Normal Mode Approximation Research Articles

On a formulation of the bending of elastic plates

A theory of the bending of linear elastic plates is developed in terms of an arbitrary set of linearly independent functions which forms a basis for the thickness parameter. A canonical system of equations is obtained in terms of stress matrix functions from which all kinematic variables, stress resultant and matrix stress resultants may be obtained. This canonical system of equations uses approximate transverse normal mode shapes which are obtained by solving an eigenvalue problem for the integrated expansion functions.

Single crystal Raman spectrum of the external phonons in 5-bromotetrazole

The six Raman active external phonons in 5-bromotetrazole have been assigned by observing four different Raman tensor elements, a(ca)c, a(cb)c, a(ba)c, and a(bb)c. In particular, the librational and translational phonons have been identified and approximate normal modes have been described for them. These results are compared to earlier work on L-alanine and imidazole single crystals. No disorder phenomena were observed in 5-bromotetrazole as in L-alanine. Harmonic force constant calculations confirm the structural prediction that, at K̄=0, no hydrogen-bond influences should be exerted on the external phonon frequencies. To our knowledge this work represents the first unambiguous single crystal Raman study of a hydrogen-bonded ring compound.

Prediction of Infrasonic Waves from Thermonuclear Explosions Using Cagniard's Integral Transform Method

The acoustic disturbance at horizontal distances of the order of 1000 km from an atmospheric nuclear blast is composed primarily of small-amplitude infrasonic waves with periods of 1 min or longer. Present methods for predicting the ground-level waveforms produce fair agreement with empirical data for most of the wavetrain; however, these methods are generally inapplicable for predicting the very early (first few cycles) portion of the waveform. If the vertical acceleration term in the equations of hydrodynamics is omitted, Cagniard's integral transform method can be used to find exact transient solutions to low-frequency wave propagation problems in atmospheres with arbitrary temperature profiles. To demonstrate the technique, solutions are derived for line and point sources in an isothermal atmosphere above a rigid ground. The results for the point source are compared with the results of Row and those of Cole and Greifinger and with the predictionsof the guided normal-mode approximation.

An Approximate Normal Mode Method for Damped Lumped Parameter Systems

An approximate normal mode method is introduced which permits any linear non-conservative system to be solved by super position of uncoupled coordinates. Accuracy of the method was found to be good when checked by digital computer for dynamic damper systems subjected to sinusoidal and white noise random excitation. A technique is presented which permits a mathematical model for a damped multimass system to be constructed entirely from experimentally obtained sinusoidal frequency response data. An expression is derived for optimum damping ratio and natural frequency ratio that will minimize the response of a two-mass system to white noise random excitation.

Molecular spectra and structure of selenophen

The infrared, Raman, and near ultraviolet spectra of selenophen have been measured. The vibrational spectra are consistent with the hypothesis of a planar structure for selenophen, a conclusion in contrast to that of previous investigations by Milazzo and co-workers. The 21 fundamentals are assigned to the irreducible representations of symmetry group C2v, on the basis of Raman intensities and polarisations, band contours in the infrared vapour spectrum, and activation in the liquid and solid infrared spectrum of A2-frequencies. The assignments to the approximate normal modes were made by analogy with those of thiophen. The near ultraviolet spectrum shows composite continuous absorption in the range 37,000–47,500 cm.–1. No traces of narrow peaks previously reported by Milazzo et al. in this spectral region were observed. The diffuse vibronic bands superposed to a continuum in the region 47,500–51,300 cm.–1 are interpreted on the basis of 405, 635, 1150, and perhaps 1350 cm.–1 excited-state totally symmetric vibrational frequencies with their origin at 47,800 cm.–1 This assignment is in disagreement with that proposed by Milazzo and co-workers.

Nonequilibrium Processes in Isotopically Disordered Crystals. Dependence on Degree of Order

A previous paper [A. A. Maradudin, G. H. Weiss, and D. H. Jepsen, J. Math. Phys. 2, 349 (1961)] has shown that approximate normal modes of isotopically disordered crystals decay irreversibly when the arrangement of isotopes is completely random. These results are generalized to crystals with an arbitrary degree of order. In particular, it is shown irreversible behavior occurs whenever the spatial correlation between isotopic species extends over a fixed finite range, as the size of the crystal tends to infinity.

An Approximation to the Influence Function for Plate‐like Wings

A general solution for the small deflexions of thin plates of slowly varying thickness under lateral loading in the form of an influence function is briefly presented. It is known that the influence function may be represented as an infinite series in terms of the eigenfunctions and eigenvalues associated with a homogeneous form of the plate differential equation. It is suggested that the series may give an acceptable approximation to the influence function when summed over a small number of terms when also the eigenfunctions and eigenvalues involved are deduced by an approximate procedure of the Rayleigh‐Ritz type. In order to test this assertion a numerical example is given for a uniform canti‐lever plate and the results are compared with experiment and with similar results deduced by an alternative theoretical procedure. Thus the calculation of a sufficient number of approximate normal vibration modes and frequencies for the plate as normally required for aeroelastic investigations may in this way be made to serve as the basis for a complete analysis of the plate. A simple approximate allowance for shear deflexion of the plate is presented and illustrated.