study the structure of the Earth and of earthquake sources.

Mechanical behaviour of hamstring muscles in low-back pain patients and control subjects

The purpose of this study was to measure and compare the passive elastic moment, the stiffness and the damping coefficient of the hip joint, as functions of the hip and knee joint angles in men with and without low-back pain. Two conventional tests, the straight-leg-raising test and the trunk forward flexion, were also performed and compared between these subjects. The passive elastic moment was measured using an isokinetic device in the passive mode. This device raised the lower limb from the horizontal position to the straight-leg-raising angle at a slow and constant angular velocity. A custom-made splint connected with the lever arm of the isokinetic device maintained the knee in extension and the ankle in the neutral position. The damping coefficient of the hip joint was measured for 0, 15, 45, 60, 75 and 90% of straight leg raising angle of each subject, using the suspension method based on small oscillation theory. To ensure that muscles were inactive during the passive hip moment tests, muscle activity was monitored with surface EMG. The stiffness was computed as the ratio of the change in passive elastic moment to the change in the hip angle. The passive elastic moment, the stiffness and the normalized trunk flexion were significantly different between the two groups respectively. There was, however, no difference between the two groups in the results of straight-leg-raise and damping coefficient of the hip. The passive elastic moment was a nonlinear function of the hip flexion angle and showed large intersubject differences, especially as the joint limit was approached. The damping coeffcient was a polynomial function of the hip flexion angle. The measured variables were analysed using a discriminant function and it was shown that the two groups were clearly discriminate in a meaningful manner.

Recent Results in Random Vibrations of Nonlinear Mechanical Systems

The influence of random vibration on the design of mechanical components has been considered within the framework of the linear theory of small oscillations. However, in some important cases this theory is inadequate and fails to predict some complex response characteristics that have been observed experimentally and which can only be predicted by nonlinear analyses. This paper describes some recent developments in the theory of nonlinear random vibration based on Markov methods and related problems in the design of dynamical systems. Research efforts have been focused on stability/bifurcation conditions, response statistics and reliability problems. Significant progress has been made in developing new analytical methods and conducting experimental testing. These developments have helped to resolve some controversies, and to enhance our understanding of difficult issues. Experimental and numerical simulations have revealed new phenomena that were not predicted analytically. These include on-off intermittency, snap-through phenomena, and the dependence of the response bandwidth on the excitation level. The main results of studying the responses of nonlinear single-and two-degree-of-freedom systems to random excitations obtained by the author and others are discussed in this paper.

Recent Results in Random Vibrations of Nonlinear Mechanical Systems

The influence of random vibration on the design of mechanical components has been considered within the framework of the linear theory of small oscillations. However, in some important cases this theory is inadequate and fails to predict some complex response characteristics that have been observed experimentally and which can only be predicted by nonlinear analyses. This paper describes some recent developments in the theory of nonlinear random vibration based on Markov methods and related problems in the design of dynamical systems. Research efforts have been focused on stability/bifurcation conditions, response statistics and reliability problems. Significant progress has been made in developing new analytical methods and conducting experimental testing. These developments have helped to resolve some controversies, and to enhance our understanding of difficult issues. Experimental and numerical simulations have revealed new phenomena that were not predicted analytically. These include on-off intermittency, snap-through phenomena, and the dependence of the response bandwidth on the excitation level. The main results of studying the responses of nonlinear single-and two-degree-of-freedom systems to random excitations obtained by the author and others are discussed in this paper.

Random excitation of nonlinear elastic structures with internal resonances

The influence of random vibration on the design of mechanical components has been restricted to the linear theory of small oscillations. However, this theory is inadequate and fails to predict the complex response characteristics which may be observed experimentally and can only be predicted by employing the nonlinear theory. This paper presents a brief overview of the basic nonlinear phenomena associated with nonlinear random vibration.

Splitting of the free-oscillation multiplets by steady flow

SUMMARY Only toroidal flow of odd harmonic degree causes first-order splitting, which is symmetrical about the degenerate frequency of a multiplet. The classical theory of small oscillations about a state of steady motion leads to a splitting matrix that is hermitean, a result that does not require the anelastic approximation. An explicit expression is given for the element of the hermitean splitting matrix.

Rotational and translational dynamics of flexible macromolecules

In two previous studies [J. Chem. Phys. 91, 1287 (1989); 95, 1234 (1991)] we had examined the dynamics of coupled, internal translational, and rotational motions in a rigid or pseudorigid N-sphere macromolecular model using the rotational–translational Brownian dynamics algorithm of Dickinson et al. [J. Chem. Soc. Faraday Trans. 2 81, 591 (1985)]. In the present study, those works are generalized to include all possible internal flexible motions in an N-sphere macromolecular model. In general, for the N-sphere system there are 6(N-1) degrees of configurational freedom, although not all modes may be active in any particular application. Using the language of small oscillation theory, the deviations in the generalized coordinates associated with the ‘‘joints’’ connecting the spheres to one another are described in terms of quadratic potentials. From these potentials, the components of the generalized forces (torques and forces) are obtained for subsequent use in the Brownian dynamics algorithm. The degree of flexibility for any particular mode in the N-sphere system can be controlled by a single constant in the associated quadratic potential function. By taking the appropriate limits of these constants, the complete range of flexibility, viz., from ‘‘torque (or force)-free’’ to ‘‘rigid’’ can be approximately realized at any point in the N-sphere system. The model given is, therefore, capable of simulating all types of macromolecular motions. As a specific example, we studied the linear elastic rotator, where all modes (translational and rotational) are constrained except for rotations about the line of center of the spheres. Brownian dynamics results expressed in terms of rotational correlation functions were in good agreement with analytical solutions obtainable for this highly symmetric system. The algorithm given here is believed to be particularly useful in the study of the dynamics of biological macromolecules where ‘‘flexibility’’ is often critical to the functionality of the macromolecule [J. Chem. Phys. 90, 3843 (1989); Macromolecules 16, 421 (1983); 15, 1544 (1982); Chem. Phys. 41, 35 (1979)].

Review of nonnegative matrices: By Henryk Minc

This book is intended as an introduction to the theory of nonnegative matrices and as a self-contained reference work for scientists and mathemati- cians. Nonnegative matrices arise in applications in probability theory, eco- nomics, numerical analysis, and the theory of small oscillations of elastic systems, to name just a few areas. In this book such matrices are studied primarily to determine their algebraic properties. The book has seven chapters. The first three chapters develop the basic Perron-Frobenius theory following the method developed by Wielandt

The experimental response of an impacting pendulum system

This paper presents experimental results obtained from a harmonically excited pendulum system. The pendulum has rigid barriers which limit the amplitude variation from its central position. It is considered in both the normal (downward) position and in the upright (inverted) position. The overall dynamics of the pendulum include impacts with the rigid constraints, and the system response to sinusoidal excitation includes non-impacting motions, stable subharmonics, and chaotic motions. These were experimentally found to occur in the parameter regions predicted by previous analytical work. This system represents an example of a deceptively simple device which can undergo extremely complicated dynamics. For example, the inverted pendulum was found to have 10 distinct possible steady-state responses at a fixed driving amplitude and frequency, each of which was obtained simply by changing the initial conditions. In addition, the normal pendulum was found to be capable of having impacting steady-state dynamics which coexist with the non-impacting steady-state motion predicted from small oscillation theory.

Geometrical interpretation of the simultaneous diagonalization of two quadratic forms

The theory of small oscillations, when cast in matrix form, leads to an eigenvalue problem involving the simultaneous diagonalization of two real symmetric matrices. While a geometrical interpretation of this process exists, its visualization is generally hampered by the multidimensional nature of the coordinate transformations involved. I consider a model problem with just two degrees of freedom for which the coordinate transformations become planar and therefore easily visualizable. A set of plane diagrams is provided showing how the diagonalization is achieved for the model problem considered.

Asymptotic behavior of the spectrum of certain problems, connected with the oscillation of fluids

One obtains the principal term of the asymptotics of the spectrum in a series of problems of the theory of small oscillations of fluids, filling partially a container. First one discusses a certain problem of general character, the “nonlocal Steklov type problem”. The considered problems are connected with the oscillations of a capillary ideal (or capillary stratified) fluid, of a heavy viscous fluid, of a capillary viscous fluid, and with the oscillations of a fluid in container with an elastic bottom.

Calculating molecular structure parameters in natural coordinates as determined by electron diffraction

The nonlinear relation between internuclear distance and the natural coordinates has been used in calculating the first three central moments for the distance distributions, with the results used to calculate the reduced molecular component in the electron-scattering intensity for a molecular gas. Small-oscillation theory employs a linear relationship between the cartesian and natural coordinates and is shown to lead to substantial errors in calculating and .

Small-oscillation theory of the one-dimensional large optic polaron.

We study the small oscillations of the one-dimensional large polaron within the Holstein molecular-crystal model. We show that additional stationary solutions are not minimal. We solve analytically the classical equations of motion for the normal modes of the displacements about the minimal polaron solution, both determining the eigenspectrum and constructing the eigenfunctions.