Parnassus: Classical Journal (Volume 6, 2018)

Parnassus is an undergraduate journal published by the College of the Holy Cross in conjunction with the Classics Department. Parnassus\u27 mission is to share the passion of Holy Cross students for the ancient world. All pieces aim to be generally understandable, allowing the field to be more accessible to non-specialists in the community.https://crossworks.holycross.edu/parnassus/1005/thumbnail.jp

Transient response of a thin cylinder with local material inhomogeneity

AbstractAnalyzed is transient response of a thin cylinder with a local material inhomogeneity to a pulse of short duration. The Galerkin method is utilized to solve the inhomogeneous dynamic equations with the eigenfunctions of the homogeneous cylinder serving as trial functions. Also treated is response of the limiting cases of a disk and of a ring with the same local inhomogeneity. All limiting cases yield to analysis when modulus E is approximated by segments of constant E along the radius for the disk and circumference for the ring. Transfer matrices relating variables at the two ends of a segment combine to satisfy continuity of variables at interfaces of segments. Curvature and axial dependence make the cylinder unique in response properties that neither disk nor ring possess

Extension of Rayleigh–Ritz method for eigenvalue problems with discontinuous boundary conditions applied to vibration of rectangular plates

The Rayleigh–Ritz (R–R) method is extended to eigenvalue problems of rectangular plates with discontinuous boundary conditions (DBC). Coordinate functions are defined as sums of products of orthogonal polynomials and consist of two parts, each satisfying the BC in its respective region. These parts are matched by minimizing the mean square error of functions and their x-derivatives at the interface between regions. Matching defines a positive definite 2N^2 × 2N^2 matrix Q whose eigenvectors form the orthogonal coordinate functions. The corresponding eigenvalues measure the matching error of the two parts at the interface. When applying the R–R method, the total error is the sum of the matching error and that arising from the finite number of coordinate functions. Although most of the coordinate functions correspond to the zero eigenvalue, these do not suffice and additional functions corresponding to small but finite eigenvalues must be included. In three examples with discontinuous BC of the clamped, simply supported and free kind, the calculated frequencies match closely those from a finely discretized finite element method