Vibration of x-braced portal frames

Both free and forced vibrations of elastic X-braced portal frames are investigated. Solutions of the Euler-Bernoulli equation for the transverse vibration coupled with the axial vibration are used. The first five natural frequencies, with the angle of inclination, α, of the bracing bars ranging from 15° to 75°, with different slenderness ratios, R, of the columns, and different stiffness of the floor beam and crossing bars, are presented along with two sets of the natural modes of the frames with α = 45°. For the forced vibration, the dynamic responses of the frames with a concentrated horizontal time dependent force acting at a top joint are studied. The responses of the frames with α = 45° are analyzed in detail.     

Hysteretic damping and causality

It is proven that linear oscillatory systems with hysteretic damping in the form of complex stiffness and/or complex elastic moduli satisfy the causality principle: the response of such a system to an arbitrary external force cannot appear earlier than the onset of the force. The proof, based on a rigorous solution to the problem of forced oscillations, is presented in detail for an oscillator with a complex stiffness, as well as in a brief explanation for a system with N mass. It is also shown that these systems are Lyapunov-unstable. A comparison is made to other linear hysteretic damping models.     

Three methods for analyzing forced vibration of a fluid-filled cylindrical shell

The forced vibration of an infinite elastic circular cylindrical shell filled with fluid is studied. Three methods are employed to analyze the forced vibration problem of this shell-fluid coupled system, that is, wave propagation approach (wave mode superposition), theorem of residues and a numerical integral method. In order to explain these methods more explicitly, before being used to investigate the vibration of an infinite fluid-filled elastic circular cylindrical shell, all these three methods are employed firstly to analyze the forced vibration problem of an infinite beam and an infinite elastic circular cylindrical shell in vacuo. Advantage and disadvantage of these three methods are discussed and their interesting relationship is revealed. That is, to any circumferential wavenumber and frequency of the external force, there is an unchangeable relationship between the general coordinates of various waves in the wave propagation approach and the residuals in the theorem of residues.     

Response of rotating rings to harmonic and periodic loading and comparison with the inverted problem

The harmonic and periodic forced vibrations of rotating rings are derived and investigated. The modal expansion technique yields the forced solution, which is characterized by four generalized co-ordinates associated with each n (circumferential wave number). The inextensional assumption is presumed, when flexural vibration is the only important component, to reduce the order of the system. The closed form solutions to the harmonic load cases, once concentrated, once distributed, are demonstrated and interpreted. The approach is then extended to periodic loads, where Fourier sine and cosine series is applied. Examples depict the numerical responses to all the cases being derived. The solutions of a stationary ring subjected to traveling loads are also solved for comparison. Their difference is investigated and interpreted from various viewpoints.     

Calculation of vibrations of the H-bonds and electrooptical parameters of the F(HF)2]− complex

The problem of calculating the vibrations of the F(HF)₂]^? complex with hydrogen bonds is considered with allowance for the anharmonicity and interaction of motions in different degrees of freedom. A systematic solution of this problem is proposed which consists in separating the total vibrational system into subsystems, obtaining sufficiently exact vibrational wave functions of subsystems, and expanding the vibrational wave functions of the total system in basis functions constructed from the wave functions of subsystems. At the first stage of our study, the stretching and bending modes of two F...HF hydrogen bonds are considered with the use of an exact kinetic energy operator and a nonempirical three-dimensional potential energy surface. It is shown that these vibrational modes of the complex are characterized by significant mechanical and electric anharmonicities. The calculated values of frequencies of the symmetric and antisymmetric vibrations of hydrogen bonds are in good agreement with the experimental findings.     

On bound states in the continuum of N-body systems and the virial theorem

We prove generalized versions of the quantum mechanical virial theorem and apply them to the investigation of the spectrum of N body Hamiltonians. We show, in particular, that for N particles interacting through 2-body potentials which may have singularities but “don't wiggle too much,” no positive energy bound state can exist. We also prove results on the absence of bound states with energy bigger than some value E⁰ ? − ∞ and extend them to the case of N particles interacting through ν-body forces (ν = 1, 2,…, N) and with an external electromagnetic field. Also some remarks for the case of a Dirac electron in an external potential are given as well as for some problems with boundary conditions. A by-product of this investigation is the unitarity of the S matrix and the strong asymptotic completeness for systems of N particles interacting by 2-body forces which are not restricted to be purely repulsive.     

Dynamic analysis of ring-stiffened circular cylindrical shells

A numerical method is developed for the dynamic analysis of ring-stiffened circular cylindrical thin elastic shells. Only circular symmetric vibrations of the shell segments and radial and torsional vibrations of the rings are considered. The geometric and material properties of the shell segments and the rings may vary from segment to segment. Free vibrations or forced vibrations due to harmonic pressure loading are treated with the aid of dynamic stiffness influence coefficients for shell segments and rings. Forced vibrations due to transient pressure loading are treated with the aid of dynamic stiffness influence coefficients for shell segments and rings defined in the Laplace transform domain. The time domain response is then obtained by a numerical inversion of the transformed solution. The effect of external viscous or internal viscoelastic damping is also investigated by the proposed method. In all the cases, the dynamic problem is reduced to a static-like form and the “exact” solution of the problem is numerically obtained.     

The interaction of light with a semiinfinite dielectric as a phonon problem; the generalized Snellius law and Fresnel formulae

Using Ewald's formulae it is possible to evaluate the electromagnetic field generated by a mechanical plane wave propagating through a semiinfinite dipole crystal. The problem of refraction and reflection of an external electromagnetic wave on a semiinfinite dipole crystal is thus reduced to the evaluation of the forced vibrations of this crystal excited by the incident electromagnetic wave. Applying this approach, the generalized Snellius law and generalized Fresnel formulae can be deduced. In the limit of long waves these general formulae change into the usual classical expressions. The Ewald extinction theorem follows directly from the lattice dynamics of our system.     

Self-induced acoustic transparency in the long-short-wave resonance mode

The effect of self-induced acoustic transparency for transverse-longitudinal pulses propagating along an external magnetic field in a system of resonance paramagnetic impurities with the effective spin S=1/2 is theoretically investigated. In this case, the short-wave transverse component of the pulse causes quantum transitions, and the longitudinal long-wave component dynamically shifts the frequency of those transitions. When the speeds of the longitudinal and transverse acoustic waves in the crystal matrix are close to each other, both components interact in the mode of the long-short-wave resonance, which is described by a system of nonlinear integro-differential equations. It is shown that this interaction results, in particular, in the modulation of the carrier frequency of the circular-polarized component of the pulse. More precisely, the frequency in the neighborhood of the signal’s maximum is less than in the vicinity of its edges. Solutions in the form of traveling 2π-pulses are analyzed analytically and numerically. It is shown that there exist solutions that include a longitudinal component and cannot be reduced to well-known transverse solitons of the sinus-Gordon equation.     

Scaling- and antiscaling-type oscillations in isoscalar and isovector nuclear monopole vibrations

Isoscalar (T = 0) and isovector (T = 1) giant monopole resonances are studied using a localscale version of the ATDHF theory developed on the basis of a rigorous energy-density functional approach. Due to the strong coupling between the bulk and surface density vibrations, the monopole collective motion is split into four normal modes. Two of them, lower in energy, correspond to scaling-type density vibrations. The other two are of antiscaling-type in which the nuclear surface oscillates opposite in phase to the scaling-type vibrations. Excitation energies, transition densities, T = 0 and T = 1 energy weighted sum rules and other properties of breathing even-even nuclei are calculated using different Skyrme-type effective forces. The strong sensitivity of the antiscaling-type vibrations to the particular form of the approximate energy-density functionals is demonstrated.     

Dynamics of a longitudinal-and-transverse acoustic wave in a crystal with paramagnetic impurities

The evolution of longitudinal-and-transverse acoustic pulses propagating along an external magnetic field through a system of resonant paramagnetic impurities with effective spin S=1/2 is studied theoretically. It is shown that, when the group velocities of longitudinal and transverse waves are equal and the impurity concentration is sufficiently small, the initial system of equations is reduced to new evolution equations, which are integrable within the framework of the inverse scattering problem approach. These equations qualitatively describe the new coherent dynamics of acoustic pulses.     

Equations for subsystems

The exp(S) equations of Coester and Kümmel are rederived from the Schrödinger equations for n-body subsystems which are specified by the amplitudes of n fermions being at certain positions in r-space and the rest moving in shell model states. This derivation needs no formal exp(S) ansatz and allows us to give physical significance to a certain method of truncating the equations. Approximations for systems with short- and long-range forces are distinguished and the application to the electron gas is briefly discussed. It is found that the mutual occurrence of long- and short-range difficulties makes it hard to truncate the equations, and a crude but simple approximation for this case is proposed. In relation to Brueckner-Bethe theory, additional self-consistency is found to be the most important difference. Its importance is stressed and discussed in relation to the concepts of a particle potential and occupation probabilities. The equations are finally studied for a solvable model which simulates long-range forces. It is seen that for this example the truncated equations are highly efficient in getting near to the exact ground state as long as the system shows no phase-transition-like phenomena.     

Reconstruction of the dynamic contact stresses in an elastic layer from the displacements of its free surface

An inverse problem on the reconstruction of the wave field of contact stresses produced by an external load in an elastic layer from the displacements of its free surface is considered for the model of forced steady-state vibrations in the approximation of plane deformations. The solution is constructed using two approaches: (1) a reduction of the problem to the Fredholm integral equation of the first kind with the use of the Tikhonov regularization and (2) an expansion of the solution in a discrete set of waves. It is shown that both approaches are approximately equivalent in the model under consideration. Possibilities for an adequate reconstruction of the source field from far-zone measurements of a finite number of propagating wave modes are analyzed.     

The dependence of pulse properties of ferrous borate single crystals on their thickness

It is established that the magnitude of the discontinuity in the slope of a pulse-switching curve defined as the ratio between the switching coefficients S _w1 and S _w2 corresponding to the first and second parts of the switching curve, respectively, decreases with an increase in the thickness of a ferrous borate single crystal. This change is generally caused by a decrease in the coefficient S _w1, whose magnitude is inversely proportional to the sample’s thickness. In order to analyze the results obtained, we used the early proposed expression τ^?1=aH _s ?,bA ², connecting the switching rate τ^?1 with the amplitude of the magnetic field H _s and the intensity A of the magnetoelastic vibrations accompanying the pulse switching. It is found that the coefficient a depends only slightly on the sample’s thickness, while the coefficient b is inversely proportional to the thickness squared. Thus, the main part of the energy losses due to magnetoelastic vibrations is associated with elastic lattice vibrations.     

Parametric-resonance-induced cable vibrations in network cable-stayed bridges. A continuum approach

There is a wealth of evidence to suggest that the bearing cables of cable-stayed bridges may experience large-amplitude oscillations, attributed in general to parametric resonance with the girder vibrations. A common coutermeasure consists of connecting the principal stays together with secondary cables to form a network and, here, optimal cable arrangments will be discussed when such a network is uniform and triangular meshed. The present approach is qualitative, and basically consists of homogenizing the cable net to an orthotropic elastic membrane, and then considering an auxiliary structure where the bridge girder, instead of being supported by the cable network, is supported by wedge-shaped membranes. The elastic solution under uniformly distributed loads, found using Lekhnitskii's approach, is the starting point for the discussion of the system in dynamic equilibrium. Having established a correspondence between the cable-net size and shape and the elastic moduli of the homogenized membrane, simple formulas are obtained to describe the global bridge vibration, as well as the local oscillations of the cables. It is then possible to estimate the girder and cable-net characteristic frequencies, to evaluate those conditions possibly leading to parametric resonance and, with respect to these variables, to determine optimal cable arrangements. This method is finally applied to the paradigmatic example of the Normandy Bridge.     

The acoustic field excited by flexural vibrations of an elastic plate with a circular inclusion

The acoustic field excited by flexural vibrations of a thin elastic plate and the perturbations of this field caused by a homogeneous circular inclusion with other elastic properties are considered. Because the density of air widely differs from that of a metal, this problem can be solved with fair accuracy in two steps: first, by considering the vibrations of the plate in a free space, and, then, by calculating the acoustic field excited by the field of plate’s vertical deflections. The main results of this work are the asymptotic expressions for the far acoustic field excited by each of the Fourier components F _m (r)cosmφ of the flexural wave scattered by the inclusion.     

A joint relaxation technique for the receptance solution of structural vibration problems

A well-established technique for the analysis of a complex vibrating system is to cut it into a convenient number of simpler subsystems. An alternative approach, suggested by recent studies of displacement excitation, is to convert the system into a set of independent subsystems by clamping it at a sufficient number of co-ordinates. Exact solutions for free and forced vibration are then obtained by considering the relaxation of the clamps. Data in the Vibration Analysis Tables [1] are used for the numerical solution of framed structures by this method. This displacement approach is in some cases simpler than the standard methods and leads to smaller frequency determinants.     

A new type of acoustic modes of vibration of thin piezoelectric plates: Quasi-longitudinal normal modes

Using ST-cut quartz crystal plates as an example, a new type of normal modes of acoustic vibrations is described. The modes propagate along the x axis with a velocity close or equal to that of longitudinal bulk waves propagating in the same direction and have a longitudinal component of elastic displacement no less than two orders of magnitude greater than the two other components (the shear-horizontal and shear-vertical ones) throughout the whole plate thickness. The domain of existence of the quasi-longitudinal modes consists of a set of limited zones that contain the “allowed” values of the plate thickness H/λ (H is the plate thickness and λ is the wavelength) and are separated by “forbidden” zones corresponding to common Lamb modes. The closeness (or coincidence) of the velocities of a quasi-longitudinal mode in the plate and a longitudinal bulk wave in an unbounded crystal is a necessary but not sufficient condition for the existence of the aforementioned type of modes in ST,x quartz.     

On the contribution of three-body forces to Nd interaction at intermediate energies

Available data on large-angle nucleon-deuteron elastic scattering Nd→dN below the pion threshold give a signal for three-body forces. The problem exists of the separation of possible subtle aspects of these forces from off-shell effects in two-nucleon potentials. By considering the main mechanisms of the process Nd→dN, we show qualitatively that in the quasi-binary reaction N+d→(NN)+N with the final spin singlet nucleon-nucleon pair in the S-state, the relative contribution of the three-nucleon forces differs substantially from the elastic channel. It gives a new testing basis for the problem in question.     

A semimicroscopic model of nuclear vibrations with separable forces and the giant dipole resonance of 12C

In the generalized ph basis generated for the realistic nuclear ground state, the nuclear hamiltonian is approximated by an expression containing the separable effective forces and terms which can be estimated from the spectroscopic data. The model hamiltonian obtained is used to describe normal nuclear vibrations. The application of this approach in the shell-model theory of nuclear reactions is discussed. The S-matrix for nucleon-nucleus scattering is calculated explicitly. The ¹²C photodisintegration calculation is made as a test example.