Quasi-periodic Boundary Conditions Research Articles

Dispersion relations and mode shapes for waves in laminated viscoelastic composites by finite difference methods

Dispersion relations and mode shapes for wave propagation in a laminated, infinite viscoelastic composite are obtained by finite difference methods. A complex modulus formulation is used. Free vibrations are studied for which the frequency is complex and the wave number real. Floquet theory is applied. Discretization of the governing differential equations and quasi periodic boundary conditions leads to a generalized eigenvalue problem for a large, sparse complex matrix. This is solved by a method involving permutation of the matrix into a convenient banded form and by writing recursion relations for the determinant. As an example, numerical results are obtained for a two medium composite with the filament material elastic and the matrix material modelled as elastic in dilatation and a standard linear solid in shear. The method proves to be very efficient and accurate for calculating frequencies as well as mode shapes.

Hindered Rotation in Methyl Alcohol

The problem of hindered rotation in methyl alcohol is discussed in relation to a model in which a rigid OH bar may rotate about the axis of a rigid pyramid representing the C${\mathrm{H}}_{3}$ group under the action of a hindering potential of the form $V=\frac{1}{2}H(1\ensuremath{-}cos3x)$. The wave equation is separated into a Mathieu equation governing the internal motion and a symmetrical rotator equation which describes the rotation of the whole molecule. The Mathieu functions, because of a coupling between the internal motion and the rotation of the entire system, obey quasi-periodic boundary conditions. A qualitative treatment is given of the behavior of the energy levels as the barrier height $H$, is raised from zero to infinity. An exact method of calculating the energy levels, the wave functions, and the transition probabilities is devised which involves finding the roots of certain continued fractions. It is shown that the levels lying well below the barrier, which may be described as vibrational levels, are each split into three components whose spread is dependent upon the penetrability of the barrier. The positions of the three components of a level are periodic functions of the variable $\ensuremath{\sigma}=\frac{K{C}_{1}}{C}$ (${C}_{1}$ and $C$ are moments of inertia and $K$ is a quantum number having integral values.) Examining levels in the order of increasing energy, the levels which lie above the barrier rapidly take on the character of states found in free rotation. A set of numerical calculations of the levels and of the resulting spectrum are made with a barrier height of 770 ${\mathrm{cm}}^{\ensuremath{-}1}$. The qualitative features of the theoretical spectrum appear to be in agreement with the observations of Borden and Barker and those of Lawson and Randall, but a comparison indicates that the barrier height is probably lower, about 470\ifmmode\pm\else\textpm\fi{}40 ${\mathrm{cm}}^{\ensuremath{-}1}$. Further experimental work, particularly in the far infra-red, should determine the barrier height more exactly in which case better correlation between the predicted and measured spectra may be expected.