Circular Frequencies Research Articles

DYNAMIC STRESS ANALYSIS OF A ROTATING SOLID DISC SUBJECTED TO CYCLIC ROTATIONS

This paper presents an analysis of the dynamic stress in a solid disc subjected to cyclically varying rotations with respect to time, that is, Ω^- (t^-) = Ω^-0 + Ω^-1 sin(Ω^-2 t^-) (Ω^-0, Ω^-1, Ω^-2 : constants) The radial displacement and stress components are derived from a general solution for a solid disc, which was obtained previously by the author. A dynamic solution leads to the existence of the two kinds of resonance circular frequencies. For Ω^-1≠0 and Ω^-0=0, the resonance circular frequency Ω^-2 agrees with half of λn which corresponds to the eigen-circular frequencies for in -plane vibrations of rotating discs. for Ω^-1 ≠0 and Ω^-0≠0, Ω^-2 agrees in both cases, λn and λn/2. It is finally discussed how the circular frequency Ω^-2 influences the stress components and their distributions.

Nonlinear free bending vibrations of plates

A general solution for the Helmholtz differential equations is obtained in the complex domain and applied to the nonlinear, free, bending vibrations of plates. The analysis is based on the decoupled nonlinear von Karman field equations by Berger assumption for the large deformations of plates. The decoupled differential equation in terms of the deflection function is a fourth order Helmholtz differential equation. Its solution, called the dynamic deflection function, is obtained in the complex domain by means of newly defined first and second kind and modified Bessel functions. The dynamic deflection function can be applied to any plates having any shape and any boundary condition under any arbitrary dynamic loads. For plates with smooth boundary, the parameters of the dynamic deflection function are determined from the boundary conditions of the plates and the initial conditions of the vibrations. The analyses of plates with piece-wise smooth boundaries are obtained on the mapped planes. The nonlinear, free vibration of circular plates are investigated by the dynamic deflection function. The effect of stretching on the natural circular frequencies are illustrated.

Third-order nonlinear spectra of a strong bichromatic field interacting with a three-level atom in a “V” configuration. II numerical results

Abstract This investigation involves a three-level atom interacting with two strong laser fields, of (circular) frequencies ω a and ω b , in a “V” configuration. Resonant and nonresonant situations are considered. The excitation spectrum, due to nonlinear three-photon processes, is computed for the frequency regions 3 ω a and ω a + 2 ω b . The excitation spectra are determined for various Rabi frequencies, η a and η b , of the two lasers; the spectra are computed from formulas previously derived by the Green function formalism. The computed spectra are interpreted in terms of previous results for three special cases: η a ⪢ η b , η a ⪡ η b and η a = η b .

Nonlinear oscillations of bubbles in compressible hydraulic oils

The nonlinear oscillation of a bubble in compressible hydraulic oils subject to periodic pulsating pressure was theoretically analyzed. According to the numerical calculations, the effect of compressibility of liquid on the oscillation of a bubble were clarified. That is, (1) the effect of compressibility is appreciable where the amplitude is large; (2) when compressibility is considered, the pressure threshold increases; (3) in the subharmonic of order 1/2, the effect of compressibility on the pressure threshold is almost constant over a wide range of initial bubble radius R0; (4) the effect of compressibility on the relation between ω/ω0 (a ratio of the circular frequencies of pulsating pressure and a bubble) and pressure threshold of the subharmonics 1/2 and 1/3 is almost constant covering the wide range of ω/ω0.

Vibrations of Annular Plates of Variable Thickness

An exact solution for the circular frequencies of a polar orthotropic annular plate of uniform and parabolic thickness under the action of in-plane forces is presented. The annular plate is reinforced with simply supported edge beams at both the inner and outer boundaries. The edge beams are subjected to uniform, radial compressive loadings. Frequency parameters are determined for both axisymmetric and higher modes and are found to be functions of the in-plane loads, plate dimensions and rigidities, edge beam stiffness and moment of inertia parameters, and the profile of the plate. The use of edge beams in this investigation provides a practical method of simulating various types of edge conditions for the polar orthotropic annular plate — including the limiting cases of the simply supported and clamped edge.

In-plane vibration of plates by continuous mass matrix method

Abstract The continuous mass matrix method derived for frameworks is extended to the analysis of in-plane vibration of plates. A continuous mass distribution which is the same as the actual mass distribution of the plate is considered over each rectangular finite element. Taking into account that the rigid body movement produces inertial forces in dynamic analysis for a rectangular plate element eight independent conditions are provided to satisfy eight independent freedoms. Each condition is obtained from an independent displacement distribution satisfying the equations of motion at any point of the element and not only at the nodes of the rectangle. The dynamic element stiffness matrix thus obtained is a function of the natural circular frequency. The limit of the dynamic element stiffness matrix when the value of the natural circular frequency tends to zero is the static, stress compatible element stiffness matrix. The analysis of plates under forcing forces is performed by modal analysis after the natural circular frequencies and the corresponding modal shapes have been obtained from the free vibrations, for all the forcing forces are assumed to be function of the same time variation. Otherwise one must recur to a numerical analysis. The effect of the sizes, number of the meshes, the additional static load on the plate and the rigidity of the boundaries on the vibration of the plate is discussed. Few example problems are solved in order to illustrate the above mentioned effects. The numerical results obtained by continuous mass matrix method are compared with those of consistent mass matrix method. The convergence in terms of the sizes of meshes and the limit of convergence are examined.

Shearing interferometry in polar coordinates

Shearing interferometers are developed that are able to display the radial component of a gradient with radial interference fringes and the azimuthal component of a gradient with circular interference fringes. Three concepts in particular are introduced. (i) A shearing interferometer is presented that has a constant azimuthal displacement between the interfering wave fronts. (ii) Emphasis is placed on the importance and convenience of using reference interference-fringe systems with radial and circular configurations to display shearing interferograms when the displacements between the wave fronts have polar-coordinate geometry. (iii) Computer-generated, holographic-type filters with circular carrier frequencies are shown to have properties of special value for shearing interferometry. Practical experimental realizations of the interferometers and techniques are discussed and results are shown.

The effect of the coupling between galvanometers on the characteristics of the Gs-11 gravimeter

With the help of equivalent constants of the system of two galvanometers of an Gs-11 gravimeter simple formulae for the amplitude and phase responses were derived. The coupling between the galvanometers is responsible for the change in the transient oscillations, which are determined by equivalent circular frequencies and damping constants (Tab. 1). As regards the main tidal terms with a diurnal and semi-diurnal period the effect of the coupling on the amplitude response is negligible, but the changes of the phase delay amount to 10–20% of the phase lag of the gravimeter (Tab. 3). With events of periods between 100 and 1000 s (e.g., surface earthquake waves and free oscillations of the Earth) the effect of the coupling is not negligible with both responses (Tab. 2).

截頭円錐殻の自由曲げ振動

On the basis of the linear bending shell theory, the axisymmetric free flexural vibrations of truncated conical shells clamped at one end and having an end-mass at the other are analyzed numerically, and the coupling between the end-mass and the mass of the shell is discussed in detail. Expanding the vibration mode into the cosine Fourier series, a family of characteristic equations for the natural circular frequencies are derived by the aid of the Rayleigh-Ritz method and solved by the use of the power method. The variation of circular frequencies and vibration modes of the shell with the end-mass, geometrical and completeness parameters are shown. And, for the cases where the mass of the shell can be ignored compared to the end-mass, the design charts for the natural circular frequency or axial spring constant of truncated conical shells are presented over a wide range of geometrical and completeness parameters.

On the Forced Vibrations of the Shaft Carrying an Unsymmetrical Rotor : Forced Vibrations Having the Circular Frequencies Differing from the Rotating Angular Velocity of the Shaft

When the unsymmetrical rotor with angular velocity ω is excited by a periodic external force having frequency ω0, two forced vibrations of frequencies ω0 and ω0'=2ω-ω0 take place simultaneously. Depending on the circumstances, the amplitudes of the vibration of frequency ω0'become remarkably larger than the amplitudes of the harmonic oscillation, i.e., the forced vibration of frequency ω0. Thus the idea generally accepted for the forced vibrations, that"the frequency of the forced vibration is equal or relates to that of the external force"is not always applicable for the forced vibrations of the shaft carrying an unsymmetrical rotor. In the present paper the response curves, the amplitude ratio between two forced vibrations, the effects of the damping forces are discussed analytically, and the obtained results are ascertained through the experiments.

The lateral damping and added mass of a horizontally oscillating shipmodel 12

For a practical range of circular frequencies forced sway and yaw tests were carried out, to obtain the values of mass and damping and their linear cross-coupling effects. Cross-coupling damping effects are considerable, but the apparent mass centre

Free, Transverse Vibrations of a Solid Elastic Mass in an Infinitely Long, Rigid, Circular-Cylindrical Tank

Making use of the field equations of elasticity, the frequency equation is derived for the free, transverse vibrations of a solid elastic mass contained by an infinitely long, rigid, circular-cylindrical tank. This frequency equation relates the natural circular frequencies and Poisson’s ratio. This relationship is plotted revealing a very interesting steplike variation of the natural frequency with Poisson’s ratio. Displacement fields are plotted for two natural frequencies in each of the first three modes.

Mixed Irregular Oscillations of Rolling and Pitching Motions of a Ship

Spectral distribution of oscillation of a ship has, in general, two peaks, which originate the resonances of pitching or heaving and rolling.This phenomena increase the irregularity of motion compared with the case when they are completely isolated with each other, and furthermore, also freqnency distribution of amplitude hastwo peaks, the larger portion of which is due to combined oscillation.According to Rice's theory of envelope, letting R be the amplitude of envelope, which is irregular, P, Q be rolling and pitching, which are supposed to be regular, then the probability density which lies between R and r+dR isp (R) = R∫∞0rJ0 (Pr) J 0 (Qr) J0 (Pr) e - (ψ0/2) r2dr.This integral can be developed further as follows : R/ψ0exp [-1/2ψ0 (P2+Q2+R2)] · {I0 (PR/ψ0) I0 (QR/ψ0) I0 (PQ/ψ0) +2∞Σm=1Im (PR/ψ0) I0 (QR/ψ0) Im (PQ/ψm)} where ψ0 is mean square of forced oscillation components.This distribution becomesR>>0.p (R) _??_I0 (PQ/ψ0) /2π√PQexp [-1/2ψ0 (R-P-Q) 2] and when there is only one resonance Q=0. p (R) =R/ψ0exp [-1/2ψ0 (P2+R2)] ·I0 (PR/ψ0) _??_1/√2πψ0 [-1/2ψ0 (R-P) 2] so that the maximum frequency shifts from R=P to R=P+Q, namely at the amplitude of sum, of P and Q.So that combined resonance oscillation sometimes causes extraordinary large amplitude.Letting circular frequencies be p andq for the amplitudes P and Q, irregularity reaches at the extreme value when : Pq=Qq

On the Forced Vibration with Non-linear Restoring Force by the Unharmonic Force (Continued)

In the previous report on the approximate solution for the forced vibration with nonlinear restoring force by the unharmonic external force, composed of two harmonic forces, we excluded the cases for ω1=3ω2 and 3ω1=ω2, where ω1 and ω2 are circular frequencies of two harmonic external forces. In the present paper the latter cases are treated theoretically and experimentally. As the conclusion we found that the approximate solution discussed in the previous report may be applied to every case for the ratio ω1/ω2.

Note on a Form of the Secular Equation for Molecular Vibrations

By starting with the Hamiltonian form of the equations of motion, the problem of the small vibrations of a polyatomic molecule can be solved with the aid of a secular equation twice the usual size. This equation, however, is readily adapted to solution by electrical circuits and has the important advantage of having as elements the force constants and inverse kinetic energy coefficients (``G'' matrix elements) directly. The force constant matrix elements occupy one off-diagonal corner of the secular equation, the G matrix elements are in the other off-diagonal corner, the circular frequencies (2πν) appear on the principal diagonal, and all other elements are zero.